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CBSE X
All India (2)
MATHS PAPER 2012

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.

SECTION A

### Question 1

1. If 1 is a root of the equations a${y}^{2}$ + ay + 3 = 0 and${y}^{2}$+ y + b = 0 then ab equals:

### Question 2

2. The sum of first 20 odd natural numbers is:

### Question 3

3. In Fig. 1, the sides AB, BC and CA of a triangle ABC, touch a circle at P, Q and R respectively. If PA = 4 cm, BP = 3 cm and AC = 11 cm, then the length of BC (in cm) is:

### Question 4

4. In Fig 2, a circle touches the side DF of ΔEDF at H and touches ED and EF produced at K and M respectively. If EK = 9 cm, then the perimeter of ΔEDF (in cm) is:

### Question 5

5. If the radius of the base of a right circular cylinder is halved, keeping the height the same, then the ratio of the volume of the cylinder thus obtained to the volume of original cylinder is:

### Question 6

6. If the area of a circle is equal to sum of the areas of two circles of diameters 10 cm and 24 cm, then the diameter of the larger circle (in cm) is:

### Question 7

7. The length of shadow of a tower on the plane ground is $\sqrt{3}$times the height of the tower. The angle of elevation of sun is:

### Question 8

8. If the coordinates of one end of a diameter of a circle are (2, 3) and the coordinates of its centre are (−2, 5), then the coordinates of the other end of the diameter are:

### Question 9

9. The coordinates of the point P dividing the line segment joining the points A (1, 3) and B (4, 6) in the ratio 2 : 1 are:

### Question 10

10. Two dice are thrown together. The probability of getting the same number on both dice is:

SECTION B

### Question 11

11. Find the value(s) of k so that the quadratic equation ${x}^{2}$−4kx + k = 0 has equal roots.

### Question 12

12. Find the sum of all three digit natural numbers, which are multiples of 11.

### Question 13

13. Tangents PA and PB are drawn from an external point P to two concentric circles with centre O and radii 8 cm and 5 cm respectively, as shown in Fig. 3. If AP = 15 cm, then find the length of BP.

### Question 14

14. In Fig. 4, an isosceles triangle ABC, with AB = AC, circumscribes a circle. Prove that the point of contact P bisects the base BC.

OR

In Fig. 5, the chord AB of the larger of the two concentric circles, with centre O, touches the smaller circle at C. Prove that AC = CB.

### Question 15

15. The volume of a hemisphere is 2425$\frac{1}{2}{\mathrm{cm}}^{3}$. Find its curved surface area. [ Use π = $\frac{22}{7}$]

### Question 16

16. In Fig. 6, OABC is a square of side 7 cm. If OAPC is a quadrant of a circle with centre O, then find the area of the shaded region. [ Use π = $\frac{22}{7}$]

### Question 17

17. If a point A (0, 2) is equidistant from the points B (3, p) and C (p, 5), then find the value of p.

### Question 18

18. A number is selected at random from first 50 natural numbers. Find the probability that it is a multiple of 3 and 4.

SECTION C

### Question 19

19. Solve for x: 4${x}^{2}$ − 4ax + (${a}^{2}$${b}^{2}$) = 0

OR

Solve for x: 3${x}^{2}$- 2$\sqrt{6}$x + 2 = 0

### Question 20

20. Prove that the parallelogram circumscribing a circle is a rhombus.

OR

Prove that opposite sides of a quadrilateral circumscribing a circle subtend supplementary angles at the centre of the circle.

### Question 21

21. Construct a right triangle in which the sides, (other than the hypotenuse) are of length 6 cm and 8 cm. Then construct another triangle, whose sides are $\frac{3}{5}$ times the corresponding sides of the given triangle.

### Question 22

22. In Fig. 7, PQ and AB are respectively the arcs of two concentric circles of radii 7 cm and 3.5 cm and centre O. If ∠POQ = 30°, then find the area of the shaded region. [ Use π = $\frac{22}{7}$]

### Question 23

23. From a solid cylinder of height 7 cm and base diameter 12 cm, a conical cavity of same height and same base diameter is hollowed out. Find the total surface area of the remaining solid. [ π = $\frac{22}{7}$]

OR

A cylindrical bucket, 32 cm high and with radius of base 18 cm, is filled with sand. This bucket is emptied on the ground and a conical heap of sand is formed. If the height of the conical heap is 24 cm, then find the radius and slant height of the heap.

### Question 24

24. The angles of depression of two ships from the top of a light house and on the same side of it are found to be 45° and 30°. If the ships are 200 m apart, find the height of the light house

### Question 25

25. A point P divides the line segment joining the points A (3, −5) and B (−4, 8) such that $\frac{\mathrm{AP}}{\mathrm{PB}}$= $\frac{k}{1}$. If P lies on the line x + y = 0, then find the value of k.

### Question 26

26. If the vertices of a triangle are (1, −3), (4, p) and (−9, 7) and its area is 15 sq. units, find the value(s) of p.

### Question 27

27. A box contains 100 red cards, 200 yellow cards and 50 blue cards. If a card is drawn at random from the box, then find the probability that it will be (i) a blue card (ii) not a yellow card (iii) neither yellow nor a blue card.

### Question 28

28. The${17}^{\mathrm{th}}$term of an A.P. is 5 more than twice its${8}^{\mathrm{th}}$term. If the ${11}^{\mathrm{th}}$term of the A.P. is 43, then find its${n}^{\mathrm{th}}$term.

SECTION D

### Question 29

29. A shopkeeper buys some books for Rs 80. If he had bought 4 more books for the same amount, each book would have cost Rs 1 less. Find the number of books he bought.

OR

The sum of two numbers is 9 and the sum of their reciprocals is $\frac{1}{2}$. Find the numbers.

### Question 30

30. Sum of the first 14 terms of an A.P. is 1505, and its first term is 10. Find its ${25}^{\mathrm{th}}$term.

### Question 31

31. Prove that the tangent at any point of a circle is perpendicular to the radius through the point of contact.

OR

A quadrilateral ABCD is drawn to circumscribe a circle. Prove that AB + CD = AD + BC.

### Question 32

32. A solid is in the shape of a cone surmounted on a hemisphere, the radius of each of them being 3.5 cm and the total height of solid is 9.5 cm. Find the volume of the solid. [ Use π = $\frac{22}{7}$]

### Question 33

33. A bucket is in the form of a frustum of a cone and it can hold 28.49 litres of water. If the radii of its circular ends are 28 cm and 21 cm, find the height of the bucket.[ Use π = $\frac{22}{7}$]

### Question 34

34. The angle of elevation of the top of a hill at the foot of a tower is 60° and the angle of depression from the top of the tower to the foot of the hill is 30°. If the tower is 50 m high, find the height of the hill.