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Chapter Page

1. Real Numbers 5

2. Polynomials 11

3. Pair of Linear Equations in two Variables 17

4. Similar Triangles 26

5. Trigonometry 39

6. Statistics 49

Sample Paper 60

[Class-X – Maths] 4

1. Euclid’s division lemma :

For given positive integers ‘a’ and ‘b’ there exist unique whole numbers ‘q’and ‘r’ satisfying the relation a = bq + r, 0 r < b.

2. Euclid’s division algorithms :

HCF of any two positive integers a and b. With a > b is obtained as follows:

Step 1 : Apply Euclid’s division lemma to a and b to find q and r such thata = bq + r , 0 r < b.

Step 2 : If r = 0, HCF (a, b) = b

Step 3 : if r 0, apply Euclid’s lemma to b and r.

3. The Fundamental Theorem of Arithmetic :

Every composite number can be expressed (factorized) as a product ofprimes and this factorization is unique, apart from the order in which theprime factors occur.

4. p , q 0 to be a rational number, such that the prime qfactorization of ‘q’ is of the form 2m5n, where m, n are non-negative integers.Then x has a decimal expansion which is terminating.

Let x

5. p , q 0 be a rational number, such that the prime factorization qof q is not of the form 2m5n, where m, n are non-negative integers. Thenx has a decimal expansion which is non-terminating repeating.

Let x

is irrational, where p is a prime. A number is called irrational if it cannot pbe written in the form q where p and q are integers and q 0.

5 [Class-X – Maths]

6.

1. 5 × 11 × 13 + 7 is a

(a)

(c)

prime number

odd number

(b)

(d)

composite number

none

2. Which of these numbers always ends with the digit 6.

(a)

(c)

4n

6n

(b)

(d)

2n

8n

where n is a natural number.

3. For a, b (a b) positive rational numbers____

(a)

(c)

4.

Rational number

a b a b is a

(b)

(d)

irrational number

0 a b 2

If p is a positive rational number which is not a perfect square then p is

(a)

(c)

integer

irrational number

(b)

(d)

rational number

none of the above.

5. All decimal numbers are–

(a)

(c)

rational numbers

real numbers

(b)

(d)

irrational numbers

integers

6. In Euclid Division Lemma, when a = bq + r, where a, b are positiveintegers which one is correct.

(a)

(c)

0 < r b

0 < r < b

(b)

(d)

0 r < b

0 r b

7. Which of the following numbers is irrational number

(a)

(c)

3.131131113...

2.35

(b)

(d)

4.46363636...

b and c both


218. The decimal expansion of the rational number

after ___ decimal places.

(a)

(c)

8

5

7 25

4 will terminate

(b)

(d)

4

never

9. HCF is always

(a)

(c)

multiple of L.C.M.

divisible by L.C.M.

(b)

(d)

Factor of L.C.M.

a and c both

10. The product of two consecutive natural numbers is always.

(a)

(c)

an even number

a prime number

(b)

(d)

an odd number

none of these

11. Which of the following is an irrational number between 0 and 1

(a)

(c)

0.11011011...

1.010110111...

(b)

(d)

0.90990999...

0.3030303...

12. pn = (a × 5)n. For pn to end with the digit zero a = __ for natural no. n

(a)

(c)

any natural number

odd number

(b)

(d)

even number

none.

13. A terminating decimal when expressed in fractional form always hasdenominator in the form of —

(a)

(c)

2m3n, m, n 0

5n 7m, m, n 0

(b)

(d)

3m5n, m, n 0

2m5n, m, n 0

14.33– 55is44

(a)

(c)

An irrational number

A natural number

(b)

(d)

A whole number

A rational number


15. If LCM (x, y) = 150, xy = 1800, then HCF (x, y) =

(a)

(c)

120

12

(b)

(d)

90

0

16. Which of the following rational numbers have terminating decimalexpansion?

(a) 912100 (b)

64455

(c)343

5 2 2 3 73(d)

2973

17.

18.

Solve 18 50. What type of number is it, rational or irrational.

Find the H.C.F. of the smallest composite number and the smallest primenumber.

If a = 4q + r then what are the conditions for a and q. What are the valuesthat r can take?

What is the smallest number by which 5 3 be multiplied to make ita rational number? Also find the number so obtained.

What is the digit at unit’s place of 9n?

Find one rational and one irrational no. between

19.

20.

21.

22.

23.

24.

3 and 5.

State Euclid’s Division Lemma and hence find HCF of 16 and 28.

State fundamental theorem of Arithmetic and hence find the uniquefactorization of 120.

Prove that25.

26.

1

2 5is irrational number..

Prove that 5 2 .3 is irrational number.7


27.

28.

29.

30.

31.

32.

Prove that 2 .7 is not rational number.

Find HCF and LCM of 56 and 112 by prime factorisation method.

Why 17 + 11 × 13 × 17 × 19 is a composite number? Explain.

Check whether 5 × 6 × 2 × 3 + 3 is a composite number.

Check whether 14n can end with the digit zero for any natural number, n.

If the HCF of 210 and 55 is expressible in the form 210 × 5 + 55y thenfind y.

33.

34.

35.

36.

Find HCF of 56, 96 and 324 by Euclid’s algorithm.

Show that the square of any positive integer is either of the form 3m or3m + 1 for some integer m.

Show that any positive odd integer is of the form 6q + 1, 6q + 5 where qis some integer.

Two milk containers contains 398 l and 436 l of milk. The milk is to betransferred to another container with the help of a drum. While transferringto another container 7l and 11l of milk is left in both the containersrespectively. What will be the maximum capacity of the drum.

Show that .7 is not a rational number.

37.

38. A sweet seller has 420 Kaju burfis and 130 Badam burfis. He wants tostack them in such a way that each stack has the same number and theytake up the least area of the tray.

i.

ii.

What is the number of burfis that can be placed in each stack forthis purpose?

Which value of the sweet seller is reflected in the question?

39. An Army contingent of 616 members is to march behind an army band of32 members in a parade, the two groups are to march in the same number


of columns. What is the maximum number of columns in which they canmarch ? What values are reflected by the members?

40. In the morning Ankita and Salma walk around a rectangular park. Ankitatakes 18 minutes to cover one round of the park while Salma takes 12minutes for the same. Suppose they both start at the same point and atthe same time and go in the same direction, find

i.

ii.

After how many minutes will they meet again at the starting point?

What values of Ankita and Salma are reflected in the question?

1.

3.

5.

7.

9.

11.

13.

15.

17.

19.

21.

24.

30.

32.

33.

35.

38.

39.

40.

b

a

c

a

b

b

d

c

30, rational

r, q whole no. 0 r < 4

1

2 × 2 × 2 × 3 × 5

Yes

4

a = 6q + r

10, Economic value

8, Work in groups, Discipline etc.

2. c

4. c

6. b

8. b

10. a

12. b

14. d

16. c

18. 2

20. 5 3 , 2

23. 4

28. HCF = 56, LCM = 112

31. No

34. a = 3q + r

36. 17 litre

HCF (210, 55) = 5, 5 = 210 × 5 + 55y y = – 19

36 minutes, Health Awareness etc.


1.

2.

3.

Polynomials of degrees 1, 2 and 3 are called linear, quadratic and cubicpolynomials respectively.

A quadratic polynomial in x with real coefficient is of the form ax2 + bx + c,where a, b, c are real numbers with a 0.

The zeroes of a polynomial p(x) are precisely the x–coordinates of thepoints where the graph of y = p(x) intersects the x-axis i.e. x = a is a zeroof polynomial p(x) if p(a) = 0.

A polynomial can have at most the same number of zeros as the degreeof polynomial.

For quadratic polynomial ax2 + bx + c (a 0)

Sum of zeros

4.

5.

ba

c .a

Product of zeros

6. The division algorithm states that given any polynomial p(x) and polynomialg(x), there are polynomials q(x) and r(x) such that :

p(x) = g(x).q (x) + r(x), g(x) 0

where r(x) = 0 or degree of r(x) < degree of g(x).

1. A real no. is a zero of the polynomial f(x) if

(a)

(c)

f() > 0

f() < 0

(b)

(d)

f() = 0

none


2. The zeroes of a polynomial f(x) are the coordinates of the points where thegraph of y = f(x) intersects

(a)

(c)

x-axis

origin

(b)

(d)

y-axis

(x, y)

3. If is a zero of f(x) then ____ is one of the factors of f(x)

(a)

(c)

(x –

(x +

(b)

(d)

(x – 2)

(2x –

4. If (y – a) is factor of f(y) then ___ is a zero of f(y)

(a)

(c)

y

2a

(b)

(d)

a

2y

5. If 1 is one of the zeroes of the polynomial x2 – x + k, the value of k is:

(a)

(c)

0

1

(b)

(d)

–2

3

6. Cubic polynomial x = f(y) cuts y-axis at atmost

(a)

(c)

one point

three points

(b)

(d)

two points

four points

7. Polynomial x2 + 1 has ___ zeroes

(a)

(c)

only one real

only two real

(b)

(d)

no real

one real and theother non-real.

8. If 1 is one of the zeroes of the polynomial ax2 – bx + 1 then.

(a)

(c)

a – b = 0

a + 1 = b

(b)

(d)

a – b – 1 = 0

a + b +1 = 0

9. If degree of polynomial f(x) is ‘n’ then maximum number of zeroes of f(x)would be –

(a)

(c)

n

n + 1

12

(b)

(d)

2n

n – 1

[Class-X – Maths]

10. If 2 is a zero of both the polynomials, 3x2 + ax – 14 and 2x – b thena – 2b = ___

(a)

(c)

–2

–8

(b)

(d)

7

–7

11. If zeroes of the polynomial ax2 + bx + c are reciprocal of each other then

(a)

(c)

a = c

b = c

(b)

(d)

a = b

a = – c

12. The zeroes of the polynomial h(x) = (x – 5) (x2 – x–6) are

(a)

(c)

–2, 3, 5

2, –3, –5

(b)

(d)

–2, –3, –5

2, 3, 5

13. If are the zeroes of the polynomial x2 + x + 1, then + =

(a)

(c)

–1

1

(b)

(d)

0

2

14.

15.

16.

If and are the zeroes of the polynomial 2x2 – 7x + 3. Find the sum ofthe reciprocal of its zeroes.

If are the zeroes of the polynomial p(x) = x2 – a (x + 1) – b such that( + 1) ( + 1) = 0 then find value of b.

If are the zeroes of the polynomial x2 – (k + 6) x + 2 (2k – 1). Find 1k if . 2

If (x + p) is a factor of the polynomial 2x2 + 2px + 5x + 10 find p.

Find a quadratic polynomial whose zeroes are 5 3 2 and 5 3 2 .

17.

18.

19.

1 and – 2 are respectively product and sum of the zeroes of a quadratic 5polynomial. Find the polynomial.

If


20. If one of the zero of the polynomial

g(x) = (k2 + 4) x2 + 13x + 4k is recoprocal of the other, find k.

21. If be the zeroes of the quadratic polynomial 2 – 3x – x2 then find thevalue of + (1 +

Form a quadratic polynomial, one of whose zero is 2 of zeroes is 4.

22. 5 and the sum

23.

24.

If sum of the zeroes of kx2 + 3k + 2x is equal to their product. Find k.

If one zero of 4x2 – 9 – 8kx is negative of the other, find k.

25.

26.

27.

28.

29.

30.

31.

32.

Find the quadratic polynomial whose zeroes are 2 and –3. Verify therelation between the coefficients and the zeroes of the polynomial.

If one zero of the polynomial (a2 + a) x2 + 13x + 6a is reciprocal of theother, find value (s) of a.

–5 is one of the zeroes of 2x 2 + px – 15. Quadratic polynomialp(x2 + x) + k has both the zeroes equal to each other. Then find k.

What should be subtracted from the polynomial 2x3 + 5x2 – 14x + 10 sothat the resultant polynomial is a multiple of (2x – 3).

If f(x) = 2x4 – 5x3 + x2 + 3x – 2 is divided by g(x), the quotient isq(x) = 2x2 – 5x + 3 and r(x) = – 2x + 1 find g(x).

If (x – 2) is one of the factors of x3 – 3x2 – 4x + 12, find the other zeroes.

If and are the zeroes of the polynomial x2 – 5x + k such that – = 1, find the value of k.

If are zeroes of quadratic polynomial 2x2 + 5x + k, find the value of 21k, such that ( 2 – =. 4

Obtain all zeroes of x4 – x3 –7x2 + x + 6 if 3 and 1 are zeroes.

If the two zeroes of the polynomial 2x4 – 2x3 – px2 + qx – 6 are –1 and2, find p and q.

33.

34.


35. If 2 3 and 2 3 are two zeroes of x4 – 4x3 – 8x2 + 36x – 9

find the other two zeroes.

36.

37.

38.

39.

What must be subtracted from 8x4 + 14x3 – 2x2 + 7x – 8 so that theresulting polynomial is exactly divisible by 4x2 + 3x – 2.

When we add p(x) to 4x4 + 2x3 – 2x2 + x – 1 the resulting polynomial isdivisible by x2 + 2x – 3 find p(x).

Find a and f if (x4 + x3 + 8x2 + ax + f) is a multiple of (x2 + 1).

If the polynomial 6x4 + 8x3 + 17x2 + 21x + 7 is divided by 3x2 + 1 + 4xthen r(x) = (ax + b) find a and b.

40. If Honesty and Dishonesty are the zeroes of the polynomial ax2 + bx + c,a 0 and are the reciprocal to each other then

i.

ii.

Find a relation between a and c

What values from the question should be adopted by the peoplein their life?

41. An NGO distributed K number of Books on moral-education to the students.If K is the zero of polynomial x2 – 100x – 20000 then

i.

ii.

How many books were distributed by the NGO?

Write any three moral values which you should adopt in your lifeby reading such books.

42. If 3 is one of the zero of the polynomial x3 – 12x2 + 47x – 60 and theremaining two zeroes are the number of plants, planted by the two studentsthen

i.

ii.

Find the total number of plants, planted by both the students.

What value is reflected in this question?

1.

3.

b

a

15

2. a

4. b

[Class-X – Maths]

5.

7.

9.

11.

13.

15.

17.

19.

21.

23.

25.

27.

29.

31.

33.

35.

37.

a

b

a

a

a

1

p = 2

6. c

8. c

10. d

12. a

14.1 17 3

16. k = 7

18. x2 – 10x + 7

x22x

15 20. 2

22. x2 – 4x – 1

24. 0

26. 5

k = – 5

23

x2 + x – 6

p 7, k 74 28. 7

30. –2, 3

32. k = 2

34. 1, –3

36. 14x – 10

38. r(x) = 0 a 1x f 7 0 a 1 and f 7

g(x) = x2 – 1

k = 6

–2, –1

± 3

61x – 65

39.

40.

41.

42.

r (x) = x + 2 = ax + b a = 1 and b = 2

a = c, Honesty

(i) 200

(i) 9

(ii) Honesty, Discipline etc.

(ii) Love towards nature, plantation etc.


1. The most general form of a pair of linear equations is :

a1x + b1y + c1 = 0

a2x + b2y + c2 = 0

Where a1, a2, b1, b2, c1, c2 are real numbers and a12 + b12 0, a22 + b22 0.

2. The graph of a pair of linear equations in two variables is represented bytwo lines;

(i)

(ii)

If the lines intersect at a point, the pair of equations is consistent.The point of intersection gives the unique solution of the equation.

If the lines coincide, then there are infinitely many solutions. Thepair of equations is consistent. Each point on the line will be asolution.

If the lines are parallel, the pair of the linear equations has nosolution. The pair of linear equations is inconsistent.

(iii)

3. If a pair of linear equations is given by a1x + b1y + c1 = 0 and a2x + b2y+ c2 = 0

(i) a1b 1 the pair of linear equations is consistent. (Uniquea2b2solution).

a1bc 1 1 the pair of linear equations is inconsistent a2b2c2(No solution).


(ii)

(iii)a1bc 1 1 the pair of linear equations is dependent anda2b2c2

consistent (infinitely many solutions).

1. Every linear equation in two variables has ___ solution(s).

(a)

(c)

no

two

(b)

(d)

one

infinitely many

2. a1bc 1 1 is the condition fora2b2c2

(a)

(c)

3.

intersecting lines

coincident lines

(b)

(d)

parallel lines

none

For a pair of linear equations in two variables to be consistent and dependentthe pair must have

(a)

(c)

no solution

infinitely many solutions

(b)

(d)

unique solution

none of these

4. Graph of every linear equation in two variables represent a ___

(a)

(c)

point

curve

(b)

(d)

straight line

triangle

5. Each point on the graph of pair of two lines is a common solution of thelines in case of ___

(a)

(c)

Infinitely many solutions

no solution

(b)

(d)

only one solution

none of these

6. If the system of equations 6x – 2y = 3 and Kx – y = 2 has unique solutionthen K is

(a)

(c)

K = 3

K 3

18

(b)

(d)

K = 4

K 4

[Class-X – Maths]

7. One of the common solution of ax + by = c and y-axis is _____

(a)c0, b

cb , 0

(b)b0, c

c0, b

(c)

8.

(d)

For x = 2 in 2x – 8y = 12 the value of y will be

(a)

(c)

–1

0

(b)

(d)

1

2

9. The pair of linear equations is said to be inconsistent if they have

(a)

(c)

only one solution

infinitely many solutions.

(b)

(d)

no solution

both a and c

10. On representing x = a and y = b graphically we get ____

(a)

(c)

parallel lines

intersecting lines at (a, b)

(b)

(d)

coincident lines

intersecting lines at (b, a)

11. A motor cyclist is moving along the line x – y = 2 and another motor cyclistis moving along the line x + y = 2. Tell whether the y

(a)

(c)

move parallel

collide somewhere

(b)

(d)

move coincidently

none

12. For 2x + 3y = 4, y can be written in terms of x as—

(a)

(c)

y 4 2x 3

4 3y 2

(b)

(d)

y

y

4 3x 2

4 2x 3x

13. For what value of p, the pair of linear equations 2x + py = 8 and x + y =6 has a unique solution x = 10, y = –4.

(a)

(c)

–3

2

19

(b)

(d)

3

6

[Class-X – Maths]

14. The point of intersection of the lines x – 2y = 6 and y-axis is

(a)

(c)

(–3, 0)

(6, 0)

(b)

(d)

(0, 6)

(0, –3)

15. Graphically x – 2 = 0 represents a line

(a)

(b)

(c)

(d)

parallel to x-axis at a distance 2 units from x-axis.

parallel to y-axis at a distance 2 units from it.

parallel to x-axis at a distance 2 units from y-axis.

parallel to y-axis at a distance 2 units from x-axis.

16. If ax + by = c and lx + my = n has unique solution then the relationbetween the coefficients will be ____

(a)

(c)

am lb

ab = lm

(b)

(d)

am = lb

ab lm

17. Form a pair of linear equations for : If twice the son’s age is added tofather’s age, the sum is 70. If twice the father’s age is added to the son’sage the sum is 95.

Amar gives9000 to some athletes of a school as scholarship everyymonth. Had there been 20 more athletes each would have got 160 less..Form a pair of linear equations for this.

Write a pair of linear equations which has the unique solution x = –1 andy = 4. How many such pairs are there?

What is the value of a for which (3, a) lies on 2x – 3y = 5.

Solve :

x + 2y – 8 = 0 2x + 4y = 16

18.

19.

20.

21.

22. Dinesh is walking along the line joining (1, 4) and (0, 6), Naresh is walkingalong the line joining (3, 4,) and (1,0). Represent on graph and find thepoint where both of them cross each other.


23. Solve the pair of linear equations

x – y = 2 and x + y = 2. Also find p if p = 2x + 3

24.

25.

Check graphically whether the pair of linear equations 3x + 5y = 15,x – y = 5 is consistent. Also check whether the pair is dependent.

For what value of p the pair of linear equations

(p + 2) x – (2 p + 1)y = 3 (2p – 1)

2x – 3y = 7

has unique solution.

26. Find the value of K so that the pair of linear equations :

(3 K + 1) x + 3y – 2 = 0

(K2 + 1) x + (k–2)y – 5 = 0 is inconsistent.

27. Given the linear equation x + 3y = 4, write another linear equation in twovariables such that the geometrical representation of the pair so formed is(i) intersecting lines (ii) parallel lines (iii) coincident lines.

Solve x – y = 4, x + y = 10 and hence find the value of p wheny = 3 x –p

Determine the values of a and b for which the given system of linearequations has infinitely many solutions:

2x + 3y = 7

a(x + y) – b(x – y) = 3a + b – 2

28.

29.

30. The difference of two numbers is 5 and the difference of their reciprocals

is 1 . Find the numbers10

31. Solve for x and y :

x 1y 1x 1y 18;92 3 3 2


32. Solve for x and y

x + y = a + b

ax – by = a2 –b2


139x 56 y 64156x 139 y 724


51 2x yx y

155 2x yx y

35. Solve the following system of equations graphically

x + 2y = 5, 2x – 3y = – 4.

Write the coordinates of the point where perpendicular from common

1 solution meets x-axis. Does the point – , 1lie on any of the lines? 2

36. Draw the graph of the following pair of linear equations

x + 3y = 6 and 2x – 3y = 12

Find the ratio of the areas of the two triangles formed by first line, x = 0,y = 0 & second line x = 0, y = 0.


1121 22 2x 3 y 7 3x 2y 74 2 for 2x + 3y 0 and 3x – 2y 02x 3 y 3 x 2 y


38. Solve for p and q

p qp q 2,6, p 0, q 0. pqpq

39. On selling a T.V. at 5% gain and a fridge at 10% gain, a shopkeeper gainsRs. 2000. But if he sells the T.V. at 10% gain & the fridge at 5% loss, hegains Rs. 1500 on the transaction. Find the actual price of the T.V. and thefridge.

40.2

x 3

y2,

4

x 9

y1 ; x 0, y 0

41. If from twice the greater of two numbers, 20 is subtracted, the result is theother number. If from twice the smaller number, 5 is subtracted, the resultis the greater number. Find the numbers.

In a deer park the number of heads and the number of legs of deer andvisitors were counted and it was found that there were 39 heads and 132legs. Find the number of deers and visitors in the park, using graphicalmethod.

A two digit number is obtained by either multiplying the sum of the digitsby 8 and adding 1; or by multiplying the difference of the digits by 13 andadding 2. Find the number. How many such numbers are there. 1In an examination one mark is awarded for every correct answer and 4mark is deducted for every wrong answer. A student answered 120 questionsand got 90 marks. How many questions did he answer correctly?

A boatman rows his boat 32 km upstream and 36 km down stream in 7hours. He can row 40 km upstream and 48 km downstream in 9 hours.Find the speed of the stream and that of the boat in still water.

In a function if 10 guests are sent from room A to B, the number of guestsin room A and B are same. If 20 guests are sent from B to A, the numberof guests in A is double the number of guests in B. Find number of guestsin both the rooms in the beginning.

In a function Madhu wished to give Rs. 21 to each person present andfound that she fell short of Rs. 4 so she distributed Rs. 20 to each and


42.

43.

44.

45.

46.

47.

found that 1 were left over. How much money did she gave and howwmany persons were there.

48. A mobile company charges a fixed amount as monthly rental which includes100 minutes free per month and charges a fixed amount there after forevery additional minute. Abhishek paid 433 for 370 minutes and Ashishpaid 398 for 300 minutes. Find the bill amount under the same plan, ifUsha use for 400 minutes.

49. Ravi used 2 plastic bags and I paper bag in a day which cost him 35while Mahesh used 3 plastic bags and 4 paper bags in a day which costhim 65.

i.

ii.

Find the cost of each bag.

Which bag has to be used and what value is reflected by using it?

50. Two schools want to award their selected students on the values ofDiscipline and Punctuality. First school wants to award these values to its3 and 2 students respectively, while second wants to award for the samevalues to its 4 and 1 students respectively. If the amount of award for eachvalue given by both the school are same and total amount spent by eachschool is 600 and 550 respectively then.

i.

ii.

Find the award money for each value.

Which more values of the students may be chosen for the award?

1.

3.

5.

7.

9.

11.

d

c

a

a

b

c

2. c

4. b

6. c

8. a

10. c

12. d


13.

15.

17.

b

b

14. d

16. a

Father’s age = x years, Son’s age = y years

x + 2y = 70, 2x + y = 95

18.

19.

21.

23.

25.

26.

28.

30.

32.

34.

36.

38.

40.

42.

44.

46.

48.

49.

50.

No. of athletes = x, No. of athletes increased = y

Infinite

Infinite

(2, 0) P = 7

p 4

20.13

22. (2, 2)

24. No

k 1, k

(7, 3), 18

19 2

29. a = 5, b = 1

31. (7, 13)

33. (2, –1)

35. (1, 2), (1, 0), yes

37. (2, 1)

39. 20000, 100000

–5, –10 or 10, 5

x = 1, y = b

(3, 2)

(6, 0), 1 : 2

1 12, 4(4, 9)

27, 12

96

100, 80

41. 15, 10

43. 41 or 14(2)

45. 2 km/hr, 10km/hr.

47. Rs. 101, 5

1Rs. 298, Rs. Rs. 4482(i) 15, 5,(ii) Environment friendly

(i) 100, 150 (ii) Honesty, regularity etc.


1. Similar Triangles : Two triangles are said to be similar if their correspondingangles are equal and their corresponding sides are proportional.

Criteria for Similarity :

in and

(i) AAA Similarity : ~ when = = and=

SAS Similarity :

2.

(ii)

ABC ~ DEF whenABBC and B EDEEF

ABACBC .DEDFEF

(iii)

3.

SSS Similarity : ABC ~ DEF ,

The proof of the following theorems can be asked in the examination :

(i) Basic Proportionality Theorem : If a line is drawn parallel to oneside of a triangle to intersect the other sides in distinct points, theother two sides are divided in the same ratio.

The ratio of the areas of two similar triangles is equal to the squareof the ratio of their corresponding sides.

Pythagoras Theorem : In a right triangle, the square of thehypotenuse is equal to the sum of the squares of the other twosides.

(ii)

(iii)


(iv) Converse of Pythagoras Theorem : In a triangle, if the square ofone side is equal to the sum of the squares of the other two sidesthen the angle opposite to the first side is a right angle.

1. ~ If DE = 2 AB and BC = 3cm then EF is equal to _______.

(a)

(c)

1.5 cm

6 cm

(b)

(d)

3 cm

9 cm

2. In AB || EW if AD = 4 cm, DE = 12cm and DW = 24 cm then thevalue of DB = ____

(a)

(c)

4 cm

12 cm

(b)

(d)

8 cm

16 cm

3.

A D

Q

c b f

Q

e

BO

a

OC E

dF

In the figure the value of cd = ________

(a)

(c)

4.

ae

bf

(b)

(d)

af

be

If the corresponding medians of two similar triangles are in the ratio 5 : 7,then the ratio of their sides is :

(a)

(c)

25 : 49

1:1

(b)

(d)

7:5

5:7


5. AD is the bisector of If BD = 4cm. DC = 3cm and AB = 6 cm. AC isequal to

(a)

(c)

4.2 cm

4.8 cm

(b)

(d)

4.5 cm

5 cm

6. In the figure, is similar to ______

B53°

53°

16 cm A

24cm

36 cm

C

D

(a)

(c)

7.

(b)

(d)

~ Also 2ar (AMB) = ar (CMD) the length of MD is

(a) 2 MB

2MB

(b) 2 MD

2MD

(c) (d)

8. If ~ and

will be PR.

(a)

(c)

10 cm

4 cm

ar ABC 9 , AB = 18 cm, BC = 15 cm whatar PQR 4

(b)

(d)

9 cm

18 cm


9. In D and E are points on side AB and AC respectively such thatDE || BC and AD : DB = 3 : 1. If EA = 3.3 cm then AC =

(a)

(c)

1.1 cm

4 cm

(b)

(d)

4.4 cm

5.5 cm

10. ABC and BDE are two equilateral triangles such that D is the midpoint ofBC. Ratio of the areas of triangles ABC and BDE is—

(a)

(c)

2:1

4:1

(b)

(d)

1:2

1:4

11. In DE || BC. In the figure, the value of x is ______

A

x

D

x+3

E

x+1

B

x+5

C

(a)

(c)

1

3

(b)

(d)

–1

–3

12. ar (PRQ )BC1 It is given that ~ withthen ar (BCA ) is equal toQR3

(a)

(c)

9

13

(b)

(d)

3

19

13. The altitude of an equilateral triangle, having the length of its side 12cm is

(a) 12 cm (b) 6 2 cm


(c)

14.

6 cm (d) 6 3 cm

A right angled triangle has its area numerically equal to its perimeter. Thelength of each side is an even number & the hypotenuse is 10 cm. Whatis the perimeter.

(a)

(c)

26 cm

30 cm

(b)

(d)

24 cm

16 cm

15. The perimeters of two similar triangles ABC and PQR are respectively 36cm and 24 cm. If PQ = 10cm then AB = ........

(a)

(c)

10 cm

25 cm

(b)

(d)

20 cm

15 cm

16. In figure ~ If BC = 8 cm, PQ = 4cm AC = 6.5 cm, AP = 2.8cm, find AB and AQ.

BP

A

Q

C

17.

18.

19.

In ADBC and BD 11 CD . Prove that CA2 = AB2 + BC 232

If is an equilateral triangle such that AD BC then prove thatAD2 = 3 DC2

An isosecles triangle ABC is similar to triangle PQR. AC = AB = 4 cm,RQ = 10 cm and BC = 6 cm. What is the length of PR? What type oftriangle is ?


20. In the adjoining figure, find AE.

A

E8 cm

4 cm

B 6 cm C 3 cm D

21. In DE || QR and DE ar PQR 1 QR . Find.4ar PDE

P

D E

Q R

22. In triangles ABC and PQR if = and

the value of

ABBC1 then what isPQQR2

PR ?AC

23. ABC is a right angled at B, AD and CE are two medians drawn from A

and C respectively. If AC = 5 cm and AD 3 5 cm. Find CE. 2

24. In the adjoining figure DE || BC. What is the value of DE.


A

10cm

D E

2cm

B3 cm

C

25. Legs (sides other then the hypotenuse) of a right triangle are of lengths16 cm and 8 cm. Find the length of the side of the largest square that canbe inscribed in the triangle.

26. In the following figure, DE || AC andBEBC . Prove that DC || APECCP

A

D

BE C P

27. Two similar triangles ABC and PBC are made on opposite sides of thesame base BC. Prove that AB = BP.


28. I a quadriatnl eral ABCD,= 90°, AD2 = AB2 + BC2 + CD2. Prove that

D

= 90°.

C

BA

29. In figure DE || BC, DE = 3 cm, BC = 9 cm and ar (ADE) = 30 cm2. Findar (trapezium BCED).

A

D3 cm

E

B9 cm

C

30. Amit is standing at a point on the ground 8m away from a house. A mobilenetwork tower is fixed on the roof of the house. If the top and bottom ofthe tower are 17m and 10m away from the point. Find the heights of thetower and house.

In a right angled triangle PRQ, PR is the hypotenuse and the other twosides are of length 6cm and 8cm. Q is a point outside the triangle suchthat PQ = 24cm RQ = 26cm. What is the measure of

PQRS is a trapezium. SQ is a diagonal. E and F are two points on parallelsides PQ and RS respectively. Line joining E and F intersects SQ at G.Prove that SG × QE = QG × SF.

Two poles of height a metres and b metres are apart. Prove that the heightof the point of intersection of the lines joining the top of each pole to the abfoot of the opposite pole ismetres.. a b


31.

32.

33.

D B

Obm

h

C x L y A

am

34.

35.

36.

37.

Diagonals of a trapezium PQRS intersect each other at the point O, PQ||RSand PQ = 3 RS. Find the ratio of the areas of triangles POQ & ROS.

In a rhombus, prove that four times the square of any sides is equal to thesum of squares of its diagonals.

ABCD is a trapezium with AE || DC. If ABD is similar to Prove thatAD = BC.

In a triangle, if the square of one side is equal to the sum of the squaresof the other two sides, then prove that the angle opposite to the first sideis a right angle.

If BL and CM are medians of a triangle ABC right angled at A. Prove that4 (BL2 + CM2) = 5 BC2.

ABCD is a rectangle in which length is double of its breadth. Two equilateraltriangles are drawn one each on length and breadth of rectangle. Find theratio of their areas.

In figure DE || BC and AD : DB = 5 : 4. Find

A

38.

39.

40.ar DEFar CFB

D

F

E

B C


41. Four friends went to a restaurant. They ordered four triangular shapedpizzas with different topings. One of them told to others they should dividetheir pizzas in four equal parts so that all can taste all the four pizzas butthe three fail to do so. Fourth friend took a straw and he make a point atmid of each side and join all mid points and cut it by a knife into four equalparts.

Prove that

i.ar (DEF )1 ar (ABC )4 if D, E, F are the midpot of sides AB, BC & CAof a triangular pizza.

Which value is described in this equation?ii.

42. A poster competition on “Save Earth” is organised by Directorate ofEducation. Medals in the shape of similar triangles were distributed to firstand second position holder, first prize in form of and second in theform of

A 12

SAVEEARTH

15 9

B

C

i. Find the value of x and y.

P

8SAVEEARTH

x

Q Y R

ii. Which mathematical concept was used in this question?


iii.

iv.

43.

What is the ratios of areas of triangle ABC and PQR.

What moral values of a child was shown in this poster competition.

From a village A to city B road passes through a mountain C. It wasconstructed so that AC CB, AC = 2x km and CB = 2 (x + 7) km. It wasproposed that a 26 km long highway will be constructed which will connectcity B to village A directly. If a person goes to city B from village A throughthis highway by bus then what distance was reduced in his journey. Writethe values of this question.

Ashok has a quadrilateral shaped field of rice. He divide it into two by onediagonal and distribute it to one son and one daughter. Each get the ricein proportion to the area which they got. Both don’t know how much areathey got. Ashok using second diagonal distribute the rice in such ratio.

44.

ar (ABC )AO ar (DBC ) DO

i. Prove the above relation

C

O

A

B D

ii.

45.

Which value of Ashok was shown in this questions?

Two advertising balloons showing “Pulse Polio” and “Save Girl Child" weretied by two wires ML and NR such that they makes an angle 40° with theground. A person standing at the point K. Length of wire ML is ‘a’ metre,MN is ‘b’ metre and NK is ‘c’ metre.

i.

ii.

Find the length of wire NR when both the balloons are on a samestraight line.

What moral values are shown in this question.


save girlchild

L

Pulsepolio

R

x40°

M b N40°

c K

1.

3.

5.

7.

9.

11.

13.

15.

18.

20.

c

a

b

a

b

c

d

d

~

55 cm

2.

4.

6.

8.

10.

12.

14.

16.

19.

21.

b

d

d

a

c

a

b

AB = 5.6 cm, AQ = 3.25 cm

20 cm , isosceles triangle 316 : 1

1 25 – 6 5 cm4

16 cm 3

22.12

2.5 cm

240 cm2

37

23.

24.

29.

25.

30. 9 m, 6 m

[Class-X – Maths]

31.

39.

41.

42.

90°

4:1

34.

40.

9:1

2581

(ii) Cooperation, healthy competition, intelligency, quick decision

ar (ABC )9 (i) x = 10, y = 6, ar (PQR )4

(ii) similar triangle

(iii) Awareness about pollution, Saving of water, Energy conservation

43.

44.

45.

(i) 8 Km (ii) Saving of time, Economic saving, Save energy.

(i) Intelligency, Self decision, Responsibility.

(i) x acb c

(ii) Save girl child, health awareness.


1. Trigonometric Ratios : In = 90°, for angle ‘A’

sin A Perpendicular Hypotenuse

C

nu otep

se

Basecos A Hypotenuse

Perpendiculartan A Base

Hy Perpendicular

Basecot A Perpendicular

Hypotenuse Base

A Base B

sec A

cosec A

2.

HypotenusePerpendicular

Reciprocal Relations :

sin 1 ,cosec 1sec ,

cosec 1sin

1cos cos sec


tan

3.

1cot , cot

1tan

Quotient Relations :

tan

4. Indentities :

sin cos , cot

cos sin

sin2 + cos2 = 1 sin2 = 1 – cos2 and cos2 = 1 – sin2

1 + tan2 = sec2 tan2 = sec2 – 1 and sec2 – tan2 = 1

1 + cot2 = cosec2 cot2 = cosec2 – 1 and cosec2 – cot2 = 1

5. Trigonometric Ratios of Some Specific Angles

0° 30° 45° 60° 90°

sin A 012

32

1

3

1

2

1

2

32

12

1

cos A 1 0

tan A 0 1 3 Not defined

cosec A Not defined 22

2 31

sec A 12

3 2 2 Not defined

cot A Not defined 3 11

30


6. Trigonometric Ratios of Complementary Angles

sin (90° – = cos

cos (90° – = sin

tan (90° – = cot

cot (90° – = tan

sec (90° – = cosec

cosec (90° – = sec

Note : In the following questions 0° 90°

1. If x = a sin and y = a cos then the value of x2 + y2 is _______

(a)

(c)

2.

a

1

(b)

(d)

a2

1a

The value of cosec 70° – sec 20° is _____

(a)

(c)

0

70°

(b)

(d)

1

20°

3. If 3 sec – 5 = 0 then cot = _____

(a)53

34

(b)45

35(c)

4.

(d)

If = 45° then sec cot – cosec tan is

(a)

(c)

0

2

(b)

(d)

1

2 2


5. If sin (90 – cos = 1 and is an acute angle then = ____

(a)

(c)

90°

30°

(b)

(d)

60°

0°

6. The value of (1 + cos (1 – cos cosec2= _____

(a)

(c)

0

cos2

(b)

(d)

1

sin2

7. is a right-angled isosceles triangle then cos T + cos R + cos Y is_____

(a) 2 (b) 2 2

(c)

8.

1 2 (d) 11

2

If sec + tan = x, then sec =

(a) x21

x

x2

(b)x

212x

1(c)

12x (d)

x2

x

9.The value of cot sin cos is _______ 22

(a)

(c)

10.

cot cos2

cos2

(b)

(d)

cot2

tan2

If sin – cos = 0, 0 90° then the value of is _____

(a)

(c)

cos

90°

(b)

(d)

45°

sin


sin 11.

1 sin 2 can be written as

(a)

(c)

cot (b)

(d)

sin

sin

cos tan

12.1 sin is equal to1 sin (a)

(c)

sec2+ tan2

sec2– tan2

(b)

(d)

sec – tan

sec + tan

13. In an isosceles right-angled = 90°. The value of 2 sin A cos Ais _____

(a) 1 (b)12

2(c)1

2(d)

2

14. Ifsin 20sin 70

2 cos 69cos 21

(a)

(c)

1

3

2

2 2 sec 60then K is ______ K

(b)

(d)

2

4

15. If tan 1

7, then

cosec sec

cosec sec 2 2

2 2

57

112

(a)

(c)

34

37

(b)

(d)


16.

17.

18.

19.

20.

21.

22.

23.

24.

In = 90° and sin R 3 , write the value of cos P.5

If A and B are acute angles and sin A = cos B then write the value ofA + B.

If 4 cot = 3 then write the value of tan + cot

Write the value of cot2 30° + sec2 45°.

Given that 16 cot A = 12, find the value ofsin A cos A sin A cos A

If = 30° then write the value of sin + cos2

If 1 tan 2 2 then what is the value of 3

3 tan 23 0.Find the value of if

If and are complementary angles then what is the value of

cosec sec – cot tan

25.

26.

27.

If tan (3x – 15°) = 1 then what is the value of x.

If sin 5 = cos 4, where 5 and 4 are acute angles. Find the value of

Simplify :

tan2 60° + 4 cos2 45° + 3 (sec2 30° + cos2 90°)

28. Evaluate

cos 582sin 32

cos 38 cosec 52 3tan 15 tan 60 tan 75

29.

30.

If sin + sin2 = 1 then find the value of cos2 + cos4

If sin 2 = cos ( – 36°), 2 and – 36° are acute angles then find thevalue of


31. Prove that

cosec4 – cosec2 = cot2 + cot4

32. If sin (3x + 2y) = 1 and cos 3x 2y

then find the value of x and y.

3 , where 0 (3x + 2y) 2

33. If sin (A + B) = sin A cos B + cos A sin B then find the value of

(a)

(b)

sin 75°

cos 15°

34. Prove that cos Acos A cos A, A 45.1 tan A 1 cot A

sec 1 sec 1

sec 1 2cosec sec 1

35.

36.

Prove that

Find the value of

sin2 5° + sin2 10° + sin2 15° + .... + sin2 85°

37. Prove that

tan sec 1cos .tan sec 1 1 sin

38. If 2 sin 3x 15 3 then find the value of2

sin

39.

2 2x10 tan x 5.

Find the value of sin 60° geometrically.

40.

41.

1Let p = tan + sec then find the value of p p .

Find the value of

tan cot 90 sec cosec 90 sin 35 sin 55 tan 10tan 20tan 30 tan 70 tan 80

2 2


42.

43.

If cos cos m andn show that (m2 + n2) cos2 = n2.cos sin

Prove that cos 1° cos 2° cos 3°.........cos 180° = 0.

44.

45.

sin cos sin cos 2 sec Prove that sin cos sin cos . 2 tan 1If A, B, C are the interior angles of a triangle ABC, show that

AAB C B C sin coscos sin1. 222 2

2

46. In the given right triangle, if base and perpendicular are represented by‘Hardwork’ and ‘Success’ and are in the ratio 1 : 1 then find Whatmathematical concept has been used? What value is depicted from theproblem?

A

B C

47. If x = sin2y = cos2

Where x represents Honesty and y represents Hardwork.

a.

b.

c.

What you will get when honesty is added with hardwork?

Which mathematical concept is used?

What value is depicted here?

48. If punctuality and regularity are two measurable quantities are numbericallyequal to A and B respectively such that


1 2 1Cos (A B ) 2

Sin(A – B )

then find A and B.

where 0° A + B 90°

Which one more value other than punctuality & regularity, would you liketo adopt in your life?

1.

3.

5.

7.

9.

11.

13.

15.

17.

19.

21.

23.

25.

27.

29.

b

c

d

a

a

d

a

a

90°

5

2.

4.

6.

8.

10.

12.

14.

16.

18.

20.

22.

24.

26.

28.

30.

a

a

b

b

b

d

d

cos P

2512

7

30°

1

10°

1

42°

35

54

30°

x = 20.

9

1


32. x = 20, y = 15

33.3 1

2 2

17 2

2 sec

,3 1

2 2

38.

41.

36.

40.

46.

47.

48.

1312

2 3

45°, Trigonometry, Hardwork and Success

(a) 1, (b) Trigonometry, (c) Honesty and Hardwork

A = 45°, B = 15°

Honesty, Hardwork, Responsibility, Cooperation


1. The mean for grouped data can be found by :

(i) The direct method X fixi .fi

(ii) The assumed mean method

X a fidi ,fi

where di = xi –a.

(iii) The step deviation method

X a

2.

fiuifi h, where u i

xi a . h

The mode for the grouped data can be found by using the formula :

f1 f 0mode l h 2f 1 f 0 f 2

l = lower limit of the modal class.

f1 = frequency of the modal class.

f0 = frequency of the preceding class of the modal class.

f2 = frequency of the succeeding class of the modal class.

h = size of the class interval.

Modal class - class interval with highest frequency.


3. The median for the grouped data can be found by using the formula :

n 2 Cf median l h f

l = lower limit of the median class.

n = number of observations.

Cf = cumulative frequency of class interval preceding the median class.

f = frequency of median class.

h = class size.

4.

5.

Empirical Formula : Mode = 3 median - 2 mean.

Cumulative frequency curve or an Ogive :

(i) Ogive is the graphical representation of the cumulative frequencydistribution.

Less than type Ogive :

•

•

(iii)

Construct a cumulative frequency table.

Mark the upper class limit on the x = axis.

(ii)

More than type Ogive :

•

•

Construct a frequency table.

Mark the lower class limit on the x-axis.

(iv) To obtain the median of frequency distribution from the graph :

• Locate point of intersection of less than type Ogive andmore than type Ogive :

Draw a perpendicular from this point on x-axis.

•

[Class-X – Maths]

The point at which it cuts the x-axis gives us the median.

50

1. Mean of first 10 natural numbers is

(a)

(c)

5

5.5

(b)

(d)

6

6.5

2. If mean of 4, 6, 8, 10, x, 14, 16 is 10 then the value of ‘x’ is

(a)

(c)

11

13

(b)

(d)

12

9

3. The mean of x, x + 1, x + 2, x + 3, x + 4, x + 5 and x + 6 is

(a)

(c)

x

x + 4

(b)

(d)

x + 3

3

4. The median of 2, 3, 2, 5, 6, 9, 10, 12, 16, 18 and 20 is

(a)

(c)

9

10

(b)

(d)

20

9.5

5. The median of 2, 3, 6, 0, 1, 4, 8, 2, 5 is

(a)

(c)

1

4

(b)

(d)

3

2

6. Mode of 1, 0, 2, 2, 3, 1, 4, 5, 1, 0 is

(a)

(c)

5

1

(b)

(d)

0

2

7. If the mode of 2, 3, 5, 4, 2, 6, 3, 5, 5, 2 and x is 2 then the value of ‘x’ is

(a)

(c)

2

4

(b)

(d)

3

5


8. The modal class of the following distribution is

Class Interval

Frequency

10–15

4

15–20

7

20–25

12

25–30

8

30–35

2

(a)

(c)

9.

30–35

25–30

(b)

(d)

20–25

15–20

A teacher ask the students to find the average marks obtained by theclass students in Maths, the student will find

(a)

(c)

mean

mode

(b)

(d)

median

sum

10. The empirical relationship between the three measures of central tendency is

(a)

(b)

(c)

(d)

3 mean = mode + 2 median

3 median = mode + 2 mean

3 mode = mean + 2 median

median = 3 mode – 2 mean

11. Class mark of the class 19.5 – 29.5 is

(a)

(c)

10

24.5

(b)

(d)

49

25

12. Measure of central tendency is represented by the abscissa of the pointwhen the point of intersection of ‘less than ogive’ and ‘more than ogive’ is

(a)

(c)

mean

mode

(b)

(d)

median

None of these

13. The median class of the following distribution is

0–10

4

10–20

4

20–30

8

30–40

10

40–50

12

50–60

8

60–70

4

Class Interval :

Frequency :


(a)

(c)

14.

20–30

30–40

(b)

(d)

40–50

50–60

The mean of 20 numbers is 17, if 3 is added to each number, then the newmean is

(a)

(c)

20

22

(b)

(d)

21

24

15. The mean of 5 numbers is 18. If one number is excluded then their meanis 16, then the excluded number is

(a)

(c)

23

25

(b)

(d)

24

26

16. The mean of first 5 prime numbers is

(a)

(c)

5.5

5.7

(b)

(d)

5.6

5

17. The sum of deviations of the values 3, 4, 6, 8, 14 from their mean is

(a)

(c)

0

2

(b)

(d)

1

3

18. If median = 15 and mean = 16, then mode is

(a)

(c)

10

12

(b)

(d)

11

13

19. The mean of 11 observations is 50. If the mean of first six observationsis 49 and that of last six observations is 52, then the sixth observation is

(a)

(c)

56

54

(b)

(d)

55

53

20. The mean of the following distribution is 2.6, then the value of ‘x’ is

Variable

Frequency

1

4

2

5

3

x

4

1

5

2


(a)

(c)

24

8

(b)

(d)

3

13

21. The mean of 40 observations was 160. It was detected on rechecking thatthe value of 165 was wrongly copied as 125 for computing the mean. Findthe correct mean.

Find ‘x’ if the median of the observations in ascending order 24, 25, 26,x + 2, x + 3, 30, 31, 34 is 27.5.

Find the mean of the following data.

10

3

12

5

14

6

16

4

18

4

20

3

22.

23.

x :

f :

24. Find the value of ‘p’, if mean of the following distribution is 7.5

3

6

5

8

7

15

9

p

11

8

13

4

Variable :

Frequency :

25. From the cumulative frequency table, write the frequency of the class20–30.

Marks

Less than 10

Less than 20

Less then 30

Less than 40

Less than 50

Number of Students

1

14

36

59

60


S.No.

26. Following is a cumulative frequency curve for the marks obtained by40 students. Find the median marks obtained by the student.

27. The following ‘more than ogive’ shows the weight of 40 students of a class.What is the lower limit of the median class.


28. The mean of the following frequency distribution is 62.8. Find the valuesof x and y.

0–20

5

20–40

x

40–60

10

60–80

y

80–100

7

100–120

8

Total

50

Class Interval :

Frequency :

29. The following frequency distribution gives the daily wages of a worker ofa factory. Find mean daily wage of a worker.

Daily Wage (in

More than 300

More than 250

More than 200

More than 150

More than 100

More than 50

More than 0

) Number of Workers

0

12

21

44

53

59

60

30. The median distance of the following data is 46. Find x & y.

Distance (m)

No. of students

10-20

12

20-30

20

30-40

x

40-50

65

50-60

y

60-70

25

70-80

18

Total

230

31. Find the mean, median and mode of the following :

0–10

6

10–20

8

20–30

10

30–40

15

40–50

5

50–60

4

60–70

2

Class Interval :

Frequency :

32. The following frequency distribution shows the marks obtained by 100students in a school. Find the mode.


Marks

Less than 10

Less than 20

Less than 30

Less than 40

Less than 50

Less than 60

Less than 70

Less than 80

Less than 90

Number of Students

10

15

30

50

72

85

90

95

100

33.

Marks :

Draw ‘less than’ and ‘more than’ ogives for the following distribution

20–30

8

30–40

12

40–50

24

50–60

6

60–70

10

70–80

15

80–90

25No. of Students :

Also find median from graph.

34. The mode of the following distribution is 65. Find the values of x and y, ifsum of the frequencies is 50.

0–20

6

20–40

8

40–60

x

60–80

12

80–100

6

100–120

y

120–140

3

Class Interval :

Frequency :

35. Following is the table people violating traffic rules under different agegroups:

18-20

63

20-22

50

22-24

35

24-26

27

26-28

16

28-30

9

Age (In yrs)

No. of personsviolating traffic rules

Find mean. Why should we obey traffic rules. What values are depictedhere?

36. The following table shows the ages of patients in a private hospital afterParliamentary Elections held in November for free treatment for those whocast their votes.


Age(In yrs)

No. ofPatients

18-28

6

28-38

11

38-48

21

48-58

23

58-68

14

68 onwards

5

Total

80

Find the mode of the above data. What values are depicted here?

37. Following table shows distance covered by 50 students of a school in ashot-put competition.

Distance (M)

No. of St.

0-20

6

20-40

11

40-60

17

60-80

12

80-100

4

Construct cumulative frequency distribution table. Find median. What valuesare depicted here?

38. The number of persons living in an old age home of a city are as follows:

50-55

10

55-60

12

60-65

17

65-70

13

70-75

16

75-80

22

Age (In yrs)

No. of old Persons

Find the mean and median. What steps should be taken to improve thecondition of old people in our society. What values are shown in thisquestion?

39. During the last two months the quantity of PNG used in a locality of 30houses was recorded as follows:

38-40

3

40-42

2

42-44

4

44-46

5

46-48

14

48-50

4

50-52

3

Quantity (M3)

No. of families

Find mean, median & mode of the above data.

40. The haemoglobin level of 35 students of a class is as follows:

Less then 8

3

Less then 10

7

Less then 12

13

Less then 14

23

Less then 16

35

Hb Level

No. of Students

Construct "less than type" of give.

What value is depicted in the question?


1.

3.

5.

7.

9.

11.

13.

15.

17.

19.

21.

23.

25.

27.

29.

31.

32.

34.

c

b

b

a

a

c

c

d

a

a

161

14.8

22

147.5

182.50

Mean = 30,

41.81

x = 10, y = 5

Median = 30.67, mode = 33.33

2.

4.

6.

8.

10.

12.

14.

16.

18.

20.

22.

24.

26.

28.

30.

b

a

c

b

b

b

a

b

d

c

x = 25

p = 3

40

x = 8, y = 12

x = 34, y = 46

33.

35.

60

(a) 22.1 years(b) discipline(c) knowledge &obeying traffic rules

36.

37.

38.

mode – 49.81 Social responsibility

m edi = 49.an41m , (A w areness f physiorcal ftiness), (I portmance of sports),(H ealhy com petton)tii

(a) (m ean) = 66.88, (m edian) = 67.3

(b) (generaton gap) (devel ng m oraliopi social values), (motivation)

(c) (moral values)

39.

40.

45.8, 46.5, 47.9

(Awarness for fitness), (Moral values).


Time allowed : 3 hours

General Instructions

1.

2.

All questions are compulsory.

Maximum Marks : 90

The question paper consists of 31 questions divided into four sections A,B, C and D. Section A comprises of 4 questions of 1 mark each. SectionB comprises of 6 questions of 2 marks each. Section C comprises of 10questions of 3 marks each and Section D comprises of 11 questions of 4marks each.

Question numbers 1 to 4 in Section A are multiple choice questions whereyou are to select one correct option out of the given four.

Use of calculator is not permitted.

3.

4.

1. In the given figure XY QR andPX 1 then :XQ 2

P

X Y

Q R


(a)

(c)

2.

XY = QR

(XY)2 = (QR)2

(b)

(d)

XY=

XY=

1 QR31 QR2

If sin (60° + A) = 1, then the value of cosec A is:

(a)

(c)

1

12

(b)

(d)

0

2

3. If sec 2. (1 + sin (1– sin = k, then the value of k is:

(a)

(c)

0

–1

(b)

(d)

1

2

4. If the mode of observation 2, 3, 5, 4, 2, 6, 3, 5, 5, 2 and x is 2, then thevalue of x is :

(a)

(c)

2

4

(b)

(d)

3

5

5.

6.

Whether 6n can end with the digit 0 for any natural number. Give reasonfor your answer.

If the product of the zeroes of the polynomial ax2 – 6x – 6 is 4, find thevalue of a.

7. arIf ~ arvalue of PR.

ABC 9 ,PQR 4 AB = 18 cm, BC = 15 cm, find the

8. If 16 cot A = 12 then find the value of

sin A cos A .sin A – cos A

9. Find HCF of 56, 96 and 324 by Euclid's algorithm.


10. Find x, if the median of the observations in ascending order

24, 25, 26, x + 2, x + 3, 30, 31, 34 is 27.5.

11.

12.

13.

If the areas of two similar triangles are equal, then prove that the trianglesare congruent.

Find the other zeroes of the polynomial for x4 – 20x3 + 23x2 + 5x – 6, iftwo of its zeroes are 2 and 3.

Use Euclid’s division lemma to show that the square of any positive integeris either of the form 3m or 3m + 1 for some integer m.

If sin (3x + 2y) = 1 and cos 3x 2 y

find the value of x and y.

3 when 0 3x + 2y 90° then2

14.

15.

16.

Divide 5x3 – 13x2 + 21x – 14 by 3 – 2x + x2 and verify the division lemma.

Solve for x and y, when x ±y

304 10x yx y 4055 13x yx y

17. The mean of the following frequency distribution is 62.8 and the sum of allthe frequencies is 50. Find the values of x and y.

0–20

5

20–40

x

40–60

10

60–80

y

80–100

7

100–120

8

Class interval

Frequency

18.

19.

The diagonals of a trapezium PQRS intersect each other at the point O,PQ||RS and PQ = 3 RS, find the ratio of the areas of and

If A, B and C are interior angles of a then show that

AB Csin cos 2 2


20. Find the median weight of the following data :

38–40

3

40–42

2

42–44

4

44–46

5

46–48

14

48–50

4

50–52

3

Weight in (kg.)

No. of Students

21.

22.

Prove that 5 is an irrational number and hence prove 2 –irrational number.

5 is an

Draw the graph of the following pair of linear equations x + 3y = 6 and2x – 3y = 12. Find the ratio of the areas of the two triangles formed by firstline, x = 0, y = 0 and second line x = 0, y = 0.

Prove that :

tan cot 1 sec . cosec .1 cot 1 tan

23.

24. Find the value of K so that the pair of linear equations:

3K 1x 3 y 2 0K 2 1x K 2y 5 0

is inconsistent.

25. Change the distribution to a more than type distribution and draw its ogive

50-55

2

55-60

8

60-65

12

65-70

24

70–75

38

75–80

16

Class Interval

Frequency

26.

27.

28.

On dividing 4x3 – 8x2 + 8x + 1 by a polynomial d(x), the quotient andremainder are (2x2 – 3x + 2) and (x + 3) respectively, find d(x).

State and prove converse of pythagoras theorem.

Prove the following identity :

cos A1 sin A 2 sec A.1 sin Acos A


29.

30.

In an isosceles with AB =AC, BD is the perpendicular drawn fromthe vertex B to the side AC. Prove that BD2 – CD2 = 2CD.AD.

Evaluate :

sec .cosec 90 – – tan . cot 90 – sin2 55 sin2 35

tan10.tan20.tan60 .tan70 .tan80

31. In a city, the number of old aged citizens lived in an old age home is asgiven below

Age (in years)

50–55

55–60

60–65

65–70

70–75

75–80

No. of People

10

12

17

13

16

22

(a)

(b)

Find mean of the above data?

Why the old people are neglected in the society and what are thevarious steps to be taken to improve the status of old people in thesociety?

What value is depicted in this question?(c)

1.

3.

6.

8.

10.

b

c

a = –3/2

7

x = 25

2.

4.

7.

9.

12.

d

a

10 cm

4

±12


13.

15.

17.

20.

24.

x = 20, y = 15

Q = 5x – 3, R = –5

x = 8, y = 12

46. 5 Kg.

K = –1, K

14.

16.

18.

22.

A = 60°, B = 30°

x = 5, y = 3

9:1

1:2

d (x) = 2x – 119

226.

230.

31.

3

(a) 66.8.

(b) Generation gap, lack of social and moral values.

(c) Impart social and moral values in the society.


S.No. Chapter Page

1. Quadratic Equations 68

2. Arithmetic Progression 77

3. Co-ordinate Geometry 84

4. Some Applications of Trigonometry 92

5. Circle 100

6. Constructions 116

7. Mensuration 120

8. Probability 133

Sample Papers 141


CHAPTER 1

QUADRATIC EQUATIONS

1.

2.

The equation ax2 + bx + c = 0, a 0 is the standard form of a quadraticequation, where a, b and c are real numbers.

A real number is said to be a root of the quadratic equation ax2 + bx+ c = 0, a 0. If a2 + b + c = 0, the zeroes of quadratic polynomial ax2

+ bx + c and the roots of the quadratic equation ax2 + bx + c = 0 are thesame.

If we can factorise ax2 + bx + c = 0, a 0 into product of two linear factors,then the roots of the quadratic equation can be found by equating eachfactors to zero.

The roots of a quadratic equation ax2 + bx + c = 0, a 0 are given by

3.

4.

b

5.

b 4ac , provided that b2 – 4ac 0.2a

2

A quadratic equation ax2 + bx + c = 0, a 0, has ___

(a)

(b)

(c)

Two distinct and real roots, if b2 – 4ac > 0.

Two equal and real roots, if b2 – 4ac = 0.

Two roots are not real, if b2 – 4ac < 0.

6. A quadratic equation can also be solved by the method of completing thesquare.

(i)

(ii)

a2 + 2ab + b2 = (a + b)2

a2 – 2ab + b2 = (a – b)2

7. Discriminant of the quadratic equation ax2 + bx + c = 0, a 0 is given byD = b2 – 4ac.


1. The roots of ax2 + bx + c = 0, a 0 are real if b2 – 4ac is

(a)

(c)

0

< 0

(b)

(d)

> 0

None of these

2. Maximum number of solutions of a quadratic equation are :

(a)

(c)

0

2

(b)

(d)

1

3

3. If the equation x2 – (2 + m) x + (– m2 – 4m – 4) = 0 has coincident roots,then

(a)

(c)

m = 0, m = 1

m = – 2, m = – 2

(b)

(d)

m = 2, m = 2

m = 6, m = 1

4. If one root of x2 – ax – 4 = is negative of the other then the value of a is:

(a)

(c)

0

2

(b)

(d)

1

4

5. Which is a quadratic equation?

(a)

(c)

x 1 2x

(b)

(d)

x2 + 1 = (x + 3)2

x (x + 2) x 1 .x

6. If the roots of a quadratic equation are 2 and 3, then the equation is:

(a)

(c)

x2 + 5x + 6 = 0

x2 – 5x – 6 = 0

(b)

(d)

x2 + 5x – 6 = 0

x2 – 5x + 6 = 0

7. Roots of the equations x2 – 3x + 2 = 0 are

(a)

(c)

1, –2

–1, –2

(b)

(d)

–1, 2

1, 2


8. If the roots of a quadratic equation are equal, then discriminant is

(a)

(c)

1

greater than 0

(b)

(d)

0

less than zero.

9. If one root of 2x2 + kx + 1 = 0 is –

(a)

(c)

3

5

1 , then the value of ‘k’ is2

–3

–5

(b)

(d)

10. The sum of the roots of the quadratic 5x2 – 6x + 1 = 0 is

(a)65

56

(b)15

15(c)

11.

(d)

The product of the roots of the quadratic equation 2x2 + 5x – 7 = 0 is

(a)

(c)

52

52

(b)

(d)

72

72

12. If the roots of the quadratic 2x2 + kx + 2 = 0 are equal then the value of‘k’ is

(a)

(c)

4

± 4

(b)

(d)

–4

± 16

13. If the roots of x2 + px + 12 = 0 are in the ratio 1 : 3 then p = _____

(a)

(c)

± 2 (b)

(d)

8

84

14. If the sum and product of roots of a quadratic equation are

respectively, then the equation is

75 and22


(a)

(c)

15.

2x2 + 7x + 5 = 0

2x2 – 7x – 5 = 0

(b)

(d)

2x2 – 7x + 5 = 0

2x2 + 7x – 5 = 0

If one root of ax2 + bx + c = 0, a 0 is the reciprocal of the other thenc = .....

(a)

(c)

b

c

(b)

(d)

a

0

16.

17.

18.

19.

If one root of the equation x2 + 7x + k = 0 is –2, then find the value of kand the other root.

If the roots of 4x2 + 3px + 9 = 0 are real and distinct, find the value of p?

For what value of m the sum of the roots of mx2 + 6x + 4m = 0 be equalto the product of the roots ?

The product of two consecutive odd integers is 63. Represent this in formof a quadratic equation.

Find the roots of the equation : x 20.

21.

22.

23.

24.

25.

26.

11 4 , x 0.x4

Find the value of c for which one roots of 4x2 – 2x + (c – 4) = 0 isreciprocal of the other.

Divide 51 in to two parts such that their product is 378.

Find ‘k’ so that (k – 12) x2 + 2 (k – 12) x + 2 = 0 has equal roots.(k 12).

If (–5) is a root of the equation 2x2 + px – 15 = 0 and the equationp(x2 + x) + k = 0 has equal roots, find values of p and k.

Find the value of K, so that the difference of the roots of x2 – 5x + 3(k – 1) is 11.

The difference of two numbers is 5 and the difference of their reciprocals 1 . Find the numbers.is 10


27.

28.

If the roots of the equation (b – c)x2 + (c – a) x + (a – b) = 0 are equal,then prove that 2b = a + c.

Find the nature of the roots of the following quadratic equations. If rootsare real, find them.

(a) 5x2 – 3x + 2 = 0. (b) 2x2 – 9x + 9 = 0.

29. Sum of two numbers is 15, if sum of their reciprocals is

numbers.

Solve the following quadratic equations:-

3 . Find the10

30.2x 3 x 3 225 5 x 3 2x 3

x 3, x 32

31.1 1 11 , a, b, x 0 and x – a b a bxabx

32.

33.

34.

35.

36.

4 3x25x 2 3 0.

ab x2 + (b2 – ac) x – bc = 0.

x 1 x 310 , x 2, x 4.x 2x 43 1111 , x 4, x 7.x 4x 730

Shikha and Mona each wish to grow 100 m2 rectangular vegetable garden.They used 30m wire each for fencing three sides of the garden lettingcompound wall of their houses act as the fourth side. Find the possiblesides of their gardens.

A shop keeper buys a number of pens for 80. If he had bought 4 morepens for the same amount, each pen would have cost him 1 less. Howwmany pens did he buy?

A two digit number is such that the product of the digits is 35, when 18 isadded to the number, the digits inter change their places. Find the number.

Three consecutive positive integers are such that the sum of the squareof the first and the product of the other two is 46, find the integers.

37.

38.

39.


40. A motor boat whose speed is 9 km/h in still water goes 12 km downstream and comes back in a total time 3 hours. Find the speed of thestream.

A train travels 360 km at uniform speed. If the speed had been 5 km/hrmore it would have taken 1 hour less for the same journey. Find the speedof the train.

The hypotenuse of right angled triangle is 6cm more than twice the shortestside. If the third side is 2 cm less than the hypotenuse, find the sides ofthe triangle.

By a reduction of 2 per kg in the price of tomato. Anita can purchase2 kg tomato more for 224. Find the original price of tomato per kg.

Two trains leave a Railway Station at the same time. The first train travelsdue west and the second train due North. Speed of first train is 5 km/hfaster than the second train.If after two hours they are 50 km apart, findthe average speed of the faster train.

A girl is twice as old as her brother. Four years hence, the product of theirages will be 160. Find their present ages.

Two years ago a man’s age was three times the square of his son’s age.Three years hence his age will be four times his son’s age. Find theirpresent ages.

In a cricket match against Sri Lanka, Sehwag took one wicket less thantwice the number of wickets taken by Unmukt. If the product of the numberof wickets taken by these two is 15, find the number of wickets taken byeach.

41.

42.

43.

44.

45.

46.

47.

48. 8 minutes. If one pipe 11takes 1 minute more than the other to fill the cistern, find the time in whicheach pipe would fill the cistern alone.

Two pipes running together can fill a cistern in 2

49. 6500 were divided equally among a certain number of students. Hadthere been 15 more students, each would have got 30 less. Find theoriginal number of students. What are the values reflected in this question?


50. A takes 10 days less than the time taken by B to finish a piece of work.If both A and B together can finish the work in 12 days, find the time takenby B to finish the work alone. What are the moral values reflected in thisquestion which are to be adopted in our life?

1.

3.

5.

7.

9.

11.

13.

15.

17.

19.

21.

23.

24.

26.

27.

29.

30.

32.

34.

b

c

a

d

a

b

b

b

p < –4 or p > 4

x2 + 2x – 63 = 0

c = 8

k = 14

2. c

4. a

6. d

8. b

10. a

12. c

14. a

16. k = 10, second root = – 5

18.

20.

–

4,

32

14

22. 9, 42

7,74 25. k = –7

10, 5

Hint : For equal roots D = 0.

5, 10

6, 1

28. (a) Not real roots.

(b) Roots are real, 3,

31. –a, –b

33.

3 .2

32 , 43

5,52

cb , ba

35. 1, 2


36.

38.

40.

42.

44.

46.

48.

50.

5 × 20, 10 × 10

57

3 km/hr.

26 cm, 24 cm, 10 cm

20 km.

29 yrs., 5 yrs.

5, 6 min

30 days, Unity

37. 16

39. 4, 5, 6

41. 40 km/hr.

43. Rs. 16

45. 12, 6

47. 3, 5

49. 50, Equality


CHAPTER 2

ARITHMETIC PROGRESSION

1.

2.

3.

Sequence : A set of numbers arranged in some definite order and formedaccording to some rules is called a sequence.

Progression : The sequence that follows a certain pattern is calledprogression.

Arithmetic Progression : A sequence in which the difference obtained bysubtracting any term from its preceding term is constant throughout, iscalled an arithmetic sequence or arithmetic progression (A.P.).

The general form of an A.P. is a, a + d, a + 2d, ..... (a : first termd : common difference).

4.

5.

General Term : If ‘a’ is the first term and ‘d’ is common difference in anA.P., then nth term (general term) is given by an = a + (n – 1) d .

Sum of n Terms of an A.P. : If ‘a’ is the first term and ‘d’ is the commondifference of an A.P., then sum of first n terms is given by

Sn n 2a n 1d 2

If ‘l’ is the last term of a finite A.P., then the sum is given by

Sn

6. (i)

(ii)

(iii)

(iv)

n a l .2

If an is given, then common difference d = an – an–1.

If sn is given, then nth term is given by an = sn – sn–1.

If a, b, c are in A.P., then 2b = a + c.

If a sequence has n terms, its rth term from the end = (n – r + 1)th

term from the beginning.


1. Three numbers in A.P. have sum 24. The middle term is—

(a)

(c)

6

3

(b)

(d)

8

2

2. If nth term of on A.P. is 2n + 7, then 7th term of the A.P. is

(a)

(c)

15

28

(b)

(d)

21

25

3. If the sum of n terms of an A.P. is

is

(a)

(c)

250

225

5 23n n , then sum of its 10 terms22

(b)

(d)

230

265

4. If nth term of the A.P. 4, 7, 10, ________ is 82, then the value of n is

(a)

(c)

29

30

(b)

(d)

27

26

5. If a, b and c are in A.P. then

(a)

(c)

a

c

b c 2

a b 2

(b)

(d)

b a c 2

b = a + c

6. 12th term of the A.P. x – 7, x – 2, x + 3 is

(a)

(c)

x + 62

x + 48

(b)

(d)

x – 48

x – 62

7. Common difference of A.P. 8

(a)

123 , 8 , 8 , ________ is888

(b)18

78

118

[Class-X – Maths]

(c)

8.

818 (d) 1

nth term of the A.P. –5, –2, 1, ________ is

(a)

(c)

3n + 5

8n – 5

(b)

(d)

8 – 3n

3n – 8

9. If nth term of an A.P. is 5 – 3n, then common difference of the A.P. is

(a)

(c)

2

–2

(b)

(d)

–3

3

10. If 5, 2k – 3, 9 are in A.P., then the value of ‘k’ is

(a)

(c)

4

6

(b)

(d)

5

–5

11. Sum of first 10 natural numbers is

(a)

(c)

50

60

(b)

(d)

55

65

12. 9th term from the end of the A.P. 7, 11, 15, _______ 147 is

(a)

(c)

135

115

(b)

(d)

125

110

13. If the sum of n terms of an A.P. is n2, then its nth term is

(a)

(c)

2n – 1

n2 – 1

(b)

(d)

2n + 1

2n – 3

14. The sum of 3 numbers in A.P. is 30. If the greatest number is 13, then itscommon difference is

(a)

(c)

4

2

(b)

(d)

3

5


15. The sum of 6th and 7th terms of an A.P. is 39 and common difference is3, then the first term of the A.P. is

(a)

(c)

2

4

(b)

(d)

–3

3

16.

17.

18.

19.

20.

21.

22.

23.

24.

25.

26.

27.

Is 2, 8, 18, .32, ______ an A.P.? If yes, then find its next two terms.

Find an A.P. whose 2nd term is 10 and the 6th term exceeds the 4th termby 12.

Which term of the A.P. 41, 38, 35... is the first negative term? Find the termalso.

The first and the last term of an AP are 4 and 81 respectively. If commondifference is 7. Find the number of terms.

Find the number of terms in an A.P. whose first term and 6th term are 3and 23 respectively and sum of all terms is 406.

How many two digits numbers between 6 and 102 are divisible by 6.

If sn the sum of first n terms of an A.P. is given by sn = 3n2 – 4n, then findits nth term and common difference.

The sum of 4th and 8th terms of an A.P. is 24 and sum of 6th and 10th termsis 44. Find A.P.

Find the sum of odd positive integers between 1 and 199.

How many terms of the A.P. 22, 20, 18, _____ should be taken so thattheir sum is zero?

4k + 8, 2k2 + 3k + 6, 3k2 + 4k + 4 are the angles of a triangle. These forman A.P. Find value of k.

If 11 times of 11th term is equal to 17 times of 17th term of an A.P. find its28th term.

Find an A.P. of 8 terms, whose first term is28.117 .and last term is26


29. The fourth term of an A.P. is equal to 3 times the first term and the seventhterm exceeds twice the third term by 1. Find the first term and commondifference of the A.P.

Find the middle term of the A.P. 20, 16, 12, ......, –176.30.

31. If 2nd, 31st and last terms of on A.P. are

Find the number of terms in the A.P.

31 113 ,and respectively..4 22

32.

33.

34.

35.

36.

37.

38.

39.

Find the number of terms of the A.P. 57, 54, 51, ______ so that their sumis 570.

The sum of three numbers in A.P. is 24 and their product is 440. Find thenumbers.

Find the sum of the first 40 terms of an A.P. whose nth term is 3 – 2n.

In an A.P., the first term is 2, the last term is 29 and the sum of the termsis 155. Find common difference ‘d’.

The sum of 5th and 9th terms of an A.P. is 8 and their product is 15. Findthe sum of first 28 terms of the A.P.

Find the sum of all the three digits numbers each of which leaves theremainder 3 when divided by 5.

The sum of first six terms of an A.P. is 42. The ratio of the 10th term to the30th term is 1 : 3. Find first term and 11th term of the A.P.

Kriti, starts a game and scores 200 points in the first attempt and sheincreases the points by 40 in each attempt. How many points will shescore is the 30th attempt?

The eight term of on A.P. is half the second term and the eleventh termexceeds one-third of its fourth term by 1. Find a15.

The sum of first 8 terms of an A.P. is 140 and sum of first 24 terms is 996.Find the A.P.

The digits of a three digits positive number are in A.P. and the sum of digitsis 15. On subtracting 594 from the number the digits are interchanged.Find the number.


40.

41.

42.

43. A picnic group for Shimla consists of students whose ages are in A.P., thecommon difference being 3 months. If the youngest student Neeraj is just12 years old and the sum of ages of all students is 375 years. Find thenumber of students in the group.

The sum of first 20 terms of an A.P. is one third of the sum of next 20terms. If first term is 1, then find the sum of first 30 terms.

The sum of first 16 terms of an A.P. is 528 and sum of next 16 terms is1552. Find the first term and common difference of the A.P.

44.

45.

46. Nidhi saves Rs. 2 on day 1, Rs. 4 on day 2, Rs. 6 on day 3 and so on.How much money she save in month of Feb. 2011? What values of Nidhiare depicted in the question?

An old lady Krishna Devi deposited Rs. 120000 in a bank at 8% interestp.a. She uses the annual interest to give five scholarships to the studentsof a school for their overall performances each year. The amount of eachscholarship is Rs. 300 less than the preceding scholarship. Find the amountof each scholarship. What values of lady are depicted here?

Ram asks the labour to dig a well upto a depth of 10 metre. Labourcharges are Rs. 150 for first metre and Rs. 50 for each subsequent metre.As labour was uneducated, he claims Rs. 550 for the whole works. Whatshould be the actual amount to be paid to the labour? What value of Ramis depicted in the question if he pays Rs. 600 to the labour?

200 logs are stacked such that 20 logs are in the bottom row, 19 in thenext now, 18 in the row next to it and so on. In how many rows are the200 logs placed? What value is depicted in the pattern of logs?

Puru and Ashu live in two different villages 165 km apart. They want tomeet each other but there is no fast means of transport. Puru travels 15kmthe first day, 14 km the second day, 13 km the third day and so on. Ashutravels 10 km the first day, 12 km the second dry, 14 km the third day andso on. After how many days will they meet. What value of Puru and Ashuis depicted here?

47.

48.

49.

50.


1.

3.

5.

7.

9.

11.

13.

15.

17.

19.

21.

23.

25.

27.

29.

31.

33.

35.

37.

39.

41.

43.

45.

47.

48.

49.

50.

b

d

b

a

b

b

a

d

4, 10, 16, ...............

12

15

–13, –8, –3, 2 ...............

23

0

3, 2

59

5, 8, 11

3

99090

1360

7, 10, 13, 16, ...............

25

3, 4

2. b

4. b

6. c

8. d

10. b

12. c

14. b

16. Yes, 50, 72

18. 15th term, –1

20. 14

22. 6n – 7, 6

24. 9800

26. 0, 2

28.1, 5 7 9 11 13 15 17 , , , , ,,,2 6 6 6 6 6 6 6

30. –76, –80

32. 19 or 20

34. –1520

136. 217, 7 d 238. First term = 2, 11th term = 22

40. 3

42. 852

44. 900

46. Rs. 812, Saving, Economy

Rs. 2520, Rs. 2220, Rs. 1920, Rs. 1620, Rs. 1320 Love for children,charity, small savings

Rs. 600, Honesty, Sincerity

16, Space saving, creativity, Reasoning, Balancing.

6 days, Affection for each other.


CHAPTER 3

CO-ORDINATE GEOMETRY

1. The length of a line segment joining A and B is the distance between two

points A (x1, y1) and B (x2, y2) is x2 x1 y 2 y1 2 2 .

2. The distance of a point (x, y) from the origin is x 2 y 2 . The distance

of P from x-axis is y units and from y-axis is x-units.

3. The co-ordinates of the points p(x, y) which divides the line segmentjoining the points A(x1, y1) and B(x2, y2) in the ratio m1 : m2 are

m 1x 2 m 2 x 1 m 1y 2 m 2 y 1 ,m mm1 m 212

we can take ratio as k : 1, k

4.

m1 .m2

The mid-points of the line segment joining the points P(x1, y1) andQ(x2, y2) is

x1 x 2 y1 y2 ,225. The area of the triangle formed by the points (x1, y1), (x2, y2) and (x3, y3)

is the numeric value of the expressions

1 x y y 3 x 2 y 3 y 1x 3 y 1 y 2 .21 2

6. If three points are collinear then we can not draw a triangle, so the areawill be zero i.e.

|x1(y2 – y3) + x2 (y3 – y1) + x3(y1 – y2)| = 0


1. P is a point on x axis at a distance of 3 unit from y axis to its left. Thecoordinates of P are

(a)

(c)

(3, 0)

(–3, 0)

(b)

(d)

(0, 3)

(0, –3)

2. The distance of point P (3, –2) from y-axis is

(a)

(c)

3 units

–2 units

(b)

(d)

2 units

13 units

3. The coordinates of two points are (6, 0) and (0, –8). The coordinates ofthe mid point are

(a)

(c)

(3, 4)

(0, 0)

(b)

(d)

(3, –4)

(–4, 3)

4. If the distance between (4, 0) and (0, x) is 5 units, the value of x will be

(a)

(c)

2

4

(b)

(d)

3

5

5. The coordinates of the point where line

(a)

(c)

(a, 0)

(0, 7b)

(b)

(d)

xy 7 intersects y-axis areab

(0, b)

(2a, 0)

6. The area of triangle OAB, the coordinates of the points A (4, 0)B (0, –7) and O origin, is

(a)

(c)

11 sq. units

28 sq. units

(b)

(d)

18 sq. units

14 sq. units

7.11 2The distance between the points P , 5 and Q , 5 is 33

(a)

(c)

6 units

3 units

85

(b)

(d)

4 units

2 units

[Class-X – Maths]

8. xy 1 intersects both the axis at P and Q, the coordinates 24of the mid point of PQ are

The line

(a)

(c)

(1, 2)

(0, 4)

(b)

(d)

(2, 0)

(2, 1)

9. A line is drawn through a point P (3, 2) parallel to x – axis. The distanceof the line from x-axis is–

(a)

(c)

2 units

13 units

(b)

(d)

3 units

none.

10. The distance between the line 2x + 4 = 0 and x – 5 = 0 is

(a)

(c)

9 units

5 units

(b)

(d)

1 unit

7 units

11. The distance between the points (5 cos 35°, 0) and (0, 5 cos 55°) is

(a)

(c)

10 units

1 unit

(b)

(d)

5 units

2 units

12. The points (–4, 0), (4, 0) and (0, 3) are the vertices of a :

(a)

(c)

right triangle

equilateral triangle

(b)

(d)

Isosceles triangle

Scalene triangle

13. The perimeter of triangle formed by the points (0, 0), (2, 0) and (0, 2) is

(a)

(c)

4 units

6 2 units

(b)

(d)

6 units

4 2 2 units

14. AOBC is a rectangle whose three vertices are A (0, 3), O (0, 0), B (5, 0).The length of its diagonal is :

(a)

(c)

5 units (b)

(d)

3 units

4 units34 units


15. If the centroid of the triangle formed by (9, a), (b, –4) and (7, 8) is (6, 8)then (a, b) is

(a)

(c)

(4, 5)

(5, 2)

(b)

(d)

(5, 4)

(3, 2)

16.

17.

18.

Find the value of a so that the point (3, a) lies on the line represented by2x – 3y = 5.

The co-ordinates of vertex P of are (–4, 2) and the point S (2, 5)is the mid point of QR. What will be the co-ordinates of centroid.

What is the value of a if the points (3, 5) and (7, 1) are equidistant fromthe point (a, 0)?

b aProve that the points 0, a , , and (b, 0) are collinear. . 2 2

19.

20.

21.

22.

23.

24.

25.

26.

27.

AB is diameter of a circle with centre at origin. What are the coordinatesof B if coordinates of A are (3, –4)?

If C is a point lying on the line segment AB joining A (1, 1) & B (2, –3)such that 3AC = BC. Find the coordinates of C.

For what value of p, are the points (–3, 9), (2, p) and (4, –5) collinear?

If the coordinates of the vertices of a triangle are A (6, 7), B (4, –5),C (x, 2x) and its area is 3 square units. Find x.

Find the coordinates of point P if P and Q trisect the line segment joiningthe points A (1, –2) and B (–3, 4).

The centre of the circle is (2 –1, 7) and it passes through the point(–3, –1). If diameter is 20 units, find

Find the abscissa of a point where ordinate is 4 and which is at a distance5 units from (5, 0).

Find a point on y-axis which is equidistant from the points (–2, 5) and(2, –3).


28. The mid point of the line segment joining the points (5, 7) and (3, 9) is alsothe mid point of the line segment joining the points (8, 6) and (a, b). Finda and b.

Find the coordinates of the point which divides the line segment joining thepoints (1, 3) and (2, 7) in the ratio 3 : 4.

If the points (x, y), (–5, –2) and (3, –5) are collinear, prove that 3x + 8y+ 31 = 0

The point K (1, 2) lies on the perpendicular bisector of the line segmentjoining the points E (6, 8) and F (2, 4). Find the distance of the point Kfrom the line segment EF.

The area of triangle ABC is 5 square units. Two of its vertices are A (2,1) and B (3, –2). The third vertex C lies on x – y – 3 = 0, find coordinatesof C.

Find the distance between the points A (a, b) and B (b, a) if a – b = 4.

Three vertices of a parallelogram taken in order are (–3, 1), (1, 1) and(3, 3). Find the coordinates of fourth vertex.

Triangle ABC is an isosceles triangle with AB = AC and vertex A lies ony-axis. If the coordinates of B and C are (–5, –2) and (3, 2) respectivelythen find the coordinates of vertex A.

If A (3, 0), B (4, 5), C (–1, 4) and D (–2, –1) are four points in a plane,show that ABCD is a rhombus but not a square.

Find the coordinates of a point which is

29.

30.

31.

32.

33.

34.

35.

36.

37.

38.

39.

40.

3 of the way (3, 1) to (–2, 5).4

The area of a triangle with vertices (6, –3), (3, K) and (–7, 7) is 15 sq. unit.Find the value of K.

Determine the ratio in which the line 3x + y = 9, divides the line segmentjoining the points A (1, 3) and B (2, 7).

A point P on the x-axis divides the line segment joining the points (4, 5)and (1, –3) in certain ratio. Find the coordinates of point P.

In right angled = 90° and AB 34 unit. The coordinates ofpoints B, C are (4, 2) and (–1, y) respectively. If ar (ABC) = 17 sq. unit,then find the value of y.

88

41.

[Class-X – Maths]

42.

43.

44.

If the point P (3, 4) is equidistant from the points A (a + b, b – a) and B(a – b, a + b) then prove that 3b – 4a = 0.

The vertices of quadrilateral ABCD are A (–5, 7), B (–4, –5), C (–1, –6)and D (4, 5). Find the area of quadrilateral ABCD.

The line segment joining the points A (2, 1) and B (5, –8) is trisected atthe points P and Q such that P is nearer to A. If P also lies on line givenby 2x – y + k = 0, find the value of k.

The line segment joining the points (3, –4) and (1, 2) is trisected at the45.

5point P and Q. If the coordinates of P and Q are (p, –2) and , q 3respectively, find the values of p and q.

46. A teacher is asked three students to stand to form a triangle at the pointsP (–1, 3), Q (1,–1) and R (5, 1). Suddenly a fourth boy came and showshis interest in participating the activity. She asked him to stand at point midway between Q & R. What is his distance from P. What values of theteacher appears when she agreed the fourth boy to participate?

To solve a riddle a girl is asked to join the three points A (7, 5), B (2, 3)and C (6, –7) with a sketchpen. After joining these points a triangle isobtained by her. What type of triangle is it? What values are depicted inthe question?

The coordinates of the houses of Mona and Nishi are (7, 3) & (4, –3)respectively. The coordinates of their school are (2, 2). If they both startfor school at the same time in the morning and reaches at the same time,who walks fast? What values are depicted from the question?

Prove that the points A (a, o), B (o, b) & C (1, 1) are collinear if1 1 1a b

47.

48.

49.

50. For social awareness regarding bad effects of pollution a school started acampaign for which 1000 students were asked to make triangular badgeswith coordinates (5, 6), (1, 5), (2, 1) If cost of 1cm2 of paper is Rs. 2 findthe total expenditure incurred in this campaign. What values are depictedin the question?


1.

3.

5.

7.

9.

11.

13.

15.

17.

20.

22.

24.

26.

28.

31.

33.

35.

c

b

c

c

a

b

d

c

(0, 4)

(–3, 4)

p = – 1

13 , 0

2. a

4. b

6. d

8. a

10. d

12. b

14. c

16. a 13

18. a = 2

5 21. , 0 3 23. x = 8

25. = 2, –4

27. (0, 1)

10 33 ,29. 77

2, 8

a = 0, b = 10

5 units

4 2 units

32. (5, 2)

34. (–1, 3)

337. 4 , 4

(0, –2)

38. K 2113 39. 3 : 4


40.

43.

45.

46.

47.

178 , 072 Square Units

41. y = –1, 5

44. K = – 8

p 7 ,q 03

5 Units, interest in Mathematics, Friendship, Cooperation

(a) Right Angled Triangle

(b) Sports, Activeness, Critical thinking.

48.

50.

(a) Mona, (b) Time bound, Reality

Rs. 11500


CHAPTER 4

SOME APPLICATIONS OF TRIGONOMETRY

1. Line of Sight : The line of sight is the line drawn from the eyes of anobserver to a point in the object viewed by the observer.

Angle of Elevation : The angle of elevation is the angle formed by the lineof sight with the horizontal, when it is above the horizontal level i.e. thecase when we raise our head to look at the object.

Angle of Depression : The angle of depression is the angle formed by theline of sight with the horizontal when it is below the horizontal i.e. casewhen we lower our head to look at the object.

2.

3.

1. The length of the shadow of a man is equal to the height of man. Theangle of elevation is

(a)

(c)

90°

45°

(b)

(d)

60°

30°

2. The length of the shadow of a pole 30m high at some instant is 10 3 m.The angle of elevation of the sun is

(a)

(c)

30°

45°

(b)

(d)

60°

90°

3. Find the angle of depression of a boat from the bridge at a horizontaldistance of 25m from the bridge, if the height of the bridge is 25m.


(a)

(c)

4.

45°

30°

(b)

(d)

60°

15°

The tops of two poles of height 10m and 18m are connected with wire. Ifwire makes an angle of 30° with horizontal, then length of wire is

(a)

(c)

10m

12m

(b)

(d)

18m

16m

5. From a point 20m away from the foot of the tower, the angle of elevationof the top of the tower is 30°. The height of the tower is

(a) 20 3 m (b)20

3

40

3

m

(c)

6.

40 3 m (d) m

1

3. The angle ofThe ratio of the length of a tree and its shadow is 1 :

elevation of the sun is

(a)

(c)

30°

60°

(b)

(d)

45°

90°

7. A kite is flying at a height of 50 3 m above the level ground, attached

to string inclined at 60° to the horizontal, the length of string is

(a)

(c)

100 m

150 m

(b)

(d)

50 m

75 m

8. In given Fig. 1 the perimeter of rectangle ABCD is

D C

10 m

30°

A BFig. 1


(a)

(c)

9.

40 m

60 m

(b)

(d)

20

10

3 1 m

3 1 m

A tree is broken at a height of 10 m above the ground. The broken parttouches the ground and makes an angle of 30° with the horizontal. Theheight of the tree is

(a)

(c)

30 m

10 m

(b)

(d)

20 m

15 m

10. The angle of elevation of the top of a 15 m high tower at a point 15 m awayfrom the base of the tower is

(a)

(c)

30°

60°

(b)

(d)

45°

75°

11. In given Fig. 2, D is mid point of BC, = 1 and = 2 thentan 1 : tan 2 is equal to

A

1

2

C D B

Fig. 2

(a)

(c)

2:1

1:1

(b)

(d)

1:2

1:3


12. In given Fig. 3, tan

(a)

(c)

16 m

32 m

8 if PQ = 16 m, then the length of PR is15

(b)

(d)

34 m

30 m

P

R Q

Fig. 3

13. The height of a tower is 50 m. When angle of elevation changes from 45°to 30°, the shadow of tower becomes x metres more, the value of x is

(a)

(c)

14.

50 m (b)

(d)

50 3 1 m

50 3 m50

3m

The angle of elevations of a building from two points on the ground 9m and16m away from the foot of the building are complementary, the height ofthe building is

(a)

(c)

18 m

10 m

(b)

(d)

16 m

12 m

15. In the given Fig. 4, If ABCD is a rectangle then AL + AM =

70(a) 70 3m (b)

3m

(c) 210 3 (d) 70 m


D

60°

45 m

L C

M60°

A 60 m B

Fig. 4

16. From a point on the ground the angle of elevations of the bottom and topof a water tank kept on the top of the 30m high building are 45° and 60°respectively. Find the height of the water tank.

The shadow of a tower standing on the level ground is found to be 60mshorter when the sun’s altitude changes from 30° to 60°, find the heightof tower.

The angle of elevation of a cloud from a point h metres above a lake isand the angle of depression of its reflection in the lake is prove that 2h sec the distance of the cloud from the point of observation is tan tan .

The angle of elevation of a bird from a point on the ground is 60°, after50 seconds flight the angle of elevation changes to 30°. If the bird is flyingat the height of 500 3 m. Find the speed of the bird.

The angle of elevation of a jet fighter plane from a point A on the groundis 60°. After a flight of 15 seconds, the angle of elevation changes to 30°.If the jet is flying at a speed of 720 km/h. Find the constant height at which

the jet is flying. Take

17.

18.

19.

20.

3 1.732 . 21. From a window 20m high above the ground in a street, the angle of

elevation and depression of the top and the foot of another house oppositeside of the street are 60° and 45° respectively. Find the height of oppositehouse.

An aeroplane flying at a height of 1800m observes angles of depressionsof two points on the opposite bank of the river to be 60° and 45°, find thewidth of the river.

96

22.

[Class-X – Maths]

23. The angle of elevation of the top of the tower from two points A and Bwhich are 15m apart, on the same side of the tower on the level groundare 30° and 60° respectively. Find the height of the tower and distance of

point B from the base of the tower. Take 3 1.732 24. The angle of elevation of the top of a 10m high building from a point P on

the ground is 30°. A flag is hoisted at the top of the building and the angleof elevation of the top of the flag staff from P is 45°. Find the length of theflag staff and the distance of the building from point P.

The angle of elevation of a bird from a point 12 metres above a lake is 30°and the angle of depression of its reflection in the lake is 60°. Find thedistance of the bird from the point of observation.

The shadow of a vertical tower on level ground increases by 10 m whensun’s attitude changes from 45° to 30°. Find the height of the tower, upto

one place of decimal

25.

26.

3 1.73 .

27. A man on a cliff observes a boat at an angle of depression of 30°, whichis approaching the shore to point ‘A’ immediately beneath the observer witha uniform speed, 12 minutes later, the angle of depression of the boat isfound to be 60°. Find the time taken by the boat to reach the shore.

A man standing on the deck of a ship, 18m above the water level observesthat the angle of elevation and depression of the top and the bottom of acliff are 60° and 30° respectively. Find the distance of the cliff from the shipand height of the cliff.

A person standing on the bank of a river observes that the angle ofelevation of the top of a tree standing on the opposite bank is 60°. Whenhe moves 40m away from the bank he finds the angle of elevation to be30°. Find the height of the tree and the width of the river.

An aeroplane, when 300 m high, passes vertically above another plane atan instant when the angle of elevation of two aeroplanes from the samepoint on the ground are 60° and 45° respectively. Find the vertical distancebetween the two planes.

The angle of depression of the top and bottom of a 10m tall building fromthe top of a tower are 30° and 45° respectively. Find the height of the towerand distance between building and tower.

A boy standing on a horizontal plane, finds a bird flying at a distance of

28.

29.

30.

31.

32.


100m from him at an elevation of 30°. A girl, standing on the roof of 20mhigh building, finds the angle of elevation of the same bird to be 45°. Boththe boy and girl are on the opposite sides of the bird. Find the distanceof bird from the girl.

33. As observed from the top of a light house, 100 m high above sea level,the angle of depression of a ship, sailing directly towards it, changes from30° to 60°. Determien the distance travelled by the ship during the period

of observation. Use

34.

3 1.732 .

The angle of elevation of a building from two points P and Q on the levelground on the same side of the building are 54° and 36° respectively. If thedistance of the points P and Q from the base of the building are 10m and

20m respectively, find the height of the building. Take 2 1.414

35. A pole of height 5m is fixed on the top of the tower. The angle of elevationof the top of the pole as observed from a point A on the ground is 60° andthe angle of depression of the point A from the top of the tower is 45°. Find

the height of tower. Take 3 1.732

36. Anand is watching a circus artist climbing a 20m long rope which is tightlystretched and tied from the top of vertical pole to the ground. Find theheight of the pole if the angle made by the rope with the ground level is30. What value is experienced by Anand?

A fire in a building 'B' is reported on telephone in two fire stations P anQ, 20 km apart from each other on a straight road. P observes that thefire is at an, angle of 60° to the road, and Q observes, that it is at an angleof 45° to the road. Which station should send its team and how muchdistance will this team has to travel? What value is depicted from theproblem?

A 1.2m tall girl spots a balloon on the eve of Independence Day, movingwith the wind in a horizontal live at a height of 88.2 m from the ground.The angle of elevation of the balloon from the of the girl at an instant is60°. After some time, the angle, of elevation reduces to 30°. Find thedistance travelled by the balloon. What value is depicted here?

37.

38.


1.

3.

5.

7.

9.

11.

13.

15.

17.

20.

22.

23.

24.

25.

27.

29.

30.

31.

32.

34.

36.

37.

38.

c

a

b

a

a

a

b

a

2. b

4. d

6. c

8. b

10. b

12. b

14. d

16. 30 3 1 m

30 3 m

2598 m

600 3

19. 20 m/sec.

21.

3 m

20 3 1 m

Height = 12.97 m, distance = 7.5 m

Length of flag staff 10 2 1 m, Distance of the building 10 3 m.

24 3 m

18 minutes

26. 13.6 m

28. 18 3 m, 72 m

Height = 34.64 m, Width of the river = 20 m.

1000 3 3 m

Height 5 3 3 m distance 5 3 3 m30 m

14.14 m

10m, Fun, Entertainment

33. 115.47 m

35. 6.83 m

Station P, 14.64, Reasoning ability. Thinking Skill.

58 3m , Fun & game, Entertainment


CHAPTER 5

CIRCLE

1.

2.

Tangent to a Circle : It is a line that intersects the circle at only one point.

There is only one tangent at a point of the circle.

1. In the given fig. 1 PQ is tangent then + is equal to

Q

o P

Fig. 1

(a)

(c)

2.

120°

80°

(b)

(d)

90°

100°

If PQ is a tangent to a circle of radius 5cm and PQ = 12 cm, Q is pointof contact, then OP is

(a)

(c)

13 cm

7 cm

(b)

(d)

17 cm

119 cm


3. In the given fig. 2 PQ and PR are tangents to the circle, = 70°, thenis equal to

Q

70°

o P

R

Fig. 2

(a)

(c)

4.

35°

40°

(b)

(d)

70°

50°

In the given fig. 3 PQ is a tangent to the circle, PQ = 8 cm, OQ = 6 cmthen the length of PS is

Q

o S P

Fig. 3

(a)

(c)

5.

10 cm

3 cm

(b)

(d)

2 cm

4 cm

In the given fig. 4 PQ is tangent to outer circle and PR is tangent to innercircle. If PQ = 4 cm, OQ = 3 cm and OR = 2 cm then the length of PR is


Q

o P

R

Fig. 4

(a)

(c)

6.

5 cm

4 cm

(b)

(d)

21 cm

3 cm

In the given fig. 5 P, Q and R are the points of contact. If AB = 4 cm,BP = 2 cm then the perimeter of is

A

B

Q

P c

R

o

Fig. 5

(a)

(c)

12 cm

10 cm

(b)

(d)

8 cm

9 cm


7. In the given fig. 6 the perimeter of is

A3cm

R

Q

2 cm

C

5 cmB P

Fig. 6

(a)

(c)

8.

10 cm

20 cm

(b)

(d)

15 cm

25 cm

The distance between two tangents parallel to each other to a circle is 12cm. The radius of circle is

(a)

(c)

13 cm

10 cm

(b)

(d)

6 cm

8 cm

9. In the given fig. 7 a circle touches all sides of a quadrilateral. If AB = 6 cm,BC = 5 cm and AD = 8 cm. Then the length of side CD is

D

C

A B

Fig. 7

(a)

(c)

6 cm

5 cm

(b)

(d)

8 cm

7 cm


10. In a circle of radius 17 cm, two parallel chords are drawn on oppositesides of diameter. The distance between two chords is 23 cm and lengthof one chord is 16 cm, then the length of the other chord is

(a)

(c)

34 cm

15 cm

(b)

(d)

17 cm

30 cm

11. In the given fig. 8 P is point of contact then is equal to

B

o

40°

P

A

Fig. 8

(a)

(c)

12.

50°

35°

(b)

(d)

40°

45°

In the given fig. 9 PQ and PR are tangents to the circle with centre O, if= 45° then is equal to

Q

o 45° P

RFig. 9

(a)

(c)

90°

135°

(b)

(d)

110°

145°


13. In the given fig. 10 O is centre of the circle, PA and PB are tangents tothe circle, then is equal to

A

Q o 40° P

B

Fig. 10

(a)

(c)

14.

70°

60°

(b)

(d)

80°

75°

In the given fig. 11 is circumscribed touching the circle at P, Q andR. If AP = 4 cm, BP = 6 cm, AC = 12 cm, then value of BC is

A

4

P R

6

B Q C

Fig. 11

(a)

(c)

15.

6 cm

10 cm

(b)

(d)

14 cm

18 cm

In the given fig. 12 is subscribing a circle and P is mid point of sideBC. If AR = 4 cm, AC = 9 cm, then value of BC is equal to

(a)

(c)

10 cm

8 cm

105

(b)

(d)

11 cm

9 cm

[Class-X – Maths]

A

R Q

B P C

Fig. 12

16. If d1, d2 (d2 > d1) be the diameters of two concentric circles and c be thelength of chord of a circle which is tangent to the other circle. Prove thatd22 = c2 + d12.

The length of tangent to a circle of radius 2.5 cm from an external pointP is 6 cm. Find the distance of P from the nearest point of the circle.

TP and TQ are the tangents from the external point T of a circle withcentre O. If = 30°, then find the measure of

In the given fig. 13 AP = 4 cm, BQ = 6 cm and AC = 9 cm. Find the semiperimeter of

A

m4c

17.

18.

19.

9cm

R P

C Q 6 cmB

Fig. 13


20. An incircle is drawn touching the sides of a right angled triangle whosesides are a, b, c where c is the hypotenuse. Prove that the radius r of the ab –ccircle is r 2

Prove that the tangent at any point of a circle is perpendicular to the radiusthrough the point of contact.

Prove that in two concentric circles the chord of the larger circle whichtouches the smaller circle is bisected at the point of contact.

In the given Fig. (14) AC is diameter of the circle with centre O and A ispoint of contact, then find x.

C

x

oB

40°

P A Q

21.

22.

23.

Fig. 14

40. In the given fig. (15) PA and PB are tangents from point P. Prove thatKN = AK + BN.

A

K

o C P

N

BFig. 15

25. In the given fig. (16) PQ is chord of length 6 cm of the circle of radius 6cm. TP and TQ are tangents. Find


P

6 cmo T

QFig. 16

26. A circle is inscribed in a having sides AB = 12 cm, BC = 8 cm andAC = 10 cm find AD, BE, CF.

C

F E

AD

B

Fig. 17

27.

28.

On the side AB as diameter of a right angled triangle ABC a circle isdrawn intersecting the hypotenuse AC in P. Prove that PB = PC.

Two tangents PA and PB are drawn to a circle with centre O from anexternal point P. Prove that = 2

A

O

PB

Fig. 18


29.

30.

If an isosceles triangle ABC in which AB = AC = 6 cm is inscribed in acircle of radius 9 cm, find the area of the triangle.

In the given fig. (19) AB = AC, D is the mid point of AC, BD is the diameterof the circle, then prove that AE = 1/4 AC.

EA

B D

C

Fig. 19

31. In the given fig. 20 OP is equal to the diameter of the circle with centreO. Prove that is an equilateral triangle.

A

o P

BFig. 20

32. In the given fig. (21) AB = 13 cm, BC = 7 cm. AD = 15 cm. Find PC.A

RB

So 4 cm

Q

C P D


33. In the given fig. 22 from an external point P, a tangent PT and a linesegment PAB is drawn to a circle with centre O. ON is perpendicular onthe chord AB.

Prove that : (i)

(ii)

(iii)

PA. PB = PN2 – AN2

PN2 – AN2 = OP2 – OT2

PA. PB = PT2

T

O

A

A

B N

Fig. 22

34. AB is a diameter and AC is a chord of a circle with centre ) such that= 30°. The tangent at c intersect extended AB at a point D. Provethat BC = BD

In the given fig. (23) PA and PB are tangents to the circle with centre O.Prove that OP is perpendicular bisector of AB.

A

35.

o P

B

Fig. 23


36. In the given fig. (24) find the radius of the circle.

A

23cm

5 cmB

o rSQ

CP D

Fig. 24

37. In the given fig. (25) if radius of circle r = 3 cm. Find the perimeter of

A

C 3 5 c mo

3 5 c m

B

Fig. 25

38.

39.

A circle touches the side BC of a at P and AB and AC producedat Q and R respectively. Prove that AQ is half the perimeter of

In the given fig. (26) XP and XQ are tangents from X to the circle withcentre O. R is a point on the circle. Prove that

XA + AR = XB + BR.


29 cmR

P

A

OR X

B

Q

Fig. 26

40. In the given fig. (27) PQ is tangent and PB is diameter. Find the value ofx and y.

P

A

o

35°

y

x yQ

B

Fig. 27

41. Distance between villages A and B is 7 km, B and C is 5 km and C andA is 8 km. Village Pradhan wants to dig a well for three villages A, B andC such that which is equidistant from all the villages.

i.

ii.

What should be the position of a well?

Which moral value is shown in this question of village Pradhan?

42. People of a circular villages wants to construct a road near-their village.Road cannot go inside the village so they wants that the road should beat minimum distance from the village.

i.

ii.

Which road will be at minimum distance from the centre of thevillage?

Which moral value of people of village reflected here?


43. In the given figure 28, four roads touch to a circular village of radius 1700m (Kanpur). Savita got a contract for constructing road AB and CD whileVijay got a construct for constructing road AD and BC.

Prove that :

i.

ii.

AB + CD = AD + BC

Which moral value was shown in this questions.

A B

KHANPUR

D C

44. Two roads starting from point P touch a circular path at A and B as shownin the Figure. Sarita walks 10 km from P to A and Ramesh goes from Pto B at the same time.

i.

ii.

If Sarita wins in this walking race them find the distance coveredby Ramesh.

What value is described here.

A

10 km

P

B

Fig. 29


45. When Rahim was returning to home he saw a circular pit and informed tomunicipal corporation for this circular pit immediately. Municipal corporationemployee fencing this it as shown in figure 30.

i.

ii.

iii.

Find the total length of fencing.

What mathematical concept was used to solve this problem?

Which value of Rahim was show here?

5 ftD C

3 ft 4 ft

A6 ft

B

Fig. 30

1.

3.

5.

7.

9.

11.

13.

15.

18.

23.

26.

b

c

b

c

d

a

a

a

60°

40°

2. a

4. d

6. a

8. b

10. d

12. c

14. b

17. 4 cm

19. 15 cm

25. 120°

AD = 7 cm, BE = 5 cm, CF = 3 cm


29.

36.

40.

41.

42.

43.

44.

45.

82 cm2

11 cm.

x = 35°, y = 55°

32. 5 cm

37. 32 cm

(i) A, B, C, will be on circumference of the circle and well lie on the centreof the circle. (ii) Social, honesty, equality, love towards humanity.

(i) tangent on circle (ii) Economic value

(ii) Gender equality

(i) 10 km (ii) Gender equality, healthy competition

(i) 36 feet (ii) tangent are equal from the external point

(iii) Moral and social responsibility, logical reasoning.


CHAPTER 6

CONSTRUCTIONS

1.

2.

Construction should be neat and clean and as per scale given in question.

Steps of construction should be provided only to those questions where itis mentioned.

1. A pair of tangents cannot be constructed to a circle at an angle of

(a)

(b)

185°

120°

(b)

(d)

90°

60°

2. To divide a line segment AB in the Ratio 2 : 3, first a ray AX is drawn sothat is an acute angle and then at equal distances. Points aremarked on the ray AX such that the minimum number of these points is

(a)

(c)

3

5

(b)

(d)

6

2

3. To divide a line segment AB in the ratio 3:8, a ray AX is drawn first suchthat is an acute angle and then points A1, A2, A3,... are located atequal distances on the ray AX and the points B is joined to

(a)

(c)

A12

A3

(b)

(d)

A11

A8

4. By geometrical constructions, which of the following is possible to dividea line segment in the given ratio?


(a)

(c)

5.

6 :2 (b)

(d)

2:1

2

2 3 : 2 – 3

m > n

m < n

3 2) : 3 –2 In the construction of triangle similar and larger to a given triangle as pergiven scale factor m : n. Then, it is possible only when

(a)

(c)

(b)

(d)

m = n

independent of Scale Factor

1.

3.

5.

a

b

a

2. c

4. b

1.

2.

Draw a line segment AB = 7 cm. Take a point P on AB such thatAP : PB = 3 : 4.

Draw a line segment PQ = 10 cm. Take a point A on PQ such thatPA2 . Measure the length of PA and AQ.PQ5

Construct a in which BC = 6.5 cm, AB = 4.5 cm and = 60°.Construct another triangle similar to such that each side of new 4triangle isof the corresponding sides of . 5

Draw a triangle XYZ such that XY = 5 cm, YZ = 7 cm and = 75°. 3Now construct a ~ with its sidestimes of the corresponding 2sides of

Construct an isoscales triangle whose base is 8 cm and altitude 5 cm and 3then construct another triangle whose sides aretimes the corresponding 4sides of the given triangle.

3.

4.

5.


6. Draw an isosceles with AB = AC and base BC = 7 cm and vertical 1angle is 120°. Construct ~ with its sides 1 times of the 3corresponding sides of

Draw in which = 90°, PQ = 6 cm, QR = 8 cm. Construct ~ with its sides equal to 2/3rd of corresponding sides of

Construct a right angled triangle in which base is 2 times of theperpendicular. Now construct a triangle similar to it with base 1.5 timesof the original triangle.

Draw an equilateral triangle PQR with side 5cm. Now construct PQ1 .such that PQ´2

Draw a circle of radius 4 cm with centre O. Take a point P outside thecircle such that OP = 6cm. Draw tangents PA and PB to the circle. Measurethe lengths of PA and PB.

Draw a line segment AB = 8 cm. Taking AB as diameter a circle is drawnwith centre O. Now draw OPAB. Through P draw a tangent to the circle.

Draw a circle of radius OP = 3 cm. Draw = 45° such that OQ = 5cm. Now draw two tangents from Q to given circle.

Draw a circle with centre O and radius 3.5 cm. Draw two tangents PA andPB from an external point P such that = 45°. What is the value of+

Draw a circle of radius 4 cm. Now draw a set of tangents from an externalpoint P such that the angle between the two tangents is half of the centralangle made by joining the points of contact to the centre.

Draw a line segment AB = 9 cm. Taking A and B as centres draw twocircles of radius 5 cm and 3 cm respectively. Now draw tangents to eachcircle from the centre of the other.

Draw a circle of radius 3.5 cm with centre O. Take point P such thatOP = 6 cm. OP cuts the circle at T. Draw two tangents PQ and PR. JoinQ to R. Through T draw AB parallel to QR such that A and B are point onPQ and PR.

7.

8.

9.

10.

11.

12.

13.

14.

15.

16.


17.

18.

Draw a circle of diameter 7 cm. Draw a pair of tangents to the circle, whichare inclined to each other at an angle of 60°.

Draw a circle with centre O and radius 3.5 cm. Take a horizontal diameter.Extend it to both sides to point P and Q such that OP = OQ = 7 cm. Drawtangents PA and QB one above the diameter and the other below thediameter.

19. A farmer wants to divide a 7 feet long sugarcane between his daughterand son equally. Using geometrical construction taking sugarcane as a linesegment (7 an long) divide it into two parts.

i.

ii.

Find the length of each part.

Which moral value was shown in the question?

20. Traffic police organize a slogan competition on a triangular sheet for awarethe people about traffic rules. It was decided that the winning slogan willbe displayed on road. Using a triangular sheet of in whichBC = 7cm, = 45° and = 105°. Ramesh win the competition.

i. Construct a similar triangle whose corresponding sides are

times the sides of triangular sheet used by Ramesh.

ii. What values traffic police wants to inculcate among people?

43

19.

20.

(i) 3.5 feet (ii) Gender equality

(ii) Obey rules, Social responsibility, Moral value, defence.


CHAPTER 7

MENSURATIONAREA OF CIRCLE, SURFACE AREAS AND VOLUMES

1.

2.

3.

Area of circle = 2 where ‘r’ is the radius of the circle.

r 2 .Area of Semi circle 2

Area enclosed by two concentric circles

= (R2 – r2)

= (R + r) (R – r ); R > r

where ‘R’ and ‘r’ are radii of two concentric circles.

R

r

4. The arc length ‘l’ of a sector of angle ‘’ in a circle of radius ‘r’ is given by

circumference of the circle 3601 =× 2r 360° 180 l r 180

l

r

r

5. If the arc subtends an angle then area of r 2 .the corresponding sector is 360 r

l


7. Angle described by minute hand in 60 minutes = 360°. Angle described by

360minute hand in 1 minute 6 . 60

8.

9.

10.

11.

Curved surface area of cylinder of radius r and height h = 2rh squareunits.

Total surface area of cylinder of radius r and height h = 2r (r + h) squareunits.

Volume of cylinder of radius r and height h = 2h cubic units.

Curved surface area of cone of radius r, height h and slant height l =

square units where l

12.

13.

14.

15.

16.

17.

r 2 h2 .

Total surface area of cone = (l + r) sq. units.

Volume of cone 1 2 r h cubic units.3

Total surface area of sphere of radius r units = 4r2 sq. units.

Curved surface area of hemisphere of radius r units = 2r2 sq. units.

Total surface area of a solid hemisphere of radius r units = 3r2 sq. units.

Volume of sphere of radius r units 4 3 r cubic units..3

2 3 .r cubic units.3

18.

19.

20.

21.

Volume of hemisphere of radius r units

Curved surface area of frustum = + R) sq. units, where l slant heightof frustum and radii of circular ends are r and R.

Total surface area of frustum = (r + R) + 2 + R2) sq. units.

Volume of Frustum

where l

1 h r 2 R2 rR cubic units..3

h2 2R r


1. If the circumference and area of a circle are numerically equal then theradius of the circle is equal to

(a)

(c)

r = 1

r = 2

(b)

(d)

r = 7

r = 0

2. The radius of a circle is 7 cm. What is the perimeter of the semi circle?

(a)

(c)

36 cm

7 cm

(b)

(d)

14 cm

14 cm

3. The radius of two circles are 13 cm and 6 cm respectively. What is theradius of the circle which has circumference equal to the sum of thecircumference of two circles?

(a)

(c)

19 cm

25 cm

(b)

(d)

19 cm

32 cm

4. The circumference of two circles are in the ratio 4 : 5 what is the ratio ofthe areas of these circles.

(a)

(c)

4:5

64 : 125

(b)

(d)

16 : 25

8 : 10

5. The area of a sector with radius 3 cm and angle of sector 40° is

(a)

(c)

2 cm2

3 cm2

(b)

(d)

cm2

4 cm2

6. An arc of a circle is of length 5 cm and the section it bounds has an areaof 10 cm2. Then the radius of circle is :

(a)

(c)

2 cm

22 cm

(b)

(d)

4 cm

8 cm

7. The area of an equilateral triangle is m2, its one side is

(a)

(c)

4 m (b)

(d)

122

33 m

2 m3 3 m 4

[Class-X – Maths]

8. If ‘C’ and ‘A’ respectively be the circumference and area of the circle ofradius ‘r’, then

(a)

(c)

C 1 Ar2

(b)

(d)

A 1 Cr2

C = 2 Ar A = 2 Cr

9. The volume of a cuboid is 440 cm3. The area of its base is 66 cm2. Whatis its height?

(a)

(c)

40 cm 3

(b)

(d)

20 cm 3

440 cm 66 cm

10. Three cubes each of side ‘a’ are joined from end to end to form a cuboid.The volume of the new cuboid is _______ cubic units

(a)

(c)

a2

a3

(b)

(d)

3a3

6a3

11. Ratio of the volumes of the cylinder and cone having same height andsame radius of the bases, is

(a)

(c)

1:1

3:1

(b)

(d)

1:2

1:3

12. The radii of the circular ends of a frustum of height 16 cm are 8 cm and20 cm. The slant height of the frustum is

(a)

(c)

28 cm

44 cm

(b)

(d)

10 cm

20 cm

13. The radius and height of a solid cylinder is r cm & h cm respectively. It isdivided into equal parts through height. The total surface area of the twoparts is—

(a)

(c)

2 rh

2r (2r + h)

(b)

(d)

2r (r + h)

2r (3r + h)


14. The ratio of the volumes of two spheres is 8: 27. It r & R are the radii ofthe two spheres respectively, then r : (R–r) is

(a)

(c)

9:4

3:1

(b)

(d)

3:2

2:1

15. During conversion of solid from one shape to another the volume of thenew solid shape will —

(a)

(c)

increase

doubled

(b)

(d)

decrease

remains the same

16.

17.

18.

19.

20.

21.

22.

What is the perimeter of a sector of angle 45° of a circle with radius 7 cm.

The perimeter of semicircular math lab protractor is 108 cm. Find thediameter of the protractor?

A bicycle wheel makes 5000 revolutions in moving 10 km. Write theperimeter of wheel.

What is the ratio of the areas of a circle and an equilateral triangle whosediameter and a side of triangle are equal.

A wire is in the form of a semicircle of radius 7 cm. It is bent into a square.Find the area of the square.

The cost of fencing a circular field at the rate of Rs. 10 per meter is Rs.440. What is the radius of the circular field?

Find the perimeter of the shaded region.4 cm

A B

6 cm

D C


23. Find the circumference of the circle with centre ‘O’.

P

o 24 cm

7 cmR Q

24.

25.

26.

27.

What is the area of the largest triangle that can be inscribed in a semicircleof radius r cm.

A piece of wire 20 cm long is bent into an arc of a circle subtending anangle of 60° at the centre then what is the radius of the circle?

The minute hand of a clock is cm long. What is the area describedby the minute hand between 8.00 a.m to 8.05 a.m.?

Find the area of shaded portion.

20 cm

20 cm 20 cm

20 cm

28. Find the perimeter of the shaded portion.

AC

14 B 14 14D

29. ABCD is a square kite of side 4 cm. What is the area of the shadedportion.


A D

4 cm4 cm

B 4 cm C

30.

31.

32.

How many cubes of side 4 cm can be cut from a cuboid measuring(16 × 12 × 8).

A cylinder, a cone and a hemisphere are of equal base and have the sameheight. What is the ratio of their volumes?

The sum of the radius of the base and the height of a solid cylinder is 15cm. If total surface area is 660 cm2. Write the radius of the base ofcylinder.

Find the height of largest right circular cone that can be cut out of a cubewhose volume is 729 cm3.

Three cubes of the same metal, whose edges are 6, 8, 10 cm are meltedto form a single cube. Find the diagonal of the single cube.

V ol e of riumght circul cylnder i 448 aris

33.

34.

35. cm3, height of cylinder is 7cm.Find the radius.

36.

37.

If lateral surface area of a cube is 64 cm2. What is its edge?

The area of a rhombus is 24 cm2 and one of its diagonal is 8 cm. Whatis other diagonal of the rhombus?

What is the length of the largest rod that can be put in a box of innerdimensions 30cm, 24 cm and 18 cm?

Curved surface area of a cylinder is 16 cm2. If radius is 4cm, then findits height.

A largest sphere is carved out of a cube of side 7 cm. Find the radius.

38.

39.

40.


41.

42.

If the semi vertical angle of a cone of height 3 cm is 60°. Find its volume.

Find the edge of cube if volume of the cube is equal to the volume ofcuboid of dimensions (8 × 4 × 2) cm.

Find the volume of cone of height 2h and radius r.

A sphere of maximum volume is cut out from a solid hemisphere of radius6 cm. What is the volume of the sphere cut out?

In the adjoining figure ABC is a right angled triangle, right angled at A.Semi circles are drawn on AB, AC as diameters. Find the area of shadedportion.

43.

44.

45.

A

3 cm4 cm

B C

46. In figure, is equilateral triangle. The radius of the circle is 4 cm. Findthe area of shaded portion.

A

4 cm

o

4

B

cm 4 cm

C


47. Four cows are tied with a rope of 7 cm at four corners of a quadrilateralfield of unequal sides. Find the total area grazed.

In the figure, find the area of the shaded portion.48.

5 cm

12 cm

49. Find the shaded area in the given figure.

28 cm

28 cm

50. Find the shaded area, in the figure.

14 cm

14 cm


51. Three cows are tied to the centre of a circular field of radius 14 m with a7 m long rope. Each one is given equal sector by putting small heightwalls. Find how much area each cow is unable to graze from the fieldgiven to her. What values of farmer is seen from his act.

The wheels of the car are of diameter 80 cm each. How many completerevolutions does each wheel make in 10 minutes when the car is travellingat a speed of 66 km/hr?

Find the perimeter of the figure in which a semicircle is drawn on BC asdiameter, = 90°.

A

5 cm

B

12 cm

C

52.

53.

54. Find the area of shaded region in the figure.

14 cm

9 cm 9 cm

14 cm

55.

56.

The volume and surface area of a sphere are numerically equal. Find theradius of the sphere.

Water flows in a tank 150 m long and 100 m broad, through a rectagularpipe whose cross-section is 2 dm × 1.5 dm and the speed of water is 15km/hr. In what time will the water be 3 m deep?

A cylindrical vessel of 36 cm height and 18 cm radius of the base is filledwith sand. The sand is emptied on the ground and a conical heap of sandis formed. The height of conical heap is 27 cm. Find the radius of base ofsand.

57.


58. The radii of circular ends of bucket are 5.5 cm and 15.5 cm and its heightis 24 cm. Find the total surface area of bucket.

Water flows out through a circular pipe whose internal diameter is 2 cmat the rate of 6m/sec. into a cylindrical tank. If radius of base of the tankis 60 cm. How much will the level of the water rise in half an hour?

A toy is in the form of a cone mounted on a cone frustum. If the radiusof the top and bottom are 14 cm and 7 cm. and the height of cone andtoy are 5.5 cm and 10.5 cm respectively. Find the volume of toy (adj. fig.)

59.

60.

5.5 cm

14 cm

7 cm

61. A solid consists of a right circular cylinder with a right circular cone at thetop. The height of cone is ‘h’ cm. The total volume of the solid is 3 timesthe volume of the cone. Find the height of the cylinder.

From a solid cylinder of height 12 cm & diameter 10 cm, a conical cavityof same height & same diameter is hollowed out. Find the total surfacearea of the remaining solid?

There is a triangular field in front of a classroom. The two sides which areperpendicular measures 6 m and 8 m. The teacher told students to constructa circular flower bed which touches the three sides of the field. Find theradius of circular flower bed. What values are developed in the studentsthrough this activity.

Naresh built a bath tub for birds in his garden which is hollow cylinder witha hemispherical cavity on the upper edge. The height of the cylinder is

130

62.

63.

64.

[Class-X – Maths]

10.5 cm

1.45 m & the diameter is 60 cm. Find the maximum capacity of wateravailable in the tub in litres. What values of Naresh do you observe. (Tot = 3.14.al

65. Dinesh installed underground cylindrical tank for rain water which is 10 mdeep and 3 m in radius. Rain water reaches this tanks from roof througha cylindrical pipe. The length & breadth of the roof are 10 m & 3.14 mrespectively. If on a particular day the rain falls up to a height of 10 cm onthe roof. What is the height of water in the cylindrical tank. (Assuming thereis no water left in the pipe). What values of Dinesh do you observe in thequestion? (Take = 3.14)

1.

3.

5.

7.

9.

11.

13.

15.

17.

19.

21.

23.

25.

27.

29.

31.

c

b

b

d

b

c

c

d

42 cm

: 3

2. c

4. b

6. b

8. b

10. b

12. d

14. d

16. 19.5 cm

18. 2m

20. 81 cm2

22. (16 + cm

24. r2

26. cm2

28. 462 cm2

30. 24

32. 7 cm

14m

25

60 / cm

86 cm2

(16–4) cm2

3: 1: 2


33.

35.

37.

39.

41.

43.

45.

47.

49.

51.

53.

55.

57.

59.

61.

63.

64.

65.

9 cm

8 cm

6 cm

2 cm

27 cm3

2 r 2h unit33

34. 12 3

36. 4 cm

38. 30 2cm

40. 3.5 cm

42. 4 cm

44. 113.14 cm3

46. (16 – 12

48.1019 cm 2 14

23 ) cm6 cm2

154 cm2

154 cm2

154 m2

373 cm7

50. 21 cm2

52. 4375

54. 49 cm2

56. 100 hour

58. 1811.07 cm2

60. 2926 cm3

62. 660 cm2

3 units

36 cm

3 m

2 h3

(a) 2m (b) working in groups, cooperation, coordination, love for nature

(a) 56. 52 L,

(a) 9 times,

(b) care for animals

(b) water conversation


CHAPTER 8

PROBABILITY

1. The Theoretical probability of an event E written as (E) is

P E

2.

3.

4.

5.

Number of outcomes favourable to ENumber of all possible outcomes of the experiment.

The sum of the probability of all the elementary events of an experimentis 1.

The probability of a sure event is 1 and probability of an impossible eventis 0.

If E is an event, in general, it is true that P(E) + P (E ) = 1.

From the definition of the probability, the numerator is always less than orequal to the denominator therefore 0 P(E) 1.

1. If E is an event then P(E) + P E = ........ ?

(a)

(c)

0

2

(b)

(d)

1

–1

2. The probability of an event that is certain to happen is :

(a)

(c)

0

1

(b)

(d)

2

–1

3. Which of the following can not be the probability of an event :

(a)

(c)

23 (b)

(d)

–32

15% 0.7


4. If P(E) is .65 what is P (Not E)?

(a)

(c)

.35

1

(b)

(d)

.25

0

5. If P(E) is 38% of an event what is the probability of failure of this event?

(a)

(c)

12%

1

(b)

(d)

62%

0

6. A bag contains 9 Red and 7 blue marbles. A marble is taken out randomly,what is the P (red marble)?

(a)

(c)

716

1816

(b)

(d)

916

1416

7. In a Survey it is found that every fifth person possess a vehicle what is theprobability of a person not possessing the vehicle?

(a)15

35

(b)45

(c)

8.

(d) 1

Anand and Sumit are friends what is the probability that they both havebirthday on 11th Nov. (ignoring leap year).

(a) 112

1365

(b)17

1366(c)

9.

(d)

The number of face cards in a well shuffled pack of cards are :

(a)

(c)

12

4

(b)

(d)

16

52


10. A die is thrown once. What is the probability of getting an even primenumber?

(a)

(c)

36

12

(b)

(d)

16

13

11. The probability of an impossible event is :

(a)

(c)

0

–1

(b)

(d)

1

12. From the letters of the word “Mobile”, a letter is selected. The probabilitythat the letter is a vowel, is

(a)

(c)

13

16

(b)

(d)

37

12

13. The probability of selecting a face card from a deck of 52 cards is:-

(a)

(c)

226

413

(b)

(d)

313

813

14. When a die is thrown, the probability of getting a number less than 7 is

(a)

(c)

16 (b)

(d)

13

1 0

15. If E is an event such that x < P(E) < 1, then the value of x is

(a)

(c)

0

0.5

(b)

(d)

1

2


16.

17.

18.

A bank A.T.M. has notes of denomination 100, 500 and 1000 in equalnumbers. What is the probability of getting a note of Rs. 1000.

What is the probability of getting a number greater than 6 in a single throwof a die.

A selection committee interviewed 50 people for the post of sales manager.Out of which 35 are males and 15 are females. What is the probability ofa female candidate being selected.

A bag contains cards numbering from 5 to 25. One card is drawn from thebag. Find the probability that the card has numbers from 10 to 15.

In 1000 lottery tickets there are 5 prize winning tickets. Find the probabilityof winning a prize if a person buys one ticket.

It is known that in a box of 600 screws, 42 screws are defective. Onescrew is taken out at random from this box. Find the probability that it isnot defective.

Two dice are rolled once. What is the probability of getting a doublet?

Two dice are rolled simultaneously. Find the probability that the sum ismore than and equal to 10.

From the well shuffled pack of 52 cards. Two Black kings and Two RedAces are removed. What is the probability of getting a face card.

In a leap year what is the probability of 53 Sundays.

A box contains cards numbered from 2 to 101. One card is drawn atrandom. What is the probability of getting a number which is a perfectsquare.

Tickets numbered from 1 to 20 are mixed up together and then a ticket isdrawn at random. What is the probability that the ticket has a numberwhich is a multiple of 3 or 7?

A bag contains 5 red balls and ‘n’ green balls. If the P(green ball) = 3 ×P (red ball) then what is the value of n.

From the data (1, 4, 9, 16, 25, 29). If 29 is removed what is the probabilityof getting a prime number.

19.

20.

21.

22.

23.

24.

25.

26.

27.

28.

29.


30. A card is drawn from an ordinary pack of playing cards and a person betsthat it is a spade or an ace. What are the odds against his winning the bet.

31. An arrow pointer is spined which is placed on a fixed circular number platenumbered from 1 to 12 at equal distance. The pointer is equally likely torest at any number. What is the probability that it will rest at

(a)

(c)

number 10

a number multiple of 3

(b)

(d)

an odd number

an even number

32. In a family having three children, there may be no girl, one girl, two girlsor three girls. Find the probability of :

(a) exactly one girl (b) less than 2 girls

33. A coin is tossed thrice then find the probability of

(i) 2 heads (ii) 2 tails (iii) 3 heads.

34. The king, queen and jack of clubs are removed from a deck of 52 playingcards and the remaining cards are shuffled. A card is drawn from theremaining cards. Find the probability of getting a card of (i) heart;(ii) queen; (iii) Clubs.

A box contains 5 Red balls, 8 white balls and 4 Green balls. One ball istaken out of the box at random. What is the probability that ball is (i) red;(ii) white; (iii) Not green.

12 defective pens are mixed with 120 good ones. One pen is taken out atrandom from this lot. Determine the probability that the pen taken out isnot defective.

A number x is selected from the numbers 1, 2, 3 and then a secondnumber y is randomly selected from the numbers 1, 4, 9. What is theprobability that the product of two numbers will be less than 9?

A box contains 90 discs which are numbered from 1 to 90. If one disc isdrawn at random from the box, find the probability that it bears (i) a twodigit number (ii) a perfect square number (ii) a number divisible by 5.

A game consists of tossing a one rupee coin 3 times and noting its outcomeeach time. Anand wins if all the tosses give the same result i.e., three


35.

36.

37.

38.

39.

heads or three tails and loses otherwise. Calculate the probability thatAnand will lose the game.

40.

41.

A die is thrown twice. What is the probability of getting : (i) The Sum of 7;(ii) The sum of greater than 10; (iii) 5 will not come up either time.

A card is drawn at randown from a well shuffled deck of playing cards.Find the probability that the card drawn is

(i)

(iii)

42.

a card of spade or an ace

either a king or a queen

(ii) a red king

(iv) neither a king nor a queen

A jar contains 24 balls, some are green and other are blue. If a ball is 2drawn at random from the jar, the probability that it is green is . Find 3the number of blue balls in the jar.

In a lottery, there are 10 prizes and 25 blanks. Find the probability ofgetting a prize? Also verify that P(E) + P E = 1

The probability of getting a bad egg in a lot of 400 is 0.035. Calculate thenumber of bad eggs in the lot. Also calculate the probability of getting agood egg from the lot.

A bag contains 12 balls out of which x are white and rest are red balls.

i.

ii.

If one ball is drawn at random, what is the probability that it will bea Red ball?

If 4 more white balls and 2 more Red balls are put in the bag thenwhat is the probability of drawing a white ball?

43.

44.

45.

46. In a class discussion, Himanshu says that probability of an event cannotbe 1.3. He shows:-

(a)

(c)

Truth value

Leadership

(b)

(d)

Economical Value

Environment Value

47. P(E) + P E = 1

The Statement depicts ____________ value.


48.

49.

E be an event associated with a random experiment and 0 < P(E) < x. Findthe maximum value of x, which value is depicted by this statement?

Ramesh got Rs. 24000 as Bonus. He donated Rs. 5000 to temple. He gaveRs. 12000 to his wife, Rs. 2000 to his servant and gave rest of the amountto his daughter. Calculate the probability of

i.

ii.

iii.

wife's share

Servant's Share

daughter's share.

Which value's are depicted by Ramesh from the way the amount isdistributed?

50. 240 students reside in a hostel. Out of which 50% go for the yoga classesearly in the morning, 25% go for the Gym club and 15% of them go forthe morning walk. Rest of the students have joined the laughing club. Whatis the probability of students who have joined laughing club? Which valueis depicted by the students?

1.

3.

5.

7.

9.

11.

13.

15.

17.

19.

b

b

b

b

a

a

b

a

0

27

2. c

4. a

6. b

8. c

10. b

12. d

14. c

16.

18.

20.

13

310

1200


21.

23.

25.

27.

29.

93100

16

27

25

22.

24.

26.

16

524

9100

28. 15

30.

112

38

517

12

38

817

13

12

913

14

1349

0

31. (i) (ii) (iii) (iv)

18

32. (a) (b)12

33. (i) (ii) (iii) 34. (i)

1317

1011

(ii) 349 (iii)

1049

35. (i)

59

34

(i)

(ii) (iii) 36.

37. 38. (i) 910

16

(ii) 110

112

(iii)15

253639.

41.

42.

44.

46.

48.

49.

50.

40. (i)

413 (ii)

126 (iii)

213 (iv)

1113

(ii) (iii)

8.

14, 0.965.

(a) Truth Value

x=1, Righteous value.

(i)12 (ii)

112 (iii)

43. 2/7

45. (i)12 x 12 (ii)

x 4 18

47. Righteous Value

5 . Social Value, Religious Value24

1 Physical Fitness10


SAMPLE QUESTION PAPER - I



1.

2.


Maximum Marks : 90



There is no overall choice.


3.

4.

5.

1. If one root of 2x2 + kx + 1 = 0 is

(a)

(c)

3

5

1 , then the value of K is2

(b)

(d)

–3

–5

2. In figure, PQ and PR are tangents to the circle with O, such that= 50°, is equal to :

Q

P O

R


(a)

(c)

3.

25°

40°

(b)

(d)

30°

50°

From the letter of the word ‘‘Education’’ a letter is selected, the probabilitythat the letter is a vowel is :

(a)

(c)

79

49

(b)

(d)

29

59

4. The mid-point of segment AB is the point (0, 4). If the co-ordinates of Bare (–2, 3), then the co-ordinates of A are :

(a)

(c)

(2, 5)

(2, 9)

(b)

(d)

(–2, –5)

(–2, 11)

5.

6.

7.

Find the value of m so that the quadratic equation x2 – 2x (1 + 3m) +7(3 + 2m) = 0 has equal roots.

How many two digit numbers are there in between 6 and 102 which aredivisible by 6 ?

A circle is inscribed in a having sides PQ = 10 cm, QR = 8 cm andPR = 12 cm. Find PS and QT.

R

U

P

T

S

Q

8. Find the area of the shaded region in the given figure.


A

7cm

7cm

B

7cm

7cm

C

9. Radius of a wheel of motor cycle is 35 cm. How many turn it should takein one minute if its speed is 66 km/hour.

10. 22 If the perimeter of a protractor is 72 cm, calculate its area. Take 7

11.

12.

13.

S olve the equaton 2 x2i – 5x + 3 = 0 by the method of completing thesquare.

Find the sum of first 15 terms of an A.P. whose nth term is 9 – 5n.

Draw a triangle ABC with side BC = 7 cm, = 45°, = 105°, Then, 4construct a triangle whose sides aretimes the corresponding sides of 3

Find a point on y-axis which is equidistant from the points (6, 5) and (–4, 3).

In what ratio does the point (½, 6) divides the line segment joining thepoints (3, 5) and (–7, 9).

Find the area of the shaded region in the given figure, if PQ = 24 cm,PR = 7 cm and O is the centre of the circle.

Q

O

14.

15.

16.

R P

17. Water in a canal, 6 m wide and 1.5 m deep, is flowing with a speed of10km/h. How much area in hectares will it irrigate in 30 minute, if 8 cm ofstanding water is needed? (1 hectare = 10000 m2)


18. The total surface area of a solid right circular cylinder is 231 cm2, its 2curved surface is rd of the total surface area. Find the radius of the base 3and height.

Entry fee in a game is 5. A coin is tossed three times in this game. IfShweta got one or two tails then entry fee is refunded. If she gets threetails, she receive twice of entry fee amount. Otherwise, she loses entryfee. Find the probability of following :

(i)

(ii)

(iii)

She loses entry fee.

She receied twice the amount of entry fee.

She received entry fee.

19.

20. Draw a circle of radius 3 cm. from a point 5 cm away from the centre drawa pair of tangents on the circle. Also measure the length of tangents.

21. Usha surveyed a class on World Food Day and observed that the numberof students who like Junk food is 4 more than the number of students wholike home made food. The sum of the squares of the number of two typesof students is 400. Find the number of students who like Junk food andwho like home made food?

If the sum of three consecutive terms of an increasing A.P. is 51 and theproduct of first and third terms is 273, find the third term.

An old lady Krishna Devi deposited Rs. 12000 in a bank at 8% simpleinterest p.a. She uses the annual interest to give five scholarships to thestudents of a school for their overall performances each year. The amountof each scholarship is 300 less than the preceding scholarship. Answerthe following

(i)

(ii)

Find amount of each scholarship?

What values of the lady are reflected?

22.

23.

24. Prove that the length of tangents drawn from an external point to a circleare equal.


25. Prove that the intercept of a tangent between a pair of parallel tangentsto a circle subtend a right angle at the centre of the circle. (Figure is given).

A P

O

B Q

26. There is a small island in the middle of a 100 meter wide river and a talltree stands on the island. P and Q are points directly opposite to eachother on two banks and in line with the tree. If the angles of elevation ofthe top of the tree from P and Q are respectively 30° and 45°, find theheight of the tree.

A bag contains some cards which are numbered between 31 and 96 areplaced. If one card is drawn from the bag, find the probability that thenumber on card is :

(i)

(iii)

(iv)

a multiple of 3

a number not more than 40.

an odd card no.

(ii) a perfect square

27.

28. The slant height of a frustum of a cone is 4 cm and the perimeter of itscircular ends are 18 cm and 6 cm. Find the curved surface area of thefrustum.

Find the value of K, if the points A (2, 3), B (4, K) and C (6, –3) arecollinear.

A cone of radius 10 cm is divided into two parts by drawing a planethrough the mid-point of its axis, parallel to its base, compare the volumeof the two parts.

As observed from the top of a 75 cm high light house from the sea level,the angles of depressions of two ships are 30° and 45°. If one ship isexactly behind the other on the same side of the light house. Find thedistance between two ships.

29.

30.

31.


1.

3.

5.

7.

9.

11.

a

d

–10 ,2 6

2. a

4. a

6. 15

8. 24.5 cm2

10. 308 cm2

12. –465

PS = 4cm, QT = cm

500

3 ,12

14.

16.

18.

20.

22.

23.

27 0, 2 15. 1 : 3

17. 56.25 hectare161.53 cm2

3.5 cm, 7 cm

4 cm

21

i 1 ii 1 iii 319. 8 8

21. 12, 16

4

(i) Rs. 2520, Rs. 2220, Rs. 1920, Rs. 1620, Rs. 1320

(ii) Love for children kindness

30.

28.

30.

36.6 m

48 cm2

1:7

27. (i) 5191 , (ii), (iii), (iv)1216642

29. K = 0

31. 75 3 1m


SAMPLE QUESTION PAPER - II



1.

2.


Maximum Marks : 90



There is no overall choice.


3.

4.

5.

1. The angle of elevation of the top of a tower of height 10 3 m from a pointon the ground, which is 30 m away from the foot of the tower is:

(a)

(c)

45°

30°

(b)

(d)

60°

90°

2. The probability of getting 53 Fridays in a leap year is:

(a)

(c)

17

47

(b)

(d)

27

57

3. If radius of a circle is doubled, its area becomes:

(a)

(c)

2 times

8 times

147

(b)

(d)

4 times

16 times

[Class-X – Maths]

4. Ratio of the radii of two spheres is 4:5. Find the ratio of the total surfaceareas of the spheres.

(a)

(c)

16:25

4:5

(b)

(d)

25:16

5:4

5.

6.

If K + 1, 3k and 4k + 2 be any three consecutive terms of an A.P., find thevalue of K.

Prove that the tangents drawn at the ends of a diameter of a circle areparallel.

22 If the perimeter of a protractor is 72m, calculate its area. 7

7.

8.

9.

Find the coordinates of point A if AB is the diameter of the circle havingcentre (2, –3) and B (1, 4)

Points (5, –2), (6, 4) and (7, –2) represents the vertices of which type oftriangle.

22 If the perimeter of a protractor is 72 cm then calculate its area. 7

10.

11. A die is thrown once. Find the probability of getting

(i)

(iii)

an even prime number

a number greater than 4

(ii) a multiple of 3

12.

13.

14.

The hypotenuse of a right angled triangle is 13 m long. If the base of thetriangle is 7 m more than the other side, find the sides of the triangle.

Find five numbers in A.P. whose sum is 12½ and the ratio of first to thelast term is 2:3.

Draw a triangle ABC in which CA = 6 cm, AB = 5 cm and = 45° 3Then, construct a triangle similar to the triangle ABC, whose sides are 5

148[Class-X – Maths]

of the corresponding sides of the triangle ABC.

15. From a point P on the ground, the angle of elevation of the top of a 10mhigh building is 30°. A flag staff is hoisted at the top of the building andthe angle of elevation of the top of flag staff from P is 45°. Find the lengthof the flagstaff.

The line segment joining the points A (3, 2) and B (5, 1) is divided at thepoint P in the ratio 1:2 and P lies on the line 3x – 18y + k = 0. Find thevalue of k.

16.

cm14 D

17. m7c

Find the area of the shaded region in the given figure:-

18. A circus tent is cylindrical to a height of 3 m and conical above it. If itsradius is 105 m and slant height of the cone is 53 m, calculate the totalarea of the canvas required.

Solve for x : 12abx 2 9a 2 8b 2 x 6ab 0

If third and nineth terms of an AP are 4 and –8 respectively then whichterm of this AP is zero.

19.

20.

21. On the independence Day, in a group of children, each child gives one giftto the other. If total numbers of gifts are 90, then find the number ofchildren in the group. What do we learn from this type of group activities?

A motor boat, whose speed is 15 km/h in still water, goes 30 km downstreamand comes back in a total of 4 hours 30 minutes. Determine the speed ofstream.

124 ; x –1, –2, –4x 1 x 2x 4

22.

23.


24.

25.

26.

Prove that the lengths of tangents drawn from an external points to a circleare equal.

Two tangents TP and TQ are drawn to a circle with centre O from anexternal point To. Prove that = 2OPQ.

From the top of a building 60 m high, the angles of depression of the topand bottom of a lamp post are observed to be 30° and 60° respectively.Find the height of the lamp post.

The king, queen and jack of clubs are removed from a pack of 52 playingcards and then the remaining pack is well shuffled. One card is selectedfrom the remaining cards. Find the probability of getting.

(i)

(ii)

a heart

a club

(ii)

(iv)

a king

a jack

27.

28. Find the area of the triangle formed by joining the mid-points of the sidesof a triangle ABC whose vertices are A (0, –1), B (2, 1) and C (0, 3). Findthe ratio of this area to the area of the given triangle ABC.

A decorative block is made of two solids - a cube and a hemisphere. Thebase of the block is the cube with edge of 7 cm and the hemisphereattached on the top has a diameter of 4.9 cm. If the block is to be painted,find the total area to be painted.

A Turkish cap is shaped like a frustum of cone. If its radius on the openside is 10 cm, radius at the upper base is 4 cm and its slant height is 15cm, find the area of the material used for making it.

A hemispherical tank full of water is emptied by a pipe at the rate of 347

29.

30.

31.

litres per second. How much time will it take to empty half of the tank, if

22 use it is 3 m in diameter? 7


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