Amity International SchoolMaths Hots

Class X

Ch :1

1. State Euclid’s division lemma.2. Why is 13/120 a non-terminating rational number?3. If 34.0356 is expressed in the form p/q, where pand q are coprime integers, then

what can you say about the prime factorization of q?4. State whether 6.34, 47.536921 and 3.21001000100001….are rational or not. Give a

reason in each case.5. Express 156 as a product of its prime factors.6. Find the HCF of 344 and 60 by the prime factorization method . Hence find their LCM ?

7. If 0.2316 is expressed in the form _____P______ for smallest i. 2n × 5m

8. Values of whole numbers n and m. Write these values of n and m ?

9. Show that 17 x 31 x 41 + 41 is a composite number ?

10. By applying the fundamental theorem of arithmetic, find the HCF and LCM of 11. positive integers 540 and 408 ?

12. By using division algorithm , find the largest number which when divides 967 and 2060 the remainders obtained are 7 and 12 respectively.

13. Prove that 2 3 – 7 2 is an irrational number ?

Ch :2

1. Write a polynomial whose zeroes are ½ and -2.2. Write a polynomial whose sum and product of zeroes are -5 and 6 respectively.3. Find a cubic polynomial with the sum, sum of the product of its zeroes taken two at a

time, and product of its zeroes as 5, -6 and -20 respectively.

4. If ά, β are the roots (zeroes) of the quadratic polynomial p(x) = x2 – (k + 6)x + 2 (2k – 1). Find the value of k, if ά + β = 1 ά β.

i. 2

5. If ά, β are the zeroes of x2 + 7x + 7, find the value of 1 + 1 – 2 ά βi. ά β

6. If m and n are the zeroes of ax2 – 5x + c, find the value of a and c when m + n = mn = 10.

7. Find a cubic polynomial with the sum, sum of the product of its zeroes taken two at a time, and product of its zeroes as 5, -6 and -20 respectively.

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8.If the polynomial f(x) = x4 – 6x3 + 16x2 – 25x + 10 is divided by another polynomial x2 – 2x + k, the reminder comes out to be x + a, find the values of k and a.

9. What must be added to the polynomial p(x) = x4 + 2x3 – 2x2 + x – 1 so that the resulting polynomial is exactly divisible by x2 + 2x – 3 ?

10 On dividing x3 – 3x2 + x + 2 by a polynomial g(x), the quotient and remainder are x – 2 and -2x + 4 respectively. Find g(x).

Ch :31. Solve graphically the system: 2x - 3y = 1 3x - 4y = 1 Does the point (3, 2) lie on any of the lines? Write its equation.

2. Find the values of a and b for which the following pairs of linear equations have an infinite number of solutions:(1) (a + b)x – 2by = 5a + 2b + 13x – y = 14(2) 2x + 3y = 7(a - b)x + (a + b)y = 3a + b – 2

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7. Using elimination method, solve the following pair of equations for x and y: (a + 2b)x + (2a - b)y = 2 (a - 2b)x + (2a + b)y = 3

8. Out of 940km journey, a part of the journey was covered by a motor car with speed of 72km/h. the rest of the journey was covered by train at a speed of 84km/h. If the total distance was covered in 12 hours, find the distance travelled by the train and the time taken by it to cover that distance.

9. Ten years ago, the sum of the ages of two sons was one-third of their father’s age. One son is two years older than the other and the sum of their present ages is 14 years less than the father’s present age. Find the present ages of all.

10.

11. Draw the graph for the system of equations x + y = 5 and 2x - y + 2=0. Shade the region bounded by these lines and the x-axis. Find the area of the shaded region.

12. Solve graphically the system: 2x - 3y = 1 3x - 4y = 1 Does the point (3, 2) lie on any of the lines? Write its equation.

Ch :4

1. If one root of a quadratic equation is ( 3 + 2√5 )/ 4,then what is the other root?

2. If -2 is the root of the equation 3x2 – 5x + 2k = 0, then what is the value of k?

3. For what value of p , will the following quadratic equation have equal roots? 2x2 – 6x + p = 0

4. If the roots of a ( b-c) x2 + b ( c-a ) x + c ( a – b ) = 0 be equal if 1 + 1 = 2 a c b5. Using quadratic formula solve ( b-c) x2 + ( c-a ) x + ( a – b ) = 06. Solve for x : a + b = a + b ax -1 bx -17. A certain number of soldiers stand for a parade , in such a way that the number of soldiers in each row is 2 more than the total number of rows.. If the number of rows are doubled and the

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number of soldiers in each row is reduced by 7 , the total number of soldiers is increased by 160. Find the original number of soldiers in each row.

8.

10. Show that the quadratic equation x2 + ax - 1 = 0 has real and distinct roots for all real values of a .

11. Solve 2 √ 2 x2 + √ 15 x + √ 2 = 0 by using the method of completing the squares.

12. For what value of α the equation (α – 3 ) x2 + 4 (α – 3 ) x + 4 = 0 represent a perfect square.

13. If a boy’s age and his father’s age amount together to 24 years , also fourth part of the product of their ages exceeds the boys age by 9 years. Find their present ages.

14. A ladder is 10 feet long leans against a wall. The bottom of the ladder is 6 feet from the wall. Then the bottom of the ladder is pulled out 3 feet further. How much does the top end move down the wall.

15. Amar , Akbar and Antony solve a given quadratic equation . Amar commits a mistake in the constant term and finds the roots as 8 and 2. Akbar commits a mistake in the coefficient of x and finds the roots as -9 and -1.If Anthony solves the equation without making any mistakes , find the correct roots.

Ch :5

1. Write the first 3 terms of an A.P. whose nth term is -5 + 2n.2. What is the common difference of the A.P.

a + b, (a + 1) + b, ( a + 1) + (b + 1), (a + 2) + (b + 1),…..3. the pth term of an AP is 1/q and the q th term is 1/p . Show that the sum pf pq terms is ½ ( pq + 1 ).4. Find the sum of all three digit numbers which leave the remainder 2 when divided by 5.

5. How many terms of the AP 72 , 69 ,66 - - - - make the sum 897. Explain the double answer.

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6.

7. S1 , S2 and S 3 are the sum of n , 2n and 3n terms of an AP. Show that S 3 = 3 (S2 - S1)

8. If 3 + 5 + 7 - - - - - - - - n terms = 7 find the value of n. 5 + 8 + 11 - - - - - -10 terms

9. First term of an AP is a and the last term is b with common difference 1 .Show that the sum of the series is ½ ( a + b ) ( 1 – a + b ).

10. The ratio of the 7th and the 3rd term of an AP is 12 : 5. Find the ratio of 13th to the . 4th term .

11. If the pth term of an AP is 1/q and the q th term is 1/p . Show that the sum pf pq terms is ½ ( pq + 1 ).

12. Find the sum of all three digit numbers which leave the remainder 2 when divided by 5.

13 How many terms of the AP 72 , 69 ,66 - - - - make the sum 897. Explain the double answer.

14. What is the common difference of the A.P. a + b, (a + 1) + b, ( a + 1) + (b + 1), (a + 2) + (b + 1),…..

15.

Ch :6

1. In a triangle ABC , N is a point on AC such that BN | AC. If BN2 = AN.NC, Proove that / .B=90º

2. In the figure XY // AC and XY divides triangular region ABC into two parts equal in area. Determine AX .

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3. In a triangle ABC , AD is a median, prove that AB ² + AC ² = 2 AD ² + 2 DC ² .

4. In a right angled triangle ABC in which C = 90 , if D is the midpoint of BC , prove that AB ² = 4 AD ² - 3 AC ² .

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6.

In triangle ABC, is AD is the bisector of A , then area ABD = AB area ACD AC

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7. Let X be any point on the side BC of a triangle ABC. If XM, XN are drawn parallel to BA and CA meeting CA , BA in M, N respectively. MN meets BC produced in T. Prove that TX ² = TB X TC.

8. In a triangle ABC right angled at B, D and E trisect BC. Prove that 8 AE² = 3 AC² + 5 AD² .

9. In triangle ABC, ABC = 135 , Prove that AC ² = AB ² + BC ² + 4 ar ( ABC) .

10. A point D is on the side BC of an equilateral triangle ABC such that DC = ¼ BC. Prove that AD ² = 13 CD ² .

11. In a triangle ABC , C is an obtuse angle. AD is perpendicular to BC and AB ² = AC ² + 3 BC ². Prove that BC = CD.

12. Prove that in an equilateral triangle three times the square of a side is equal to four times the square of its altitude.

13. In a triangle ABC , AD is a median, prove that AB ² + AC ² = 2 AD ² + 2 DC ² .

Ch :7

1. What is the distance of the point P(4,2) from the x-axis?2. If the end points of a diameter of a circle are (-2,3) and (4,-5), find the coordinates of its centre3. Find the distance between the points A ( cos θ , sin θ ) and (sin θ , - cos θ )

4. If ( x , y ) is a point on the line joining ( a , 0 ) and ( 0 , b ) then show that

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x/a + y/b = 1

5. The line segment joining the points ( 3 , -4) and ( 1 , 2) is trisected at points P and Q . If the co-ordinates of P and Q are ( p , -2 ) and ( 5/3 , q ) respectively , find The values of p and q.

6. Find the are of the triangle formed by the points ( p + 1, 1 ) , ( 2p + 1 , 3 ) & ( 2p + 2 , 2p ) and show that these points are collinear if p = 2 or ½.

7. If the distance between the points (0,y) and (8,9) is 8 units, find the value of y.

8. Determine the ratio in which the point P ( a , -2 ) divides the join of A ( -4 , 3 ) and B ( 2 , - 4 ) Also find the value of a.

9. If a is the length of one of the sides of an equilateral triangle ABC , base BC lies on the x – axis and vertex B is at the origin , find the co ordinates of the vertices.

10. The opposite angular points of a square are ( 5 , 4 ) and ( -1 , 6 ). Find the co ordinates of the remaining vertices.

11. Four points A ( 6 , 3 ) , B ( -3 , 5 ) , C ( 4 , -2) and D ( x , 3x ) are given in such a way that ar ∆ DBC : ar ∆ ABC = 1 : 2 , find x.

Ch :8

1. If sin θ + cos θ = 1, then find the value of sin θ cos θ.

2. a) If Sin 3 A =1 find tan2 A-

b) If cot 2 A -3 = 0 find sin 2 A

3. If sin = find the value of tan + sec .

4. If tan , prove that = .

5. If is an acute angle and sin = cos , find the value of 3 tan2 + 2 sin2 -1.

6. Find the angles A, B and C, if tan(A + B - C) = 1, sin(B + C - A) = 1 and cos (C + A - B) = 1.

7. If sec = x + , then prove that sec +tan = 2x or .8. If tan + sin = m and tan - sin = n, then show that m2 – n2 = 4 .

9. If cosec + cot = p, show that (p2 + 1) cos = p2 -1

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10.

Ch :9

1) A vertical tower stands on a horizontal plane and is surmounted by a vertical flag Staff of height h. At a point on the plane , the angle of elevation of the bottom of the flag staff is α and that of the top of the flag staff is β. Prove that the height of the tower is h tan α Tan β – tan α2. The angle of elevation of a cloud from a point 200 metres above a lake is 30 0 andThe angle of depression of its reflection in the lake is 60 0. Find the height of the cloud.

3. A man standing on a window of the first floor of a building observes that the angle of depression of dustbin which is 10 m from the foot of the building is 45º.He climbs to the window of the second floor, directly above the first floor and observes the angle of depression of dustbin to be 60º. Calculate the height of the first floor and the second floor.

4. ) A round balloon of radius a subtends an angle θ at the eye of the observer while angle of elevation of its centre is φ . Prove that the height of the centre of the balloon is a sin φ . cosec θ/2

5. At the foot of a mountain , the elevation of its summit is 450 . After ascending 1 km towards the mountain up at an inclination of 30 0 , the elevation changes to 60 0.Find the height of the mountain.

6. The height of a house subtends a right angle at the opposite window. The angle of Elevation of the window from the base of the house is 60 0 . If the width of the road is 6 m , find the height of the house.

7. A ladder rests against a wall at an angle α to the horizontal. Its foot is pulled away From the wall through a distance a , so that it slides a distance b down the wall Making an angle β with the horizontal. Show that a = cos β - cos α b sin α – sin β

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10.A man standing on a window of the first floor of a building observes that the angle of depression of dustbin which is 10 m from the foot of the building is 45º.He climbs to the window of the second floor, directly above the first floor and observes the angle of depression of dustbin to be 60º. Calculate the height of the first floor and the second floor.

Ch :10

1. Two tangents PA and PB are drawn to a circle with centre O from an external point P. Prove that APB = 2 OAB.

2. AB is a chord of length 6 cm of circle of radius 5 cm. The tangents at A and B intersect at a point X. Find length of XA.

3. A circle is touching the side BC of ABC at P and touching AB and AC produced at Q and R respectively. Prove that AQ =AR =

(Perimeter of ABC). Given AQ = 5cm, find the perimeter of ABC.

4.

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A

B

OX Y5 cm3cm

5. Calculate the value of ‘x’ in the figure below, if TN is a tangent.

6. Two tangent segments BC and BD are drawn to a circle C (O, r) such that CBD = 120º. Prove that BO = 2BC.

7. A circle is inscribed in a ABC having sides 8 cm, 10 cm, 12 cm, as shown in the adjoining figure. Find AD, BE and CF.

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

9. .Two circles touches each others externally at C. Prove that the common tangent at C bisects the other two common tangents.

10. The radii of two concentric circles are 13 cm and 8 cm. AB is a diameter of the bigger circle. BD is a tangent to the smaller circle, touching it at D. Find the length AD

Ch:11

Q.1 Draw a line segment of length 6cm and divide it in the ratio 2:3.Measure the two

parts.

Q.2 Draw a line segment of length 8cm and divide in the ratio 5:3. Measure the two

parts.

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A B

C

10 cm 8 cm

D

F E

12 cm

30ºA

B

O

N

x

T

Q.3 Construct a triangle similar to the given triangle with sides 5cm, 6cm and 8cm

and whose sides are

i) 4/7th, ii) 7/4th of the corresponding sides of the given triangle.

Q.4 Construct a Δ ABC in which AB=5cm, ∟B=600 and the altitude CD=3cm.

Construct a Δ AQR similar to Δ ABC such that each side of Δ AQR is 1.5 times the

corresponding side of ΔABC.

Q.5 Construct a ΔABC in which BC=6cm, ∟A=600 and median through A is 4.5cm

long. Draw a ΔA’BC’similar to ΔABC, having base BC’ =7.5cm. Write the steps of

construction involved.

Q.6 Construct an isosceles triangle whose base is 8cm and altitude 4cm, and then

triangle whose sides are 1½ times the corresponding sides of the isosceles triangle.

Q.7 Draw a triangle ABC with side BC=7cm, ∟B=450, ∟A=1050. Then, construct a

triangle whose sides are 4/3 times the corresponding sides of ΔABC.

Q.8 Draw a right triangle in which the sides(other than hypotenuse) are of lengths 4cm

and 3cm. Then, construct another triangle whose sides are 5/3 times the corresponding

sides of the given triangle.

Q.9 Construct a circle of radius 3.5cm. From a point on the concentric circle of radius

6.5cm, draw a tangent to the first circle and measure the length of the tangent drawn.

Also, find the length of the tangent by actual calculation.

Q.10 Let ABC be a right triangle in which AB=6cm.BC=8cm and ∟B=900. BD is the

perpendicular from B on AC. The circle through B,C,D is drawn. Construct the tangents

from A to this circle.

Q.11 Draw a circle with the help of a bangle. Take a point outside the circle. Construct

the pair of tangents from this point to the circle.

Ch :12

1. Two circles touch each other externally. The sum of their areas is 149 cm ² and the distance between their centres is 17 cm. Find the radii of the circles.

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2. The wheels of a car are of diameter 70 cm. each. How many complete revolutions does each wheel make in one minute when the car is traveling at a speed of 52.8 km. per hour.

3. PQRS is a diameter of a circle of radius 6 cm. The lengths PQ, QR and RS are equal. Semicircles are drawn on PQ and QS as diameters. Find the perimeter of the shaded region.

4. In the given figure OPQR is a rhombus, three of whose vertices lie on a circle with centre O. If the area of the rhombus is 32 3 cm ². Find the radius of the circle ?

5. An equilateral triangle ABC is inscribed in a circle of radius 32 cm. Prove that the Area of the shaded region is 256 X (4 - 3 3 ) cm ² .

6. Two circles touch each other externally. The sum of their areas is 149 cm ² and the distance between their centres is 17 cm. Find the radii of the circles.

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7. The wheels of a car are of diameter 70 cm. each. How many complete revoluitions does each wheel make in one minute when the car is traveling at a speed of 52.8 km. per hour.

8. The short and long hands of a clock are 4 cm. and 6 cm. respectively. Find the sum of the distances traveled by their tips in two days.

9. If a chord of a circle of radius r subtends an angle of 60 at the centre of the circle, prove that the area of the corresponding segment of the circle is [ - 3 r sq units. 6 4

10. AB and CD are two diameters opf a circle perpendicular to each other. A smaller circle is drawn on OB as diameter. If AB = 16 cm. prove that the area of the shaded region is equal to 16 ( 3 - 4 ) cm ² .

Ch :13

1. The adjoining figure shows the cross-section of an ice-cream consisting of a cone surmounted by a hemisphere. The radius of the hemisphere is 3.5 cm and the height of the cone is 10.5 cm. The outer shell ABCDFE is shaded and is not filled with ice cream. AE = DC = 0.5 cm, AB ║ EF and BC ║ FD. Calculate.

2. A solid is in the form of a right circular cone mounted on a hemisphere. The radius of the hemisphere is 3.5 cm and the height of the cone is 4 cm. The solid is placed in a cylindrical tub, full of water, in such a way that the whole solid is submerged in water. IF the radius of the cylinder is 5 cm and its height is 10.5 cm, find the volume of water left in the cylindrical tub.

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4.

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5. A comical vessel of radius 6cm and height 8 cm is completely filled with water. A sphere is lowered into water and its size is such that when it touches the sides, its just immersed. What fraction of the water overflows?

6. Two solid spheres made of the same metal have weights 5920 and 740 gm, respectively. Determine the radius of the heavier sphere, if the diameter of the lighter one is 5 cm.

7. The total surface area of a solid right circular cylinder is 23 cm2. Its curved surface is 2/3 of the total surface. Determine the radius of its base and height.

8. A solid iron pole consists of a cylinder of height 220 cm and base diameter 24 cm, which is surmounted by another cylinder of height 60 cm and radius 8 cm. Find the mass of the pole, given that 1 cu cm of iron has approximately 8 g mass. (Use π = 3.14)

9. A solid toy is in the form of a hemisphere surmounted by a right circular cone. The height of the cone is 2 cm and the diameter of the base is 4 cm. Determine the volume of the toy. If a right circular cylinder circumscribes the toy, find the difference in the volumes of the cylinder and the toy. (Use π = 3.14)

10. An exhibition tent is in the form of a cylinder surmounted by a cone. The height of the tent above he ground is 85 m and the height of the cylindrical part is 50 m. If the diameter of the base is 168 m, find the quantity of canvas required to make the tent. Allow 20% extra for folds and for stitching. Give your answer correct to the nearest sq m.

11. The adjoining figure shows the cross-section of an ice-cream consisting of a cone surmounted by a hemisphere. The radius of the hemisphere is 3.5 cm and the height of the cone is 10.5 cm. The outer shell ABCDFE is shaded and is not filled with ice cream. AE = DC = 0.5 cm, AB ║ EF and BC ║ FD. Calculate.

(i) the volume of the ice cream in the cone ( the unshaded portion including the hemisphere) in cu cm, and(ii) the volume of the outer shell ( the shaded portion) in cu cm.Give your answers correct to the nearest cu cm.

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Ch 14

Q.1 A survey was conducted by a group of students as a part of their environment awareness programme, in which they collected the following data regarding the number of plants in 20 houses in a locality. Find the mean number of plants per house.

Number of Plants0 – 2 2 – 4 4 – 6 6 – 8 8 – 10 10 – 12 12 - 14

Number of houses1 2 1 5 6 2 3

Which method did you use for finding the mean, and why?

Q.2 In a retail market, fruit vendors were selling mangoes kept in packing boxes. These boxes contained varying number of mangoes. The following was the distribution of mangoes according to the number of boxes.

Consider the following distribution of daily wages of 50 workers of a factory:

Number of mangoes50 – 52 52 – 54 54 – 56 56 – 58 58 – 60

Number of boxes15 110 135 115 25

Find the mean number of mangoes kept in a packing box. Which method of finding the mean did you choose?

Q.3 A survey conducted on 40 households in a locality by a group of students resulted in the following frequency table for the number of family members in a household:

Family size1 – 3 3 – 5 5 – 7 7 – 9 9 – 11

Number of families14 16 4 4 2

Find the mode of this data.

Q.4 The following table shows the ages of the patients admitted in a hospital during a year:

Age (in years)5 – 15 15 – 25 25 – 35 35 – 45 45 - 55 55 – 65

Number of patients6 11 21 23 14 5

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Find the mode and the mean of the data given above. Compare and interpret the two measures of central tendency.

Q.5 Find the mode of the following distribution:

Wages (in Rs) less than200 300 400 500 600 700 800 900

Number of Workers5 18 38 70 90 95 98 100

Q.6 100 surnames were randomly picked up from a local telephone directory and the frequency distribution of the number of letters in the English alphabets in the surnames was obtained as follows:

Number of letters1 – 4 4 – 7 7 – 10 10 – 13 13 – 16 16 – 19

Number of surnames6 30 40 16 4 4

Determine the median number of letters in the surnames. Find the mean number of letters in the surnames. Also, find the modal size of the surnames.

Q.7 A life insurance agent found the following data for distribution of ages of 100 policy-holders. Calculate the median age, if policies are given only to persons having age 18 years onwards but less than 60 years.

Age (in years) Number of policy-holders

Below 20

Below 25

Below 30

Below 35

Below 40

Below 45

Below 50

2

6

24

45

78

89

92

17

Below 55

Below 60

98

100

Q.8 If the median of the following frequency distributions is 46, find the missing frequencies:

Variable Frequency

10 – 20

20 – 30

30 – 40

40 – 50

50 – 60

60 – 70

70 – 80

12

30

f1

65

f2

25

18

Total 229

Q.9 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

Number of farms2 8 12 24 38 16

Change the distribution to a more than type distribution, and draw its Ogive.

Q.10 Draw the less than Ogive and more than Ogive for the following distribution and find the median graphically:

Daily wages (in Rs)150 – 155 155 – 160 165 – 170 165 – 170 170 – 175 175 – 180

Number of workers6 10 22 34 16 12

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Ch :15

1. A jar contains 24 marbles, some green and other blue. If a marble is drawn at random from the jar, the probability that it is green is 2. Find the number of blue marbles in the jar.

2. A box contains 90 discs which are numbered form 1 to 90. If one disc is drawn at random from the box, fins the probability that it bears (i) a 2 – digit number (ii) a perfect square number, and (iii) a number divisible by 5.

3. The king, queen and jack of clubs are removed from a pack of 52 playing cards. The remaining cards are them well shuffled and one card is selected at random. Find the probability of getting

(i) a heart. (ii) a king.(iii) a club (iv) the 10 of hearts.

4. All the tree face cards of spades are removed from a well – shuffled pack of 52 cards. A card is then drawn at random form the remaining pack. Find the probability of getting.

(i) a black face cards. (ii) a queen.(iii) a black card.

5. A game consists of tossing a Re 1 coin three times and noting its outcome each time. Hanif wins if all the tosses give the same realty, i.e., tree heads or three tails, and loses otherwise. Calculate the probability that Hanif will lost the game.

6. A box contains 90 discs which are numbered form 1 to 90. If one disc is drawn at random from the box, fins the probability that it bears (i) a 2 – digit number (ii) a perfect square number, and (iii) a number divisible by 5.

7. The king, queen and jack of clubs are removed from a pack of 52 playing cards. The remaining cards are them well shuffled and one card is selected at random. Find the probability of getting

(i) a Spade . (ii) a queen.

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9. Find the probability of getting 53 Fridays in a non leap year.

10. Two coins are tossed simultaneously. Find the probability of getting

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a) at least one head b) atmost one head c) Exactly one head d) no heads

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