slc _ opt math _ all

Downloaded From:- www.bhawesh.com.np


SLC Model Question SLC _ OPT Math _ Polynomials

[Optional Mathematics]

A. Short Answer Questions

• Evaluate f(x).g(x) for the following.

• f(x) = x+4 g(x) = x+6

• f(x) = x2-2x+1 g(x) = x-1

• f(x) = 3x2+6x+4 g(x) = 4x2+x-2

• Divide f(x) by g(x) for the following:

• f(x) = x3+6x2+10x+16 and g(x) = x+3

• f(x) = x4+1 and g(x) = x-1

• Define remainder theorem and find the remainder when 2x3 – 7x2 +5x + 4 is divided by x – 3

• Define factor theorem. Is x – 2 a factor of f(x) = x3 – 6x2 +11x – 6?

• Use synthetic division method to divide f(x) by g(x) in the following:

• f(x) = x4+1 & g(x) = x-1

• f(x) = 2x3-10x2+19x-10 & g(x) = x -

• f(x) = (6x2 - 7x + 12) & g(x) = (x - 2)

• f(x) = (3x4 - 8x2 - 20) & g(x) = (x2 - 2)

• f(x) = (2x3 - 9x2 + 4x - 6 ) & g(x) = ( 2x - 3)

• f(x) = (2x3 + 3x2 - 4x - 5) & g(x) =

• f(x) = (2x4 - 7x3 - 4x - 5) & g(x) = (x - 3)

• f(x) = (4x4 - 3x3 + 2x - 6) & g(x) = ( x - 2)

• f(x) = (6x4 - 5x3 + 7x2 - 3x + 2) & g(x) = x2 – 2

• Use remainder theorem to find remainder when f(x) = x3-6x2+11x-6 is divided by

•

• x-1

• x-4

• x+2

• 3x – 5

• 5x + 7

• 3x + 11

• If x -1 is a factor of a2 x2-3ax+3a-1, find the value of a

• Find the value of k if the polynomial 3x3+ kx2+x-2 leave a remainder 7.

• x – 1 is the factor of g(x) = x2+bx+2. Is x – 2 the factor of g(x)?

• For what values of a will 3x5 + 9x4 - 5x2 - 4ax + 3a2 contain x - 1 as a factor?

• Find the value of for a for which x7 + 9x4 - 7x3 + 11ax + 5a2 may contain (x + 1) as a factor.

• Find the values of the polynomial f(x) = 2x3 - 5x2 + x + 2 at x = 0, 1, 2, 3.What are the factors of f(x)?

• Show that 2x + 4 is a factor of f(x) = x(x + 1) (x + 3) (x + 4) - 4



• If f(x) = 2x4 - 3x2+6x + k & f(1) = 0 then find the value of k.

• What must be added to x3-3x2+4x-15 to make it exactly divisible by x – 3?

B. Long Answer Questions

• State and prove factor theorem.

• State and prove remainder theorem.

• If x -1 is a factor of a2 x2-3ax+3a-1, find the value of a

• A graph of a polynomial f(x) is shown. Find the polynomial and its roots if f(x) = 0.


• Factorize the following & solve each of them when they are equated to zero

• x3+2x2-5x-6 x3-9x2+26x-24

• x2 + 4x2 + x – 6

• 3x2 + 13x + 12

• x3 - 9x2 + 24x – 20

• x3 + 2x2 - 5x – 6

• x3 - 4x2 + x + 6

• x(x+1)(x + 3) (x + 4) – 4

• x3 - 13 x - 12

• 6x3 + 7x2 - x – 2

• x3 + 3x2 - 9x + 5



• (x - 1)( x - 3)( x - 5)(x - 7) + 16

• x3 + 2x2 - 5x – 6

• x4 - x3 - 19x2 + 49x – 30

• x3-6x2+11x-6=0


• Solve the following polynomial equations:

• x4-4x3-x2+16x-12 = 0

• x3-2ax2-a2x+2a3 = 0

• x3-7x2+7x+15= 0

• x3 - 9x2 + 23x - 15 = 0

• x3 + 5x2 - 2x - 24 = 0

• 5x2 - 23x + 12 = 0

• 2x4 - 17x3 + 37x2 - 7x - 15 = 0

• x3 - 6x2 + 3x + 10 = 0

• 8x3 – 12x2 -2x + 3 = 0

• y = x3 – 4x2 + x + 8 & y = 2



SLC Model Question SLC _ OPT Math _ Compound Angles


# Short-Answer Questions:

• Find the values of

• Sin15o

• Tan750

• Sin750

• Cos75o

• Tan105o

• Cos150+Cos75

• Show that:

•

• Cos400- Sin400 = Sin50

• Cos15o-Sin15o =

• Sin105o+Cos105o =

• Cos500 + Cos700 = Cos100

• Tan15o + Cot15o = 4

• Sin105o + Cos75o =

• Tan75o +Cot75o = 4

• Cos(400+) – Sin(500-) = 0

• = tan100

• = Cot500

• Solve the following

•

• If CosA = and CosB = , find the value of:

• Sin (A-B)

• Cos (A-B)

• If CosA = and CosB = ,find the value of:

• Cos (A+B)

• Sin (A-B)



# Long-Answer Questions:

• Prove the following identities:

•

•

•

• • Prove that:

• Sin (45o+ ) +Cos (45o+ ) = Cos

• Cot (45o+ ) =

• Sin (A-45o) =

• Tan (45o+A) tan (45o-A) = 1

• Sin2 (45o+A) +Sin2 (45o-A) = 1

•

• Solve the following problems.

• If tan A = and tanB = , show that A+B =

• If tanA = and tanB = , show that A+B =

• If tanA = , show that A+B =

• If tanA = show that A+B =

• Prove the following:

•

• Cos18o-Sin18o = Sin27o

• Cos55o + Cos35o = Cos10o

• Prove the following:

•

•

• • Show that:

• Tan20o+tan25o+tan20otan25o=1

• Tan20o+tan72o+tan88o = tan20o.tan72otan88o

• Tan3A-tan2A-tanA = tan3A.tan2A.tanA

• If (A+B) = 45o, show that :

• (1+tanA) (1+tanB) = 2

• (CotA-1) (CotB-1) = 2



SLC Model Question SLC _ OPT Math _ Functions


# Short Answer Questions

• If f(x) = x+2 , g(x) = x -4 and h(x) = 3x, find the following:

• fog(x)

• goh(x)

hof fog(x)

• If f(x) = 3x+4 find the following functions.

• f1(x)

• f-1(4)

• f(2) .f-1(2)

• f(2) f-1(2)

• If f and g are the functions defined by f = {(1, 5), (2, 4) , (3, 6)} and g {(5, 3), (4, 1), (6, 2)} Find the composite

function gof and fog

• If f(x) = 3x + 2 and g(x) = 2x – 1, find gof(x).

• If f(x) = 14x + 12 and g(x) = 5x – 21, find fog(x).

• If f(x) = x + 12 and g(x) = 2x + 12, find fog(x).

• If f(x) = 3x - 2 and g-1(x) = 2x +1, find gof(x) and fog(x).

• If f(x) = 3x + 12 and g-1(x) = x -10, find gof(x) and fog(x).

• If f and g are the functions defined by f = {(1, 2), (3, 5) , (4, 1)} and g {(2, 3), (5, 1), (1, 3)} Find the composite

function gof and fog

• If f: x 3x +1 and h: x4x, Find the equation of foh(x) and hof(x) and value of hof (2), foh (2) and ff (-5).

• If f: R R is defined by f(x) = 2x and g: R R is defined by f(x) = x + 1. Find gof and fog.

• If f : {(x, 3x)} and g : {x, 2x + 5} find fog-1 (x) and gf-1(x)

• If f(x) = x2 and g(x) = x- 3 Find (i) fog(x) (ii) gof(x) (iii) fog(5) (iv) gof(3)

• If f(x) = 2x2 and g(x) = 5x+ 3 Find (i) fog(x) (ii) gof(x) (iii) fog(4) (iv) gof(-1)

• Find the inverse of the following function f

•

•

•

•

•

• If f(x) = 2x + 4 is function. Find f1 and draw the graphs of both the functions.

• If f-1(x) = 2x - 1 is function. Find f(x) and draw the graphs of both the functions.



# Long Answer Questions

• If f(x) = x2 – 2x, g(x) = 2x + 3 and fog-1(x) = 3 then find the value of x.

• If f(x) = and g(x) = 2 +a, evaluate fog(x) = g-1(x) to find the value of a.

• Let A = {-1, 0, 1} and B = {0, 1}.The function f: A→B is defined by f(x) = x2. Can f-1(x) be defined?

• It is given that f(x) = 4x – 17 and . If f(x) = g-1(x), find the value of x

• It is given that f(x) = 5x – 9 and . If f(x) = g-1(x), find the value of x

• It is given that f(x) = 3x – 7 and . If f-1(x) = g(x), find the value of x

• It is given that f(x) = 4x –5 and . If f(x) = g-1(x), find the value of x

• It is given that f(x) = 9x +15 and . If f-1(x) = g-1(x), find the value of x • If f(x) = 1 + 2x and g(x) =, find g-1 and fog (-1). Is fog(x) = gof(x)? • If f(x) = 10 -3x and g(x) =, find g-1 and fog (5). Is f-1og(x) = gof-1(x)? • If f(x) = 3x + a and fof (6) = 10 find the value of a and f-1 (4).

• If f(x) = 5x - k and fof (-3) = 1 find the value of k and f-1 (-5).

• Find the inverse function of the following functions and show that fof-1 is an identity.

• g(x) = 8x - 17

• p(x) = x + 9

• f(x) = 3x – 1

• f(x) = 5x + 7

• If f = {(x, 3x - 4): x X R}, XR}, g = {(x, -2x + 1): x XR} and h = {(x, x+3): x XR} are functions.

Prove that fof1, gog-1, h-1oh are identity function.

•

• Given that f(x) = 2x + 3 and fog(x) = 6x + 13, find g(x)

• Given that g(x) = 3x + 5 and fog(x) = 6x + 13, find f(x)

• Given that g(x) = 3x - 5 and fog(x) = 4x + 3, find f(x)

• Given that f(x) = 2x + 3, g(x) = 3x + 5 and gohof(x) = 6x + 17, find h(x).



SLC Model Question SLC _ OPT Math _ Geometric Progression


# Short Answer Questions

• Find the 5th term of the sequence 3,-6, 12,-24, …

• Find the 7th term of the series 3,6,12,24,…….

• Find the arithmetic mean and geometric mean between the two numbers 6 and 54.

• If the first term of a geometric series is 4 and the sum of first two term is 36, find the common ratio.

• How many terms are there in the series ? • In a GP, the first term is 7 and the last term is 448 and sum 889, find the common ratio.

• Find the 8th term of the series 5,-10,-15, 20 …

• If the 2nd and 4th term of a GP are -10 and 20 respectively then find first term, common ratio and 12th term.

• If 2nd term and 5th term of a GP are 4 and 32 respectively. Find 8th term

• In a GP if the common ratio is 2 and the 8th term is 384, find the first term

• Which term of the series is ? • Find the sum of the following series:

• +1+4+…… 7 terms

• 8 terms

• 10terms

• 8-12+18-……..

• How many terms of the series 32+48+72+….. will add up to 665 ? • Evaluate the following

•

•

•

• • In a GP if S6 = 28 and S3 = 1, find a and r.

• The first three terms of a G.P. are 2x, 2x + 3 & 2x + 9. Find x & the value of 5th term. • If 4th term of a G.P. is 54 and 6th term is 24, which term is 7 ?



# Long Answer Questions

• If the second and fifth term of a GS is 4 and 32, find the series.

• Given series is 9+3+1+………..+ . • How many terms are there in the series ? • Calculate the sum

• For two unequal numbers show that AM> GM

• If the AM and GM between two positive and unequal numbers are 5 and 4 respectively, what are the numbers ?

• Find two numbers whose arithmetic mean is 5 and geometric mean is 4.

• If the AM and GM between two positive and unequal numbers are 13 and 5 respectively, what are the numbers ?

• If the AM and GM between two positive and unequal numbers are and 6 respectively, what are the numbers ?

• The sum of first four terms is 40 and the sum of first two terms is 4 of a geometric series whose common ratio is

positive, find the sum of first eight terms.

• If the AM of two unequal positive real numbers a and b (a>b) be twice as great as their GM, show that

• Anisha borrows Rs 4368 which she promises to pay in 6 annual instilments, each installment being treble of the

preceding one. Find the first and the last installment.

• The product of three numbers in a GP is 216 and the sum of products of the numbers taken in pairs is 156. Find

the numbers.

• x + 6, x and x - 3 are the first three terms of a geometric series. Find the value of x and its fifth term.

• If x, 6, y, 24, P are in GS, find the values of x, y, p given that common ratio is +ve. • If 4th term of a G.P. is 54 and 6th term is 24, which term is 7 ? • Insert 3 geometric means between and 9.

• Insert 4 GM between and 16; also find the sum of the series.

• Insert 2 GM between 6 and 48.

• Insert 3 GM between 5 and 3125.

• The 2nd term of a GP is 6 and its 5th term is 162. Find the sum of 1st 5 terms.

• If ,x,y,z are in G.P., find the values of x, y & z.

• If 5, a, b, 135 are in GP, find a and b.

• If 2, m, n, s, are in GP, find m, n and s.

• The sum of 3 consecutive terms of G.P. is 28 and their product is 512. Find the terms

• The sum of three terms in AP is 30. If 5 is added to the third term it becomes a GP. Find the terms.

• Find two numbers whose AM is 34 & GM is 16.

• If a, b, c are in AP and x, y, z are in G.P. prove that.

• In a G.P. tp = a, tq = b & tr = c. Then show that aq-r br-p cp-q = 1.



SLC Model Question SLC _ OPT Math _ Natural Numbers


# Formulas

1. Sum of 1st n natural numbers Sn = 1+2+3+……. to n terms =

2. Sum of 1st n even numbers Sn = 1 + 2 + 4 + to n terms = .

3. Sum of 1st n odd numbers Sn = 1 + 3 + 5 +…… to n terms = n2

4. Sum of square 1st n natural numbers Sn= 12 + 22 + 32 +…………to n terms =

5. Sum of cubes of 1st n natural numbers Sn = 13 + 23 + 33 +………to n terms= [ ] 2

6. =

7. 2 =

8. 3 = [ ] 2

# Rules to find General term

To find general terms we use following rules:

1) If the series/sequence is AS then tn= a + (n-1) d

2) If the series/sequence is GS then tn = arn-1

3) If the series/sequence is not both AS and GS we find a pattern.

4) If we cannot use all these methods mentioned above then tn = an2 + bn2 + c.

# Illustrative Examples

• Find the sum of first 5 natural numbers.

Solution:

Here n =5

So,

• Find the sum of first 10 even numbers.

Solution:

Here n = 10

So,

• What is the sum of first 7 odd numbers?

Solution:

Here n = 7

So,

• Calculate the sum of square of first three natural numbers?

Solution:

Here n = 3



• Find the sum of cubes of first 5 natural numbers.

Solution:

Here, n = 5

So, = = 152 = 225

# Important Questions for SLC Examination

• Find the sum of the following series

• 1 + 3 + 5 + ………….. 25 terms

• 2 + 4 + 6 + ……….. 21 terms

• 1 + 2 + 3 + …………. 25 terms

• 2 + 4 + 6 + ……… 30 terms

• 1 + 3 + 5 + ………… 50 terms

• Find the sum of the following series

• 13 + 23 + 33 …………… +103

• 13 + 23 + 33 + …………… 8 terms

• 12 + 22 + 32 + …………….. + 102

• 12 + 22 + 32 + ……….. 12 terms

• Find the nth term and sum of n terms of following series.

• 22 + 42 + 62 + ……………… n terms

• (1×2) + (2×3) + (3×4) + ……….. n terms

• (2×4) + (3×5) + (4×6) + ………….. n terms



SLC Model Question SLC _ OPT Math _ Arithmetic Progression


# Important Questions for SLC

A. Short Answer Questions

• If the first term is 3 and 7th term is 15 then find the common difference of the arithmetic series.

• If the first term is 13 and 3rd term is 22 then find the common difference of the arithmetic series.

• Find the sum of to 20 terms.





• Find the sum of the series to 15 terms

• If 3, x, y,-9 are in AP, find the values of x and y.

• If 1, x, y,13 are in AP, find the values of x and y.

• If 0, x, y,21 are in AP, find the values of x and y.

• In an AP, 10th term is 17; find the sum of first 19 terms.

B. Long Answer Questions

• If the third term of an A.P. is 1 and its fifth term is 7, find the sum of first ten terms of the series.

• If the fourth term of an AP is 1 and the sum of its first eight terms is 18, find the tenth term of the series.

• The last term of an arithmetic series of 20 terms is 195 and the common difference 5. Calculate the sum

of the series.

• The first and last terms of an arithmetic series are -24 and 72 respectively. If sum of all terms of the series is

600, find the number of terms and the common difference of the series.

• The sum of first three terms of an arithmetic series is 21. If the sum of first two terms is subtracted from the

third term then it would be 9, find the three terms of the series.

• If the fourth and eighth term of an AP are 4 and 32 respectively, find the sum of first ten terms.

• If the sum of first three terms of an AS is 42 and that of the first five terms is 80, find the 20th term of the

series.

• The first and the last term of an As are -24 and 72 respectively. If the sum of all the terms of the series is

600, find the number of terms and the common difference.

• In an arithmetic series, the sum of the first ten terms is 520. If its seventh term is double of its third term,

calculate the first term and the common difference of the series.

• Find two numbers whose arithmetic mean is 5 and geometric mean is 4.

• In an AP, 10th term is 19; find the sum of first 21 terms.

• In an arithmetic sequence sum of first 31 terms is 310. Find the 16th term.

• In an A.P. seven times of seventh term and eleven times of eleventh term are equal, find 18th term.

• If the first 3 terms of A.P. are x + 2, 2x - 1 and x + 6, find x and sum of its first five terms.

• If the first 3 terms of A.P. are x + 3, 5x and x + 13, find x and sum of its first ten terms



• If the fifth term and 10th term of an A.P. are 17 and 42 respectively. Find the sum of first 12 terms.

• Find three terms of AP whose sum is 21 and product is 315.

• Find three terms of AP whose sum is 27 and product is 693.

• Find 3 arithmetic means between 140 and -20

• Find 5 arithmetic means between 10 and 70

• Insert 5 arithmetic means between 2 and 18.

• Insert three arithmetic means between 13 and 29.

• Insert 4 arithmetic means between 3 and 33

• There are 6 A.M's between -3 and 32 find them.

• There are 4 A.M's between 3 and 18 find them.

• If the 6th term of an AP is 4 times the second term and the sum of the first 24 terms is 1704. Find the sum of

first 48 terms.

• If 3rd term and 13th term of an AP are -40 and 0 respectively. Find the first term and common difference

of AP. is -72 a term of the AP?

• 3 AMs are inserted between two numbers a and b. If the first and 3rd means are 16&34 respectively. Find

the 2nd mean and also the value of a & b.

• A taxi meter reads Rs. 7 at the time of starting and Rs. 10 for each additional kilometer. Find the charge

read by the taxi meter when the distance covered is 19 kilometer.

• The cost of 175 calls of telephone per moths is Rs. 200 and Rs. 2 for additional calls, if the total calls in a

particular month are 250. Find the total cost of that particular month

• There are n A.M.'s between 3 and 39. Find n so that the third mean: the last mean = 3 : 7

• There are n arithmetic means between 12 and 33. If the fourth arithmetic mean is 24. Find the value of n.

• The sum of three numbers in A.P. is 27 and the sum of their squares is 293. Find the numbers.

• The interior angles of a polygon are in arithmetic progression. The smallest angle is 520 and the common

difference is 80. Find the number of sides of the polygon.

• The pth term of an A.P. is q and the qth term p; find the (p+q)th term and (p+q-1)th terms

• If a, b and c be respectively mth, nth and pth terms of a GP then prove that an-pbp-m,cm-n = 1

• If the arithmetic mean between a and b is twice as greater as their geometric mean show that a:b = (2 +

) : (2 - )

• Divide 15 into three parts which are in A.P. and The product of the three numbers is 120.



slc _ opt math _ all

Documents

x fx

x3 4x2 x

x3 4x2 x

x2 4x2 x

2x3 5x2 x

6x3 7x2 x

polynomial 3x3 kx2 x

factor of fx