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CBSE X
All India
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. The roots of the quadratic equation 2${x}^{2}$âˆ’x âˆ’ 6 = 0 are _________.

### Question 2

2. If the ${n}^{\mathrm{th}}$term of an A.P. is (2n + 1), then the sum of its first three terms is _________.

### Question 3

3. From a point Q, 13 cm away from the centre of a circle, the length of tangent PQ to the circle is 12 cm. The radius of the circle (in cm) is

### Question 4

4. In Figure 1, AP, AQ and BC are tangents to the circle. If AB = 5 cm, AC = 6 cm and BC= 4 cm, then the length of AP (in cm) is

### Question 5

5. The circumference of a circle is 22 cm. The area of its quadrant (in ${\mathrm{cm}}^{2}$) is _________.

### Question 6

6. A solid right circular cone is cut into two parts at the middle of its height by a plane parallel to its base. The ratio of the volume of the smaller cone to the whole cone is

### Question 7

7. A kite is flying at a height of 30 m from the ground. The length of string from the kite to the ground is 60 m. Assuming that there is no slack in the string, the angle of elevation of the kite at the ground is

### Question 8

8. The Distance of the point (âˆ’3, 4) from the x-axis is __________.

### Question 9

9. In Figure 2, P (5, âˆ’3) and Q (3, y) are the points of trisection of the line segment joining A (7, âˆ’2) and B (1, âˆ’5). Then y equals

### Question 10

10. Cards bearing numbers 2, 3, 4, ..., 11 are kept in a bag. A card is drawn at random from the bag. The probability of getting a card with a prime number is

SECTION B

### Question 11

11. Find the value of p for which the roots of the equation px (x âˆ’ 2) + 6 = 0, are equal.

### Question 12

12. How many two-digit numbers are divisible by 3 ?

### Question 13

13. In Figure 3, a right triangle ABC, circumscribes a circle of radius r. If AB and BC are of lengths of 8 cm and 6 cm respectively, find the value of r.

### Question 14

14. Prove that the tangents drawn at the ends of a diameter of a circle are parallel.

### Question 15

15. In Figure 4, 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. (Use Ï€ = 3.14)

OR

From a rectangular sheet of paper ABCD with AB = 40 cm and AD = 28 cm, a semi-circular portion with BC as diameter is cut off. Find the area of the remaining paper.(Use Ï€ = 3.14)

### Question 16

16. A solid sphere of radius 10.5 cm is melted and recast into smaller solid cones, each of radius 3.5 cm and height 3 cm. Find the number of cones so formed. ( Use Ï€ = $\frac{22}{7}$)

### Question 17

17. Find the value of k, if the point P (2, 4) is equidistant from the points A(5, k) and B (k, 7).

### Question 18

18. A card is drawn at random from a well-shuffled pack of 52 cards. Find the probability of getting

(i) a red king.

(ii) a queen or a jack.

SECTION C

### Question 19

19. Solve the following quadratic equation for x:

${x}^{2}$âˆ’4ax âˆ’${b}^{2}$+${4a}^{2}$ = 0

OR

If the sum of two natural numbers is 8 and their product is 15, find the numbers.

### Question 20

20. Find the sum of all multiples of 7 lying between 500 and 900.

### Question 21

21. Draw a triangle ABC with BC = 7cm, âˆ  B = 45Â° and âˆ C = 60Â°. Then construct another triangle, whose sides are $\frac{3}{5}$ times the corresponding sides of Î”ABC.

### Question 22

22. In Figure 5, a circle is inscribed in a triangle PQR with PQ = 10 cm, QR = 8 cm and PR = 12 cm. Find the lengths of QM, RN and PL.

### Question 23

23. In Figure 6, O is the centre of the circle with AC = 24 cm, AB = 7 cm and âˆ BOD = 90Â°. Find the area of the shaded region. [Use Ï€ = 3.14]

or

In Figure 7, find the area of the shaded region, if ABCD is a square of side 14 cm and APD and BPC are semicircles.

### Question 24

24. A hemispherical bowl of internal radius 9 cm is full of water. Its contents are emptied in a cylindrical vessel of internal radius 6 cm. Find the height of water in the cylindrical vessel.

### Question 25

25. The angles of depression of the top and bottom of a tower as seen from the top of a 60$\sqrt{3}$m high cliff are 45Â° and 60Â° respectively. Find the height of the tower.

### Question 26

26. Find the coordinates of a point P, which lies on the line segment joining the points A (âˆ’2, âˆ’2), and B (2, âˆ’4), such that AP = $\frac{3}{7}$AB.

OR

Find the area of the quadrilateral ABCD whose vertices are A (âˆ’3, âˆ’1), B (âˆ’2, âˆ’4), C (4, âˆ’1) and D (3, 4).

### Question 27

27. If the points A (x, y), B (3, 6) and C (âˆ’3, 4) are collinear, show that x âˆ’ 3y + 15 = 0.

### Question 28

28. All kings, queens and aces are removed from a pack of 52 cards. The remaining cards are well shuffled and then a card is drawn from it. Find the probability that the drawn card is

(i) a black face card.

(ii) a red card.

SECTION D

### Question 29

29. The numerator of a fraction is 3 less than its denominator. If 1 is added to the denominator, the fraction is decreased by $\frac{1}{15}$. Find the fraction.

OR

In a flight of 2800 km, an aircraft was slowed down due to bad weather. Its average speed is reduced by 100 km/h and time increased by 30 minutes. Find the original duration of the flight.

### Question 30

30. Find the common difference of an A. P. whose first term is 5 and the sum of its first four terms is half the sum of the next four terms.

### Question 31

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

### Question 32

32. A hemispherical tank, full of water, is emptied by a pipe at the rate of $\frac{25}{7}$ litres per sec. How much time will it take to empty half the tank if the diameter of the base of the tank is 3 m?

OR

A drinking glass is in the shape of the frustum of a cone of height 14 cm. The diameters of its two circular ends are 4 cm and 2 cm. Find the capacity of the glass. [Use Ï€ = $\frac{22}{7}$]

### Question 33

33. A military tent of height 8.25 m is in the form of a right circular cylinder of base diameter 30 m and height 5.5 m surmounted by a right circular cone of same base radius. Find the length of the canvas used in making the tent, if the breadth of the canvas is 1.5 m.

### Question 34

34. The angles of elevation and depression of the top and bottom of a light-house from the top of a 60 m high building are 30Â° and 60Â° respectively. Find

(i) the difference between the heights of the light-house and the building.

(ii) the distance between the light-house and the building.

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