Time allowed: 180 minutes; Maximum Marks: 90

General Instructions:
1)  All questions are compulsory.
2) The question paper consists of thirty questions divided into 4 sections A, B, C and D. Section A comprises of ten questions of 01 mark each, Section B comprises of five questions of 02 marks each, Section C comprises ten questions of 03 marks each and Section D comprises of five questions of 06 marks each.
3)  All questions in Section A are to be answered in one word, one sentence or as per the exact requirement of the question.
4)  There is no overall choice. However, internal choice has been provided in one question of 02 marks each, three questions of 03 marks each and two questions of 06 marks each. You have to attempt only one of the alternatives in all such questions.
5)  In question on construction, drawing should be near and exactly as per the given measurements.
6) Use of calculators is not permitted.


Question 1

1. The first three terms of an A.P. respectively are 3y - 1, 3y + 5 and 5y + 1. Then y equals:

Question 2

2. In Fig, QR is a common tangent to the given circles, touching externally at the point T. The tangent at T meets QR at P. If PT = 3.8 cm, then the length of QR (in cm) is :

Question 3

3. In Fig, PQ and PR are two tangents to a circle with centre O. If ∠QPR = 46°, then ∠QOR equals:

Question 4

4. A ladder makes an angle of 60° with the ground when placed against a wall. If the foot of the ladder is 2 m away from the wall, then the length of the ladder (in metres) is:

Question 5

5. If two different dice are rolled together, the probability of getting an even number on both dice, is:

Question 6

6. A number is selected at random from the numbers 1 to 30. The probability that it is a prime number is:

Question 7

7. If the points A(x, 2), B(−3, −4) and C(7, − 5) are collinear, then the value of x is:

Question 8

8. The number of solid spheres, each of diameter 6 cm that can be made by melting a solid metal cylinder of height 45 cm and diameter 4 cm, is:


Question 9

9. Solve the quadratic equation 2x2 + ax − a2 = 0 for x.

Question 10

10. First and the last terms of an A.P. are 5 and 45 respectively. If the sum of all its terms is 400, find its common difference.

Question 11

11. Prove that the line segment joining the points of contact of two parallel tangents of a circle, passes through its centre.

Question 12

12. If from an external point P of a circle with centre O, two tangents PQ and PR are drawn such that ∠QPR = 120°, prove that 2PQ = PO.

Question 13

13. Rahim tosses two different coins simultaneously. Find the probability of getting at least one tail.

Question 14

14. In fig, a square OABC is inscribed in a quadrant OPBQ of a circle. If OA = 20 cm, find the area of the shaded region. (Use π = 3.14)


Question 15

15. Solve the equation 4x - 3 = 52x+3; x≠0, −32, for x.

Question 16

16. If the seventh term of an A.P. is 19 and its ninth term is 17, find its 63rdterm.

Question 17

17. Draw a right triangle ABC in which AB = 6 cm, BC = 8 cm and ∠B = 90°. Draw BD perpendicular from B on AC and draw a circle passing through the points B, C and D. Construct tangents from A to this circle.

Question 18

18. If the point A (0, 2) is equidistant from the points B (3, p) and C (p, 5), find p. Also find the length of AB.

Question 19

19. Two ships are there in the sea on either side of a light house in such a way that the ships and the light house are in the same straight line. The angles of depression of two ships as observed from the top of the light house are 60° and 45°. If the height of the light house is 200 m, find the distance between the two ships. [Use 3 = 1.73]

Question 20

20. If the points A(−2, 1), B(a, b) and C(4, −1) are collinear and a − b = 1, find the values of a and b.

Question 21

21. In Fig., a circle is inscribed in an equilateral triangle ABC of side 12 cm. Find the radius of the inscribed circle and the area of the shaded region. [Use π = 3.14 and 3 = 1.732]

Question 22

22. In Fig, PSR, RTQ and PAQ are three semicircles of diameters 10 cm, 3 cm and 7 cm respectively. Find the perimeter of the shaded region. [Use π = 3.14]

Question 23

23. A farmer connects a pipe of internal diameter 20 cm from a canal into a cylindrical tank which is 10 m in diameter and 2 m deep. If the water flows through the pipe at the rate of 4 km per hour, in how much time will the tank be filled completely?

Question 24

24. A solid metallic right circular cone is 20 cm high and whose vertical angle is 60°, is cut into two parts at the middle of its height by a plane parallel to its base. If the frustum so obtained be drawn into a wire of diameter 112 cm, find the length of the wire.


Question 25

25. The difference of two natural numbers is 5 and the difference of their reciprocals is 110. Find the numbers.

Question 26

26. Prove that the lengths of the tangents drawn from an external point to a circle are equal.

Question 27

27. The angles of elevation and depression of the top and the bottom of a tower from the top of a building, 60 m high, are 30° and 60° respectively. Find the difference between the heights of the building and the tower and the distance between them.

Question 28

28. A bag contains cards numbered from 1 to 49. A card is drawn from the bag at random, after mixing the cards thoroughly. Find the probability that the number on the drawn card is:

(i) an odd number

(ii) a multiple of 5

(iii) a perfect square

(iv) an even prime number

Question 29

29. Find the ratio in which the point P(x, 2) divides the line segment joining the points A (12, 5) and B (4, - 3). Also find the value of x.

Question 30

30. Find the values of k for which the quadratic equation (k + 4) x2+ (k + 1) x + 1 = 0 has equal roots. Also find these roots.

Question 31

31. In an A.P. of 50 terms, the sum of first 10 terms is 210 and the sum of its last 15 terms is 2565. Find the A.P.

Question 32

32. Prove that a parallelogram circumscribing a circle is a rhombus.

Question 33

33. Sushant has a vessel, of the form of an inverted cone, open at the top, of height 11 cm and radius of top as 2.5 cm and is full of water. Metallic spherical balls each of diameter 0.5 cm are put in the vessel due to which 25thof the water in the vessel flows out. Find how many balls were put in the vessel. Sushant made the arrangement so that the water that flows out irrigates the flower beds. What value has been shown by Sushant?

Question 34

34. From a solid cylinder of height 2.8 cm and diameter 4.2 cm, a conical cavity of the same height and same diameter is hollowed out. Find the total surface area of the remaining solid. (Take π =227)