ASSIGNMENTS FOR WEEK STUDENTS

1.REAL NUMBERS

1. Use Euclid’s division algorithm to find the HCF of 135 and 225.2. Use Euclid’s division lemma, to show that the cube of any positive integer is of the form 9m, 9m

+ 1 or 9m + 8.3. Prove that √2 is an irrational number.4. Prove that 3+2√5 is irrational. If √5 is an irrational.5. Without actually performing the long division, state whether the following rational number will

have a terminating decimal expansion or a non-terminating repeating decimal expansion :6. (i) 13/3125 (ii) 17/8 (iii) 64/455 (iv) 15/1600 (v) 29/343 .7. Explain why 7 × 11 × 13 + 13 and 7 × 6 × 5 × 4 ×3 × 2 × 1 + 5 are composite numbers?8. Find the LCM and HCF of the following pairs of integers and verify that LCM × HCF = Product of

the integers. (i) 63 and 168 (ii) 144 and 160 (iii) 510 and 92 (iv) 252, 488 .9. Find the HCF of 96 and 404 by prime factorization method. Hence, find their LCM.10. The product of two numbers is 20736 and their HCF is 54. Find their LCM.

2. POLYNOMIALS

1. Find the zeroes of the quadratic polynomial x2 – 2x – 8, and verify the relationship between the zeroes and their coefficients.

2. Find a quadratic polynomial whose sum and product of the zeroes is ¼ and –1 respectively.3. Divide the polynomial f(x) = x4 – 3x2 + 4x + 5 by the polynomial 1 – x + x2 and verify the division

algorithm.4. Obtain all the zeroes of the polynomial f (x) = 3 x4+ 6x3 – 2x2 – 10x – 5,, if two of its zeroes are

√5/3 and – √5/3.5. If the zeroes of the polynomial x3 – 3x2 + x + 1 are a – b, a, a + b, find a and b.6. Verify that 3, -1, -1/3 are the zeroes of the cubic polynomial p(x) = 3x3 – 5x2 – 11x – 3, and then

verify the relationship between the zeroes and the coefficients.7. If α and β are the zeroes of the polynomial x2 – 5x + 6, find the value of 1/α + 1/ β.

3. PAIR OF LINEAR EQUATIONS IN TWO VARIABLES

1. Find the values of a and b so that the following system of linear equations has an infinite number of solutions : 2x – 3y = 7 & (a + b)x – (a + b – 3) y = 4a + b.

2. The sum of a two-digit number and the number obtained by reversing the order of digits is 165. If the digits differ by 3, find the number.

3. A man travels 370 km partly by train and partly by car. If he covers 250 km by train and the rest by car, it takes him 4 hours. But, if he travels 130 km by train and the rest by car, he takes 18 minutes longer. Find the speed of the train and that of the car.

4. Solve the following system of equations using the method of elimination (by equating the coefficients). x+y=7 & 12x + 5y = 7.

5. Solve the following system of equations using the method of cross-multiplication.2x + 3y = 17 & 3x – 2y = 6.

6. Solve the system of equation:- 1/2x – 1/y = -1 & 1/x + 1/2y = 8.7. Two years ago, a father was five times as old as his son. Two years later, his age will be 8

more than three times the age of the son. Find the present age of the father and son.8. A two-digit number is such that the product of the digits is 20. If 9 is added to the

number, the digits interchange their places. Find the number.9. A boat goes 12 km upstream and 40 km downstream in 8 hours. It can go 16 km

upstream and 32 km downstream in the same time. Find the speed of the boat in still water and the speed of the stream.

10. Points A and B are 70 km apart on a highway. A car starts from A and another car starts from B at the same time. If they travel in the same direction, they meet in 7 hours, but if they travel towards each other they meet in one hour. What are their speeds?

11. A man travelled 300 km by train and 200 km by taxi, it took him 5 hours 30 minutes. But if he travels 260 km by train and 240 km by taxi he takes 6 minutes longer. Find the speed of the train and that of the taxi.

4. QUADRATIC EQUATIONS

1. Find the roots of the quadratic equation 2x2-7x +3 = 0 by the method of completing the square.2. Find the roots of the quadratic equation 8x2 – 22x – 21 = 0 by factorization.3. A two digit number is such that the product of its digits is 18. When 63 is subtracted from the

number, the digits interchange their places. Find the number. 4. A plane left 30 minutes later than the schedule time and in order to reach its destination 1500

km away in time it has to increase its speed by 250 km/hr from its usual speed. Find its usual speed.

5. Determine the discriminant of quadratic equation:- 5x2 + 4x – 1 = 0 .6. Solve the quadratic equations using Quadratic Formula:- x2 – 6x + 9 = 0.7. Find the value(s) of k for which the roots of quadratic equation are real and equal:-

5x2 – 6x + k = 0.8. The speed of a boat in still water is 8 km/hr. It can go 15 km upstream 22 km downstream in 5

hours. Find the speed of the stream.9. A train covers a distance of 90 km at a uniform speed. Had the speed been 15 km per hour

more, it would have taken 30 minutes less for the journey. Find the original speed of the train.10. Seven years ago Varun’s age was five times the square of Swati’s age. Three years hence Swati’s

age will be two-fifth of Varun’s age. Find their present ages.

5. ARITHMETIC PROGRESSION

1. Which term of the A.P. 5, 15, 25, .... will be 130 more than its 31st term?2. Find the sum of the first 25 terms of an A.P. whose nth term is given by an = 7 – 3n.3. How many terms of the series 54, 51, 48, ..... be taken so that their sum is 513?4. In an A.P., the sum of first n terms is 3n2/2 + 5n/2. Find its 25th term.5. The 10th term of an A.P. is 52 and 16th term is 82. Find the 32nd term and the general

term.6. For what value of n is the nth term of the following A.P.’s the same?

1, 7, 13, 19, ..... and 69, 68, 67...............7. Which term of the A.P. 3, 10, 17, .... will be 84 more than its 13th term?8. Find the 8th term from the end of the A.P. 7, 10, 13, ..... , 184.

9. If the mth term of an A.P. be and its nth term be . Show that its (mn)th term is 1.

10. Find: ….n terms.

11. In an A.P. given a = 2, d = 8, Sn = 90, find n and an.

12. The sums of n-terms of two A.P.’s are in the ratio 7n + 1 : 4n + 27. Find the ratio of their 11th terms.

6. TRIANGLES

1. Prove that Basic proportionality theorem (Thales theorem) : If a line is drawn parallel to one side of a triangle to intersect the other two sides in distinct points, then the others two sides are divided in the same ratio.

2. Prove that the ratio of the areas of two similar triangles is equal to square of the ratio of their corresponding sides.

3. Prove that (Pythagoras Theorem) In a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.

4. Prove that (Converse of Pythagoras Theorem) If in a triangle, square of one side is equal to the sum of the squares of the other two sides, then the angle opposite to first side is a right angle.

5. ABCD is a trapezium such that AB || DC. The diagonals AC and BD intersect at O. Prove that OA/OB = OC/OD.

6. If the diagonals of a quadrilateral divide each other proportionally, then it is a trapezium. 7. Prove that area of an equilateral triangle described on one side of a square is equal to half the

area of the equilateral triangle described on one of its diagonal.8. Prove that the sum of the squares of the sides of a rhombus is equal to the sum of the squares

of its diagonals9. In an equilateral triangle ABC, D is a point on side BC such that BD= 1/3BC.

Prove that 9AD2 = 7AB2.

10. In Triangle ABC, DE || BC and AD/DB = 3/5, If AC = 4.8 cm, find AE.11. Two right triangles ABC and DBC are drawn on the same hypotenuse BC on the same side of BC.

If AC and DB intersect at P, prove that AP × PC = BP × PD.12. In a right angled triangle ABC, A = 90° and AD BC. Prove that AD2 = BD × CD.13. The areas of two similar triangles are 100 cm2 and 49 cm2 respectively. If the altitude of the

bigger triangle is 5 cm, find the corresponding altitude of the other.14. The areas of two similar triangles are 121 cm2 and 64 cm2 respectively. If the median of the first

triangle is 12.1 cm, find the corresponding median of the other.

15. Equilateral triangles are drawn on the sides of a right triangle. Show that the area of the triangle on the hypotenuse is equal to the sum of the areas of triangles on the other two sides.

16. ABC is an isosceles triangle right angled at C. prove that AB2 = 2AC2.17. The perpendicular AD on the base BC of a ABC intersects BC at D so that DB = 3CD. Prove that

2AB2 = 2AC2 + BC2.18. ABC is a right triangle right-angled at B. Let D and E be any points on AB and BC respectively.

Prove that AE2 + CD2 = AC2 + DE2.19. In a ABC, AD BC and AD2 = BC × CD. Prove that ABC is a right triangle.20. A point O in the interior of a rectangle ABCD is joined with each of the vertices A, B, C and D.

Prove that OB2 + OD2 = OC2 + OA2.

7. CO-ORDINATE GEOMETRY

1. Find value(s) of y for which the distance between the points P (2, –3) and Q (10, y) is 10 units.2. Show that (1, –2), (3, 0), (1, 2) and (–1, 0) are the vertices of a square.3. Show that the points (1, –1), (5, 2) and (9, 5) are collinear.4. Find the co-ordinates of the points of trisection of the line joining the points (4, –1) and

(–2, –3).5. If A(5, –1), B(–3, –2) and C(–1, 8) are the vertices of ABC, find the length of medians through A

and the coordinates of the centroid.6. If the points (3, 2) and (2, –3) are equidistant from (a, b) show that a + 5b = 0.7. If the points (5, 1) and (–1, 5) are equidistant from the points (x, y), show that 3x = 2y.8. If the points (5, 1) and (–1, 5) are equidistant from the points (x, y), show that 3x = 2y.9. What is the area of triangle formed by the points (–2, 0), (4, 0) and (2, 3).10. Three vertices of a parallelogram, taken in order are (3, 1), (2, 2) and (–2, 1) respectively. Find

the coordinates of fourth vertex.11. Find the area of the quadrilateral, the coordinates of whose vertices are (1, 2), (6, 2), (5, 3) and

(3, 4).12. If (3, 2), (4, 4) and (1, 3) are the mid-points of the sides of a triangle, find the coordinates of the

vertices of the triangle.8. INTRODUCTION TO TRIGONOMETRY

1. Cotθ= 20/21, find all other trigonometric ratios.2. If 3 tan A = 4, check whether 1–cot2 A/1+cot2A = cos2A – sin2A or not.3. Evaluate : 5cos 60°+4sec 30° –tan 45°/ sin230° + cos230°.4. If tan (A + B) = √3 and tan (A – B) = 1/√3 ; 0° < A + B <90°; A > B, find A and B.5. Prove that (sinθ + cosecθ )2 + (cosθ + secθ)2 = 7 + tan2θ + cot2θ.6. (tan A/1-cotA) + (cot A/1-tanA) = 1+tan A + cot A = 1+secAcosecA.7. If tan θ + sinθ= m and tan θ – sinθ= n, show that m2 - n2 =4√mn.8. Simplify the following expressions:

(i) (1 + cosA) (cosec A – cot A)

(ii) sin3 A+cos3 AsinA+cos A

.

9. Prove that sinA−2 sin3 A2cos3 A−cos A

= tanA.

10. Prove that sinA

cotA+cossecA = 2+sinA

cotA−cossecA .

9. SOME APPLICATIONS OF TRIGONOMETRY

1. The angle of elevation of the top of the building from the foot of the tower is 30° and the angle of elevation of the top of the tower from the foot of the building is 60°. If the tower is 50 m high, find the height of the building.

2. Two pillars are of equal height and on either side of a road, which is 100 m wide. The angles of elevation of the top of the pillars are 60° and 30° at a point on the road between the pillars. Find the position of the point between the pillars and the height of each pillar.

3. A tree breaks due to storm and the broken part bends so that the top of the tree touches the ground making an angle of 30° with the ground. The distance from the foot of the tree to the point where the top touches the ground is 10 m. Find the height of the tree before it was broken.

4. The angles of elevation of the top of a tower from two points at distances a and b meters from the base and in the same straight line with it are complementary. Prove that the height of the tower is ab meters.

5. A straight highway leads to the foot of a tower. A man standing at the top of the tower observes a car at an angle of depression of 30°, which is approaching the foot of the tower with a uniform speed. Six minutes later, the angle of depression of the car is found to be 60°. Find the time taken by the car to reach the foot of the tower.

6. A man standing on the deck of a ship, which is 10 m above water level. He observes the angle of elevation of the top of a hill as 60° and the angle of depression of the base of the hill as 30°. Calculate the distance of the hill from the ship and the height of the hill.

7. The angle of elevation of the top of a tower from a point A on the ground is 30°. On moving a distance of 20 metres towards the foot of the tower to a point B, the angle of elevation increases to 60°. Find the height of the tower and the distance of the tower from the point A.

8. A 7 m long flagstaff is fixed on the top of a tower on the horizontal plane. From a point on the ground, the angle of elevation of the top and the bottom of the flagstaff are 45° and 30° respectively. Find the height of the tower correct to one place of decimal.

9. The angle of elevation of a tower from a point on the same level as the foot of the tower is 30°. On advancing 150 metres towards the foot of the tower, the angle of elevation of the tower becomes 60°.Show that the height of the tower is 129.9 metres. (use ( √3 = 1.732).

10. From the top of a building 15 m high the angle of elevation of the top of a tower is found to be 30°. From the bottom of the same building, the angle of elevation of the top of the tower is found to be 60°. Find the height of the tower and the distance between the tower and the building.

10. CIRCLES

1. Prove that Tangent to a circle at a point is perpendicular to the radius through the point of contact.

2. Prove that The lengths of the two tangents drawn from an external point to a circle are equal.3. If all the side of a parallelogram touch a circle, show that the parallelogram is a rhombus4. Prove that the angle between the two tangents drawn from an external point to a circle is

supplementary to the angle subtended by the line-segment joining the points of contact at the Centre.

5. The radius of the incircle of a triangle is 4 cm and the segments into which one side is divided by the point of contact are 6 cm and 8 cm. Determine the other two sides of the triangle.

6. A quadrilateral ABCD is drawn to circumscribe a circle (fig-2). Prove that, AB + CD = AD + BC.

7. A point P is 13 cm from the Centre of the circle. The length of the tangent drawn from P to the circle is 12 cm. Find the radius of the circle.

8. Prove that the perpendicular at the point of contact to the tangent to a circle passes through the Centre.

9. Prove that the tangents drawn at the ends of a chord of a circle make equal angles with the chord.

10. Two tangents TP and TQ are drawn to a circle with Centre O from an external point T. Prove that PTQ = 2.OPQ.

11. CONSTRUCTIONS

1. Draw a line segment of length 7 cm and divide it in the ratio 2 : 3.2. Construct a triangle similar to a given triangle with sides 7 cm, 9 cm and 10 cm and whose sides

are 57 th of the corresponding sides of the given triangle.

3. Draw a right triangle in which the sides (other than hypotenuse) are of lengths 4 cm and 3 cm.

Then construct another triangle whose sides are 53 times the corresponding sides of the given

triangle.4. Draw a pair of tangents to a circle of radius 3 cm which are inclined to each other at an angle of

60°.5. Construct an isosceles triangle whose base is 6 cm and altitude 3.5 cm and then another

triangle whose sides are 1 23 times of the corresponding sides of the isosceles triangle.

6. Draw a circle of radius 4.5 cm. From a point 10 cm away from its Centre, construct the pair of tangents to the circle.

7. Draw a pair of tangents to a circle of radius 4.5 cm which are inclined to each other at an angle of 45°.

8. Construct a tangent to a circle of radius 4 cm from a point on the concentric circle of radius 6 cm and measure its length. Also, verify the measurement by actual calculation.

9. Draw a circle of radius 3.5 cm. Take two points A and B on one of its extended diameter each at a distance of 8 cm from its Centre. Draw tangents to the circle from these two points A and B.

10. Draw a line segment AB of length 11 cm. Taking A as Centre, draw a circle of radius 4 cm and taking B as Centre, draw another circle of radius 3 cm. Construct tangents to each circle from the Centre of the other circle.

12. AREAS RELATED TO CIRCLES

1. The wheels of a car are of diameter 80 cm each. How many complete revolutions does each wheel make in 10 minutes when the car is travelling at the speed of 66 km per hour?

2. A horse is tied to a peg at one corner of a square shaped grass field of side 15 m by means of a 5 m long rope (see figure). Find (i) the area of that part of the field in which the horse can graze. (ii) the increase in grazing area if the rope were 10 m long instead of 5m. (Use = 3.14).

3. The area of an equilateral triangle ABC is 17320.5 cm2. With each vertex of the triangle as Centre, a circle is drawn with radius equal to half the length of the side of the triangle. Find the area of the shaded region. [Use π = 3.14 and √ √3 = 1.73205].

4. In fig, Find the area of the shaded region, enclosed between two concentric circle of

radii 7 cm and 14 cm, Where ∠AOC =40°.

5. In the given, OACB is a quadrant of a circle with Centre O and radii 3.5 cm. If OD = 2

cm, Find the area of the shaped region.

6. In given figure, ABCD is a rectangle of dimensions 21 cm × 14 cm. As semi-circle is

drawn with BC diameter. Find the area and the perimeter of the shaped region.

7. In fig, ABCD is the quadrant of a circle of radius 14 cm and a semi-circle is drawn with

BC as the diameter. Find the area of the shaded region.

8. In fig, ABCD is a right-angled triangle, right- angled at A. Semi-circle are drawn on AB,

AC, and BC as diameter. Find the area of shaded region.

9. In fig, AB and CD are two perpendicular diameter of a circle with Centre O. If OA = 7

cm, Find the area of the shaded region.

10. Find the area of the shaded region in fig, where arcs drawn with centers A,B,C and D

intersect in pairs at mid-points P,Q,R and S of the sides AB,BC,CD and DA respectively

of a square ABCD , where the length of a each side of square is 14 cm.

11. Find the perimeter of shaded region in fig, if ABCD is a square of side 14 cm and APB

and CPD are semicircle.

12. In fig, ABCD is a square of side 4 cm. A quadrant of a circle of radius 1 cm is drawn at

each vertex of the square and a circle of diameter 2 cm is also drawn. Find the area of

the shaded region.

13. SURFACE AREAS AND VOLUMES

1. A toy is in form of a cone mounted on a hemi-sphere of radius 3.5 cm. The total height of the toy is 15.5 cm. Find its total surface area.

2. A wooden article was made by scooping out a hemisphere from each end of a solid cylinder, as shown in the given figure. If the height of the cylinder is 10 cm, and its base is of radius 3.5 cm, find the total surface area of the article.

3. A solid consisting of a right circular cone of height 120 cm and radius 60 cm standing on a hemisphere of radius 60 cm is placed upright in a right circular cylinder full of water such

that it touches the bottom. Find the volume of water left in the cylinder, if the radius of the cylinder is 60 cm and its height is 180 cm.

4. A sphere, of diameter 12 cm, is dropped in a right circular cylindrical vessel, partly filled with water. If the sphere is completely submerged in water, the water level in the

cylindrical vessel rises by 3 59 cm. find the diameter of cylindrical vessel.

5. A solid right circular cone of diameter 14 cm and height 8 cm is melted to form a hollow sphere. If the external diameter of the sphere is 10 cm, find the internal diameter of the sphere.

6. A circus tent is cylindrical up to a height of 3 m and conical above it. If the diameter of the base is 105 m and the slant height of the conical part is 53 m, find the total canvas used in making the tent.

7. A solid is composed of a cylinder with hemispherical ends. It the whole length of the solid is 104 cm and the radius of each of the hemispherical ends is 7 cm, find the cost of polishing its surface at the rate of Rs. 10 per dm2.

8. A cylindrical tub of radius 5 cm and length 9.8 cm is full of water. A solid in the form of a right circular cone mounted on a hemisphere is 3.5 cm and height of the cone outside the hemisphere is 5 cm, find the volume of the water left in the tub.

9. A toy is in the shape of a right circular cylinder with a hemisphere on one end and a cone on the other. The radius and height of the cylindrical part are 5 cm and 13 cm respectively. The radii of the hemispherical and conical parts are the same as that of the cylindrical parts. Find the surface area of the toy if the total height of the toy is 30 cm.

10. A hemispherical bowl of internal radius 9 cm is full of liquid. This liquid is to be filled into cylindrical shaped small bottled each of diameter 3 cm and height 4 cm, How many bottles are necessary to empty the bowl?

11. The diameters of the internal and external surfaces of a hollow spherical shell are 6 cm and 10 cm respectively. If it is melted and recast into a solid cylinder of diameter 14 cm, find the height of the cylinder.

12. Four right circular cylindrical vessels each having diameter 21 cm and height 38 cm are full of ice-cream. The ice-cream is to be filled in cones of height 12 cm and diameter 7 cm having a hemispherical shape at the top. Find the total number of such cones which can be filled with ice-cream.

13. A metallic sphere of radius 10.5 cm is melted and then recast into small cones , each of radius 3.5 cm and height 3 cm. Find how many cones are obtained.

14. The base radius and height of a right circular solid cone are 2 cm and 8 cm respectively. It is melted and recast into spheres of diameter 2 cm each. Find the number of spheres so formed.

15. If the radii of the circular ends of a conical bucket which is 45 cm high be 28 cm and 7 cm, find the capacity of the bucket.

16. A bucket made of aluminium sheet is of height 20 cm and its upper and lower ends are of radius 25 cm and 10 cm respectively. Find the cost of making the bucket, if the aluminium sheet costs Rs. 70 per 100 cm2.

17. If a cone of radius 10 cm is divided into two parts by drawing a plane through the mid-point of its axis, parallel to its base. Compare the volumes of the two parts.

18. A bucket of height 8 cm and made up of copper sheet is in the form of a frustum of a right circular cone with radii of its lower and upper ends as 3 cm and 9 cm respectively. Calculate (i) the height of the cone of which the bucket is a part.

(iii) the volume of water which can be filled in the bucket. (iv) (iii) the area of copper sheet required to make the bucket.19. The radii of the circular ends of a solid frustum of a cone are 33 cm and 27 cm and its slant

height is 10 cm. Find its total surface area.20. If the radii of the circular ends of a conical bucket which is 45 cm high be 28 cm and 7 cm,

find the capacity of the bucket.

14. STATISTICS

1. The following table shows marks secured by 140 students in an examination :

Marks 0-10 10-20 20-30 30-40 40-50No.of students 20 24 40 36 20

Calculate mean marks by using :

(i) direct method (ii) assumed mean method (or short-cut method) (iii) step-deviation method

2. Find the mode of the following distribution table:-

Class interval

0-20 20-40 40-6- 60-80 80-100 100-120

frequency 10 35 52 61 38 29

3. The lengths of 40 leaves of a plant are measured correct to the nearest millimetre and the data obtained is represented in the following table :

Length (in mm)

118-126 127 -135 136 -144 145-153 154-162 163-171 172-180

No. of leaves

3 5 9 12 5 4 2

Find the median length of the leaves.

4. During the medical check-up of 35 students of a class, their weights were recorded as follows :

Weight (in kg No. of studentsLess than 38 0Less than 40 3Less than 42 5

Less than 44 9Less than 46 14Less than 48 28Less than 50 32Less than 52 35

5. The following table gives production yield per hectare of wheat of 100 farms of a village :

Production yield

(in kg/ha)

50-55 55-60 60-65 65-70 70-75 75-80

No. of farms 2 8 12 24 38 16

Change the distribution to a more than type distribution and draw its ogive .6. If the mean of the following distribution is 54, find the value of p :

Class 0-20 20-40 40-60 60-80 80-100Frequency 7 p 10 9 13

7. The mean of the following frequency table is 50. But the frequencies f1 and f2 in class 20-40 and 60-80 are missing. Find the missing frequencies.

Class 0-20 20-40 40-60 60-80 80-100 totalFrequency 17 F1 32 F2 19 120

8. The median of the following distribution is 35, find the value of a and b.

Class 0-10 10-20 20-30 30-40 40-50 50-60 60-70 totalFrequency 10 20 A 40 B 25 15 170

9. The table given below shows the distribution of the daily wages, earned by 160 workers in a building site:

Wages (in Rs)

0-10 10-20 20-30 30-40 40-50 50-60 60-70 70-80

No of Workers

12 20 30 38 24 16 12 8

Draw a cumulative frequency curve by using :

(i) ‘less than type’ and (ii) ‘more than type’. (ii) Hence, estimate the median wages using graph.

10. Calculate the modal height from the following table:

Height (in cm) No. of boys135-140140-145145-150

49

18

150-155155-160160-165165-170170-175

28241052

15. PROBABILITY

1. An unbiased die is thrown. What is the probability of getting : (i) an odd number (ii) a multiple of 3 (iii) a perfect square number (iv) a number less than 4.

2. Find the probability that a leap year selected at random will contain 53 Sundays.3. One card is drawn from a pack of 52 cards, each of the 52 cards being equally likely to be drawn.

Find the probability that the card drawn is :(i) an ace (ii) either red or king(iii) a face card (iv) a red face card

4. Cards marked with the numbers 2 to 101 are placed in a box and mixed thoroughly. One card is drawn from this box. Find the probability that the number of the card is :(i) an even number (ii) a number less than 14(iii) a number which is a perfect square (iv) a prime number less than 20.

5. A bag contains 3 red balls and 5 black balls. A ball is drawn at random from a bag. What is the probability that the ball drawn is :(i) red (ii) not red

6. A bag contains 12 balls out of which x are white.(i) If one ball is drawn at random, what is the probability that it will be a white ball?(ii) If 6 more white balls are put in the bag, the probability of drawing a white ball will be double than that in (i). Find x.

7. Two dice are thrown simultaneously. Find the probability of getting :(i) a doublet i.e. same number on both dice.(ii) the sum as a prime number.

8. The probability that it will rain tomorrow is 0.86. What is the probability that it will not rain tomorrow?

9. A die is thrown once. Find the probability of getting : (i) a multiple of 3 (ii) a multiple of 2 or 3 (iii) a prime number

10. Two coins are tossed simultaneously. Find the probability of getting : (i) two heads (ii) exactly one tail (iii) no tail

11. Three unbiased coins are tossed simultaneously. Find the probability of getting :(i) one head (ii) two heads (iii) All heads (iv) at least two heads (v) at least one head and one tail

12. A die is thrown once. What is the probability of getting : (i) an even number (ii) an odd number (iii) a number 3 (iv) a number 5 or 6 (v) a number > 6

13. In a simultaneous throw of a pair of dice, find the probability of getting :

(i) 7 as a sum (ii) a doublet of odd numbers (iii) not a doublet (iv) an odd number on the first die (v) a sum less than 6 (vi) a sum more than 10 (vii) neither 9 nor 11 as the sum of the numbers on the faces (viii) a total of at least 10 (ix) a multiple of 3 as the sum (x) a doublet of prime numbers

14. What is the probability that an ordinary year has 53 Mondays? 15. A card is drawn at random from a pack of 52 cards. Find the probability that the card drawn is :

(i) a jack, queen or a king (ii) a face card (iii) a black king (iv) black and a king (v) neither a heart nor a king (vi) spade or an ace (vii) a queen of diamond (viii) either a black card or a king (ix) an ace of heart (x) neither an ace nor a king (xi) ‘10’ of black suit (xii) a club (xiii) neither a red card nor a queen (xiv) a ‘8’ of heart (xv) an ace of red colour

16. Two black kings and two black jacks are removed from a pack of 52 cards. Find the probability of getting: (i) a card of hearts (ii) a card of clubs (iii) a king (iv) a black card (v) either a red card or a king (vi) a red king (vii) neither an ace nor a king (viii) a jack, queen or a king

17. A bag contains 4 red balls, 5 black balls and 3 green balls. A ball is drawn at random from a bag. Find the probability that the ball drawn is : (i) a red ball (ii) a black ball (iii) not a green ball (iv) a black or a green ball

18. A bag contains 5 red balls and some black balls. If the probability of drawing a black ball is double that of a red ball, find the number of black balls in the bag.

19. A bag contains 5 red, 8 white and 7 black balls. A ball is drawn at random from the bag. Find the probability that the ball drawn is ; (i) red or white (ii) not black (iii) neither white nor black

20. 17 cards numbered 1, 2, 3, ...., 17 are put in a box and mixed thoroughly. One person draws a card from the box. Find the probability that the number on the card is : (i) odd (ii) a prime (iii) divisible by 3 (iv) divisible by 2 and 3 both

21. Balls marked with numbers 2 to 101 are placed in a box and mixed thoroughly. One ball is drawn at random from this box. Find the probability that the number on the ball is : (i) an even number (ii) an odd number (iii) a number less than (iv) a number which is a perfect square (v) a number which is a perfect cube (vi) a number divisible by 9 (vii) a prime number less than 41 (viii) a number which is divisible by 3 or 5. 15.

22. Find the probability that a number selected from the number 1 to 25 is not a prime number when each of the given numbers is equally likely to be selected.

23. Find the probability that a number selected at a random from the numbers 1, 2, 3, ...., 35 is a (i) prime number (ii) multiple of 7 (iii) a multiple of 3 or 5

24. Out of 400 bulbs in a box, 15 bulbs are defective. One bulb is taken out at random from the box. Find the probability that the drawn bulb is not defective

25. A bag contains 3 red, 5 black and 7 white balls. A ball is drawn from the bag at random. Find the probability that the ball drawn is : (i) white (ii) red (iii) not black (iv) red or white

26. A card is drawn from a well shuffled pack of 52 cards. Find the probability that the card is neither a red card nor a queen.

27. A bag contains 5 white balls, 7 red balls, 4 black balls and 2 blue balls. One ball is drawn at random from the bag. What is the probability that the ball drawn is : (i) white or blue (ii) red or black (iii) not white (iv) neither white nor black

28. A card is drawn at random from a well-shuffled deck of playing cards. Find the probability that the card drawn is : (i) a card of spade or an ace (ii) a red king (iii) neither a king nor a queen (iv) either a king or a queen

29. Cards marked with numbers 3, 4, 5, ...., 50 are placed in a box and mixed thoroughly. One card is drawn at random from the box. Find the probability that number on the card drawn is : (i) divisible by 7 (ii) a number which is a perfect square

30. A bag contains 5 red balls and some blue balls. If the probability of drawing a blue ball from the bag is four times that of a red ball, find the number of blue balls in the bag.

31. 1000 tickets of a lottery were sold and there are 5 prizes on these tickets. If Saket has purchased one lottery ticket, what is the probability of winning a prize?

32. A child has a block in the shape of a cube with one letter written on each face as shown below :

A B C D E A

The cube is thrown once. What is the probability of getting : (i) A ? (ii) D ? 33. Two customers are visiting a particular shop in the same week (Tuesday to Saturday). Each is

equally likely to visit on any one day as on another. What is the probability that both will visit the shop on : (i) the same day (ii) different days? (iii) Consecutive days

34. The letters A, B, C, A, D, E, B, A, B and F are marked on different cards such that one card has only one letter on it. A man draws one card. Find the probability that the card drawn is marked : (i) letter A (ii) letter B (iii) letter B or D (iv) letter A or C

35. A bag contains 36 balls out of which x are black. (i) If one ball is drawn at random, what is the probability of getting a black ball?(ii) If 12 more black balls are put in the bag, the probability of drawing a black ball is ½. Then find the value of x.

36. From a pack of 52 playing cards jacks, queens, kings and aces of red colour are removed. From the remaining, a card is drawn at random. Find the probability that the card drawn is : (i) a black queen (ii) a red card (iii) a black jack (iv) a picture card (Jacks, queens and kings

are picture cards) 37. A letter is chosen at random from the letters of the word ‘MATHEMATICS’. Find the probability

that the letter chosen is a (i) vowel (ii) consonant. 38. Ram and Shyam are friends. What is the probability that both will have : (i) different birthday ?

(ii) the same birthday? (ignore the leap year). 39. In a family, there are 3 children. Assuming that the chances of a child being a male or a female

are equal, find the probability that : (i) there is one girl in the family. (ii) there is no male child in the family. (iii) there is at least one male child in the family.

40. A number x is selected from the numbers 1, 2, 3 and then a second number y is selected randomly from the numbers 1, 4, 9. What is the probability that the product xy < 9?

41. A number is selected randomly from all possible 3-digit numbers. What is the probability that the number selected is : (i) an odd number (ii) having all 3 digits same (iii) divisible by 3

42. If x and y are natural numbers such that 1≤x≥4 and 3 ≤y≤6. What is the probability that :43. (i) x +y ≥8 (ii) xy is even.