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**CBSE Sample Papers for Class 10 SA2 Maths Solved 2016 Set 9**

** Section A**

1.If 2 is a root of the equation x^{2} + bx + 12 = 0 and the equation has equal roots, find the value of b.

4.If the perimeter and the area of a circle are numerically equal, then find the radius of the circle.

** Section B**

5.If sin a and cos a are the roots of the equation ax^{2} + bx + c = 0, then prove that a^{2} + 2ac = b^{2}.

6.Determine k so that 3k – 3, 2k^{2} – 5k + 7 and Ak + 2 are the three consecutive terms of an AP.

7.In figure, two tangents PA and PB are drawn to a circle with centre O from an external point P. Prove that triangle APB = 2 triangle OAB.

8.Ace, Jack and queen of diamonds are removed from a deck of 52 playing cards. One card is selected from the reamaining cards. Find the probability of getting a card of diamond.

9.In given figure, PQR is a triangle right angled at P, with PQ = 21 cm and PR = 42 cm with the vertices P, Q and R as centres, arcs are drawn each of radius 10 cm. Find the area of the shaded region. (Use pi = 3.14)

10.A cone and a sphere have equal radii and equal volume. What is the ratio of the diameter of the sphere to the height of the cone?

** Section C**

11.At t minutes past 2 p.m. the time needed by the minutes hand of a clock to show 3 p.m. was found to be 3 minutes less than t^{2}/4 minutes. Find t.

12.Solve for x :6X2 – 6(a + b)x + (4/3 a^{2}+10/3ab+4/3 b^{2})= 0

13.The sum of n terms of an AP whose first term is 6 and common difference is 40 is equal to the sum of 2n terms of another AP whose first term is 40 and common difference is 6. Find n.

14.Draw a pair of tangents to a circle of radius 4 cm which are inclined to each other at an angle of 45 °.

15.Cards bearing numbers 1, 4, 7, 10,….58 are kept in a bag. A card is drawn at random from the bag. Find

the probability of getting a card bearing

(i)a prime number less than 18.

(ii)a number divisible by 4.

16.Six cards – ace, jack, queen, king, ten and nine of spade are well shuffled with their face downwards. One card is then picked up at random.

(i) What is the probability that the card is an ace?

(ii) If the ace is drawn and put aside, what is the probability that the second card picked up is

(a) a jack,(b) an ace.

.

17.If P is a point lying on the line segment QR, joining Q(-l, -1) and R(4, -1) such that PQ = 3/7 QR, then find the coordinates of P.

18.For what value of k, (k > 0) the area of triangle with vertices (k, 1), (3k, 1) and (2, 4) is 6 sq. units.

19.A bucket is raised from a well by means of a rope which is wound round a wheel of diameter 154 cm. If bucket ascends in 2 minutes 56 seconds with a uniform speed of 2.2 m/s, then calculate the number of

complete revolutions the wheel makes in raising the bucket. (Use pi = 22/7)

20.A solid cone of base radius 30 cm is cut into two parts through the mid-point of its height by a plane parallel to its base. Find the ratio of the volumes of the two parts of the cone.

** Section D**

21.A man bought a certain number of toys for Rs 180; he kept one for his own use and sold the rest for one rupee each more than he gave for them, besides getting his own toy for nothing he made a profit of Rs 10. Find the number of toys.

22.A ladder has rungs 50 cm apart. The rungs decrease uniformly in length from 90 cm at the bottom to 50 cm at the top. If top and bottom rungs are 5 m apart, what is the length of the wood required for the rungs?

23.ABC is a right-angled triangle, right angled at A. A circle is inscribed in it. The lengths of two sides containing the right angle are 24 cm and 10 cm. Find the radius of the incircle.

24.In figure the common tangent, AB and CD to two circles with centres O and O’ intersect at E.Prove that the points O, E and O’ are collinear.

25.In the given figure, the diameters of two wheels have measures 4 cm and 2 cm. Determine the lengths of the belts AD and BC that pass around the wheels if it is given that belts cross each other at right angles.

26.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, 6 seconds 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 from this point.

27. A fire in a building B is reported on telephone to two fire stations P and Q, 20 km apart from each other on a straight road. P observes that the fire is at an angle of 60 ° to the road and Q observes that it is at an angle of 45 ° to the road. ‘

(a) Which station should send its team and how much will this team have to travel?

(b) What according to you, are the values displayed by the teams at fire stations P and Q?

28.The opposite angular points of a square are (2, 0) and (5, 1). Find the remaining points.

29.In the given figure, three circles of radius 2 cm touch one another externally. These circles are circumscribed by a circle of radius R cm. Find the value of R and the area of the shaded region.

30.The area of the base of a cone is 770 cm2 and the curved surface area is 814 cm2. Find the volume of the cone.

31.An agricultural field is in the form of a rectangle of length 20 m and width 14 m. A pit 6 m long, 3 m wide and 2.5 m deep is dug in the corner of the field and the earth taken out of the pit is spread uniformly over the remaining area of the field. Find the extent to which the level of the field has been raised.

**Answers**

** Section A**

**1.If 2 is a root of the equation x ^{2} + bx + 12 = 0 and the equation has equal roots, find the value of b.**

**Ans.**

**Ans.**

**Ans.**

**4.If the perimeter and the area of a circle are numerically equal, then find the radius of the circle.**

**Ans.**

** Section B**

**5.If sin a and cos a are the roots of the equation ax ^{2} + bx + c = 0, then prove that a^{2} + 2ac = b^{2}.**

**Ans.**

**6.Determine k so that 3k – 3, 2k ^{2} – 5k + 7 and Ak + 2 are the three consecutive terms of an AP.**

**Ans.**

**7.In figure, two tangents PA and PB are drawn to a circle with centre O from an external point P. Prove that triangle APB = 2 triangle OAB.**

**Ans.**

**8.Ace, Jack and queen of diamonds are removed from a deck of 52 playing cards. One card is selected from the reamaining cards. Find the probability of getting a card of diamond.**

**Ans.**

**9.In given figure, PQR is a triangle right angled at P, with PQ = 21 cm and PR = 42 cm with the vertices P, Q and R as centres, arcs are drawn each of radius 10 cm. Find the area of the shaded region. (Use pi = 3.14)**

**Ans.**

**10.A cone and a sphere have equal radii and equal volume. What is the ratio of the diameter of the sphere to the height of the cone?**

**Ans.**

** Section C**

**11.At t minutes past 2 p.m. the time needed by the minutes hand of a clock to show 3 p.m. was found to be 3 minutes less than t ^{2}/4 minutes. Find t.**

**Ans.**

**12.Solve for x :6X2 – 6(a + b)x + ((4/3) a ^{2}+(10/3)ab+(4/3) b^{2})= 0**

**Ans.**

**13.The sum of n terms of an AP whose first term is 6 and common difference is 40 is equal to the sum of 2n terms of another AP whose first term is 40 and common difference is 6. Find n.**

**Ans.**

**14.Draw a pair of tangents to a circle of radius 4 cm which are inclined to each other at an angle of 45 °.**

**Ans.**

**15.Cards bearing numbers 1, 4, 7, 10,….58 are kept in a bag. A card is drawn at random from the bag. Find**

**the probability of getting a card bearing**

**(i)a prime number less than 18.**

**(ii)a number divisible by 4.**

**Ans.**

**16.Six cards – ace, jack, queen, king, ten and nine of spade are well shuffled with their face downwards. One card is then picked up at random.**

**(i) What is the probability that the card is an ace?**

**(ii) If the ace is drawn and put aside, what is the probability that the second card picked up is**

**(a) a jack,(b) an ace.**

**Ans.**

**17.If P is a point lying on the line segment QR, joining Q(-l, -1) and R(4, -1) such that PQ = 3/7 QR, then find the coordinates of P.**

**Ans.**

**18.For what value of k, (k > 0) the area of triangle with vertices (k, 1), (3k, 1) and (2, 4) is 6 sq. units.**

**Ans.**

**19.A bucket is raised from a well by means of a rope which is wound round a wheel of diameter 154 cm. If bucket ascends in 2 minutes 56 seconds with a uniform speed of 2.2 m/s, then calculate the number of ****complete revolutions the wheel makes in raising the bucket. (Use pi = 22/7)**

**Ans.**

**20.A solid cone of base radius 30 cm is cut into two parts through the mid-point of its height by a plane parallel to its base. Find the ratio of the volumes of the two parts of the cone.**

**Ans.**

** Section D**

**21.A man bought a certain number of toys for Rs 180; he kept one for his own use and sold the rest for one rupee each more than he gave for them, besides getting his own toy for nothing he made a profit of Rs 10. Find the number of toys.**

**Ans.**

**22.A ladder has rungs 50 cm apart. The rungs decrease uniformly in length from 90 cm at the bottom to 50 cm at the top. If top and bottom rungs are 5 m apart, what is the length of the wood required for the rungs?**

**Ans.**

**23.ABC is a right-angled triangle, right angled at A. A circle is inscribed in it. The lengths of two sides containing the right angle are 24 cm and 10 cm. Find the radius of the in circle.**

**Ans.**

**24.In figure the common tangent, AB and CD to two circles with centres O and O’ intersect at E.Prove that the points O, E and O’ are collinear.**

**Ans.**

**25.In the given figure, the diameters of two wheels have measures 4 cm and 2 cm. Determine the lengths of the belts AD and BC that pass around the wheels if it is given that belts cross each other at right angles.**

**Ans.**

**26.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, 6 seconds 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 from this point.**

**Ans.**

**27. A fire in a building B is reported on telephone to two fire stations P and Q, 20 km apart from each other on a straight road. P observes that the fire is at an angle of 60 ° to the road and Q observes that it is at an angle of 45 ° to the road. ‘**

**(a) Which station should send its team and how much will this team have to travel?**

**(b) What according to you, are the values displayed by the teams at fire stations P and Q?**

**Ans.**

**28.The opposite angular points of a square are (2, 0) and (5, 1). Find the remaining points.**

**Ans.**

**29.In the given figure, three circles of radius 2 cm touch one another externally. These circles are circumscribed by a circle of radius R cm. Find the value of R and the area of the shaded region.**

**Ans.**

**30.The area of the base of a cone is 770 cm2 and the curved surface area is 814 cm2. Find the volume of the cone.**

**Ans.**

**31.An agricultural field is in the form of a rectangle of length 20 m and width 14 m. A pit 6 m long, 3 m wide and 2.5 m deep is dug in the corner of the field and the earth taken out of the pit is spread uniformly over the remaining area of the field. Find the extent to which the level of the field has been raised.**

**Ans.**

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