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CBSE X
All India
MATHS PAPER 2011

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 equation${x}^{2}-3x-m\left(m+3\right)=0$, where m is a constant, are:

Question 2

2. If the common differences of an A.P. is 3, then ${a}_{20}-{a}_{15}$is:

Question 3

3. In figure 1, O is the centre of a circle, PQ is a chord and PT is the tangent at P. If ∠POQ = 70°, then ∠TPQ is equal to ________.

Question 4

4. In Figure 2, AB and AC are tangents to the circle with centre O such that ∠BAC = 40°. Then ∠BOC is equal to ________.

Question 5

5. The perimeter (in cm) of a square circumscribing a circle of radius a cm, is _________.

Question 6

6. The radius (in cm) of the largest right circular cone that can be cut out from a cube of edge 4.2 cm is _________.

Question 7

7. A tower stands vertically on the ground. From a point on the ground which is 25 m away from the foot of the tower, the angle of elevation of the top of the tower is found to be 45°. Then the height (in meters) of the tower is _________ m.

Question 8

8. If P($\frac{a}{2}$, 4) is the mid-point of the line-segment joining the points A (−6, 5) and B(−2, 3), then the value of a is

Question 9

9. If A and B are the points (−6, 7) and (−1, −5) respectively, then the distance

2AB is equal to __________.

Question 10

10. A card is drawn from a well-shuffled deck of 52 playing cards. The probability that the card will not be an ace is ________.

SECTION B

Question 11

11. Find the value of m so that the quadratic equation mx (x − 7) + 49 = 0 has two equal roots.

Question 12

12. Find how many two-digit numbers are divisible by 6.

Question 13

13. In Figure 3, a circle touches all the four sides of a quadrilateral ABCD whose sides are AB = 6 cm, BC = 9 cm and CD = 8 cm. Find the length of the side AD.

Question 14

14. Draw a line segment AB of length 7 cm. Using ruler and compasses, find a point P on AB such that $\frac{\mathrm{AP}}{\mathrm{AB}}$= $\frac{3}{5}$.

Question 15

15. Find the perimeter of the shaded region in Figure 4, if ABCD is a square of side 14 cm and APB and CPD are semicircles. [Use π = $\frac{22}{7}$]

Question 16

16. Two cubes each of volume 27 ${\mathrm{cm}}^{3}$are joined end to end to form a solid. Find the surface area of the resulting cuboid.

OR

A cone of height 20 cm and radius of base 5 cm is made up of modeling clay. A child reshapes it in the form of a sphere. Find the diameter of the sphere.

Question 17

17. Find the value of y for which the distance between the points A(3, −1) and B(11, y) is 10 units.

Question 18

18. A ticket is drawn at random from a bag containing tickets numbered from 1 to 40. Find the probability that the selected ticket has a number which is a multiple of 5.

SECTION C

Question 19

19. Find the roots of the following quadratic equation:${x}^{2}-3\sqrt{5}x+10=0$

Question 20

20. Find the A.P. whose fourth term is 9 and the sum of its sixth term and thirteenth term is 40.

Question 21

21. In Figure 5, a triangle PQR is drawn to circumscribe a circle of radius 6 cm such that the segments QT and TR into which QR is divided by the point of contact T, are of lengths 12 cm and 9 cm respectively. If the area of ΔPQR = 189 ${\mathrm{cm}}^{2}$, then find the lengths of sides PQ and PR.

Question 22

22. Draw a pair of tangents to a circle of radius 3 cm, which are inclined to each other at an angle of 60°.

OR

Draw a right triangle in which the sides (other than hypotenuse) are of lengths 4 cm and 3 cm. Then construct another triangle whose sides are $\frac{3}{5}$ times the corresponding sides of the given triangle.

Question 23

23. A chord of a circle of radius 14 cm subtends an angle of 120° at the centre. Find the area of the corresponding minor segment of the circle. Use π = $\frac{22}{7}\mathrm{and}\sqrt{3}=1.73$

Question 24

24. An open metal bucket is in the shape of a frustum of a cone of height 21 cm with radii of its lower and upper ends as 10 cm and 20 cm respectively. Find the cost of milk which can completely fill the bucket at Rs. 30 per litre. [Use π = $\frac{22}{7}$]

Question 25

25. Point P(x, 4) lies on the line segment joining the points A(−5, 8) and B(4, −10). Find the ratio in which point P divides the line segment AB. Also find the value of x.

Question 26

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

OR

Find the area of triangle formed by joining the mid-points of the sides of the triangle whose vertices are A(2, 1), B(4, 3) and C(2, 5).

Question 27

27. From the top of a vertical tower, the angles of depression of two cars, in the same straight line with the base of the tower, at an instant are found to be 45° and 60°. If the cars are 100 m apart and are on the same side of the tower, find the height of the tower. [Use $\sqrt{3}$= 1.73]

Question 28

28. Two dice are rolled once. Find the probability of getting such numbers on the two dice, whose product is 12.

OR

A box contains 80 discs which are numbered from 1 to 80. If one disc is drawn at random from the box, find the probability that it bears a perfect square number.

SECTION D

Question 29

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

Question 30

30. The first and the last terms of an A.P. are 8 and 350 respectively. If its common difference is 9, how many terms are there and what is their sum?

OR

How many multiples of 4 lie between 10 and 250? Also find their sum.

Question 31

31. A train travels 180 km at a uniform speed. If the speed had been 9 km/hour more, it would have taken 1 hour less for the same journey. Find the speed of the train.

OR

Find the roots of the equation.

$\frac{1}{2x-3}+\frac{1}{x+5}=1,x\ne \frac{3}{2},5$

Question 32

32. In Figure 6, three circles each of radius 3.5 cm are drawn in such a way that each of them touches the other two. Find the area enclosed between these three circles (shaded region). [Use π = $\frac{22}{7}$]

Question 33

33. Water is flowing at the rate of 15 km/hour through a pipe of diameter 14 cm into a cuboidal pond which is 50 m long and 44 m wide. In what time will the level of water in the pond rise by 21 cm?

Question 34

34. The angle of elevation of the top of a vertical tower from a point on the ground is 60°. From another point 10 m vertically above the first, its angle of elevation is 30°. Find the height of the tower.

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