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CBSE X
Delhi
MATHS PAPER 2010

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 the sum of first p terms of an Î‘.P. is ${\mathrm{ap}}^{2}+\mathrm{bp}$, find its common difference.

### Question 2

2. In the given fig. , S and T are points on the sides PQ and PR, respectively of Î”PQR, such that PT = 2 cm, TR = 4 cm and ST is parallel to QR. Find the ratio of the areas of Î”PST and Î”PQR.

### Question 3

3. In the given figure, Î”AHK is similar to Î”ABC. If AK = 10 cm, BC = 3.5 cm and HK = 7 cm, find AC.

### Question 4

4. If α, Î² are the zeroes of a polynomial, such that α + Î² = 6 and αÎ² = 4, then write the polynomial.

### Question 5

5. Has the rational number $\frac{441}{\left({2}^{2}.{5}^{7}.{7}^{2}\right)}$ a terminating or a non-terminating decimal representation?

### Question 6

6. If cosec Î¸ = 2x and cot Î¸ = $\frac{2}{x}$, find the value of 2(${x}^{2}$-$\frac{1}{{x}^{2}}$)

### Question 7

7. A card is drawn at random from a well shuffled pack of 52 playing cards. Find the probability of getting a red face card.

### Question 8

8. The slant height of frustum of a cone is 4 cm and the perimeters (circumferences) of its circular ends are 18 cm and 6 cm. Find the curved surface area of the frustum.

[Use Ï€ = $\frac{22}{7}$]

### Question 9

9. If A(1, 2), B(4, 3) and C(6, 6) are the three vertices of the parallelogram ABCD, find the coordinates of the fourth vertex D.

### Question 10

10. If P(2, p) is the mid-point of the line segment joining the points A(6, âˆ’ 5) and B(âˆ’ 2, 11), find the value of p.

SECTION B

### Question 16

16. Prove that (2$\sqrt{3}$-1) is an irrational number.

### Question 17

17. In figure, ABC is a right triangle, right angled at C and D is the mid-point of BC. Prove that ${\mathrm{AB}}^{2}=4{\mathrm{AD}}^{2}âˆ’3{\mathrm{AC}}^{2}$.

### Question 18

18. Prove the following

$\frac{\mathrm{tan}A}{\left(1âˆ’\mathrm{cot}A\right)}$ + $\frac{\mathrm{cot}A}{\left(1âˆ’\mathrm{tan}A\right)}$= 1+ tan A + cot â¡A

OR

Prove the following

(coses A - cos A)(sec A - cos A) = $\frac{1}{\left(\mathrm{tan}A+\mathrm{cot}A\right)}$

### Question 19

19. In an A.P., the sum of first ten terms is âˆ’150 and the sum of its next ten terms is âˆ’550. Find the A.P.

### Question 20

20. The sum of numerator and denominator of a fraction is 3 less than twice the denominator. If each of the numerator and denominator is decreased by 1, the fraction becomes $\frac{1}{2}$. Find the fraction.

SECTION C

### Question 11

11. If $\sqrt{5}$ and -$\sqrt{5}$ are two zeroes of the polynomial ${x}^{3}+3{x}^{2}âˆ’5xâˆ’15$, find its third zero.

### Question 12

12. If all the sides of a parallelogram touch a circle, show that the parallelogram is a rhombus.

### Question 13

13. Without using trigonometric tables, find the value of the following expression:

$\frac{\mathrm{sec}\left(90Â°âˆ’\mathrm{Î¸}\right).\mathrm{cos}\mathrm{ec}\mathrm{Î¸}âˆ’\mathrm{tan}\left(90Â°âˆ’\mathrm{Î¸}\right).\mathrm{cot}\mathrm{Î¸}+{\mathrm{cos}}^{2}25Â°+{\mathrm{cos}}^{2}65Â°}{3\mathrm{tan}27Â°.\mathrm{tan}63Â°}$

OR

Find the value of cosec 30Â°, geometrically.

### Question 14

14. Find the value of k for which the following pair of linear equations has infinitely many solutions: 2x + 3y = 7; (k âˆ’ 1) x + (k + 2)y = 3k.

### Question 15

15. In an A.P., first term is 2, the last term is 29 and sum of the terms is 155. Find the common difference of the A.P.

### Question 21

21. Construct a triangle PQR in which QR = 6 cm, âˆ Q = 60Â° and âˆ R = 45Â°. Construct another triangle similar to Î”PQR such that its sides are $\frac{5}{6}$ of the corresponding sides of Î”PQR.

### Question 22

22. Cards bearing numbers 1, 3, 5, ..., 35 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 15.

(ii) a number divisible by 3 and 5.

### Question 23

23. If the point P (m, 3) lies on the line segment joining the points A ($âˆ’\frac{2}{5}$, 6 ) and B (2, 8), find the value of m.

### Question 24

24. Point P divides the line segment joining the points A (2, 1) and B (5, âˆ’8) such that $\frac{\mathrm{AP}}{\mathrm{AB}}=\frac{1}{3}$. If P lies on the line 2x âˆ’ y + k = 0, find the value of k.

### Question 25

25. In figure, the boundary of shaded region consists of four semicircular arcs, two smallest being equal. If diameter of the largest is 14 cm and that of the smallest is 3.5 cm, calculate the area of the shaded region. [Use Ï€ = $\frac{22}{7}$]

Or

Find the area of shaded region in figure 5, if AC = 24 cm, BC = 10 cm and O is the centre of the circle. [Use Ï€ = 3.14]

SECTION D

### Question 26

26. A milk container is made of metal sheet in the shape of frustum of a cone whose volume is 10459 $\frac{3}{7}{\mathrm{cm}}^{3}$. The radii of its lower and upper circular ends are 8 cm and 20 cm respectively. Find the cost of metal sheet used in making the container at the rate of Rs. 1.40 per square centimetre. Use Ï€ = $\frac{22}{7}.$

Or

A toy is in the form of a hemisphere surmounted by a right circular cone of the same base radius as that of the hemisphere. If the radius of base of the cone is 21 cm and its volume is $\frac{2}{3}$ of the volume of the hemisphere, calculate the height of the cone and the surface area of the toy. Ï€ = $\frac{22}{7}$.

### Question 27

27. Prove that, in a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.

Using the above, prove the following:

Point D is the mid-point of the side BC of a right triangle ABC, right angled at C. Prove that 4${\mathrm{AD}}^{2}=4{\mathrm{AC}}^{2}$+ ${\mathrm{BC}}^{2}$

### Question 28

28. Three consecutive positive integers are such that the sum of the square of the first and the product of the other two is 46, find the integers.

Or

The difference of squares of two numbers is 88. If the larger number is 5 less than twice the smaller number, then find the two numbers.

### Question 29

29. From the top of a 7 m high building, the angle of elevation of the top of a tower is 60Â° and the angle of depression of the foot of the tower is 30Â°. Find the height of the tower.

### Question 30

30. Find the mean, mode and median of the following frequency distribution: