# Class X-2008

MATHEMATICS PAPER 2008.

Time allowed: 3 hours; Maximum Marks: 80

 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. Write  a rational number between √2 and √3

### Question 2

2. Write the number of zeroes of the polynomial y = f(x) whose graph is given in the figure

### Question 3

3. Is x=-2 a solution of the equation x²-2x+8=0

### Question 4

4. Write the next term of the A.P. √8, √18 and √32

### Question 5

5. D, E and F are midpoints of sides AB, BC and CA respectively of Δ ABC . Find ar(Δ  DEF)/ar(Δ  ABC)

### Question 6

In the figure, if <ATO= 40°, find <AOB

### Question 7

If sin θ=cos θ, find the value of θ

### Question 8

Find the perimeter of the given figure where AED is a semi circle and ABCD is a rectangle.

### Question 9

A bag contains 4 red and 6 black balls,  a ball is taken out of the bag at random. Find the probability of finding a black ball.

### Question 10

Find the median class of the following data.

SECTION B

### Question 11

Find the quadratic polynomial sum of whose zeroes is 8 and their products is 12. Hence , find the zeroes of the polynomial.

### Question 12

In the figure, OP is equal  to diameter of the circle. Prove that ABP is an equilateral triangle.

Question 13

13. Without using trigonometric tables, evaluate the following

(sin² 25°+ sin² 65°) + √3 (tan 5° tan 15° tan 30° tan 75° tan 85°)

### Question 14

14. For what value of k are the points (1,1) ; (3,k) and (-1,4) collinear?

### Question 15

15. Cards marked with  numbers 5 to 50, are placed in a box  and shuffled thoroughly. A card is drawn from the box at random.  Find the probability  that the number on the card is

i) a prime number less  than 10

ii) a number which is a perfect square.

SECTION C

### Question 16

16. Prove that √3 is an irrational number.

### Question 17

17. Use Euclids division Lemma  to show the square  of any positive integer is either of the form 3m or 3m+1 for some integer m.

Question 18

18. The sum of the 4th and 8th terms of an A.P. is 24 and the sum of  the 6th and 10 th terms is 44. Find the first three terms of  the A.P.

### Question 19

19. Solve for x and y.

(a-b)x + (a+b) y = a²-2ab-b²

### Question 20

20. Prove that (sinθ + cosec θ)² + (cosθ + sec θ)²= 7+ tanθ  +cot θ

### Question 21

21. If the point P (x,y) is equidistant from the points A(3,6) and B (-3,4) prove that 3x+y-5=0.

### Question 22

22. The point R divides the line segment AB, where A ( -4,0)  and B (0,6)  are such that  AR= 3/4 AB. Find the coordinates of R.

### Question 23

23. In the figure ABC is a right angled triangle, right angled at A. Semicircles are drawn on AB, AC and BC as diameters.Find the area of the shaded region.

Question 24

24. Draw ΔABC with side BC= 6 cm, AB = 5 cm and < ABC= 60. Construct a triangle A’B'C’ wherein the sides are 3/4 of the sides of ABC.

### Question 25

25. D and E are points on the sides CA and CB respectively of ΔABC , right angled at C. Prove that AE²+BD²= AB²+ DE²

SECTION D

### Question 26

26. A motor boat whose speed is 18kmph in still water takes 1 hour more to go 24 km upstream than to return downstream to the same spot. Find the speed of the stream.

### Question 27

27. Prove that the ratio of the areas of 2 similar traingles is equal to the ratio of the squares on their corresponding sides. Use the above, do the following :

The diagonals of a trapezium ABCD intersect at a point O with AB  parallel CD. If AB= 2CD , find the ratio ofΔ AOB toΔ COD.

Question 28

28. A tent consists of frustum of a cone, surmounted by a cone. If the diameter of the upper and lower circular ends of the frustum be 14m and 26m respectively, the height of the frustum be 8m and the slant height of the surmounted conical portion be 12m, find the area of canvas required to make the tent. (Assume that the radii of the upper circular end of the frustum and the base of surmounted conical portion are equal).

Question 29
29. The angle of elevation of a jet fighter from a point A on the ground is 600. after a flight of 15 seconds, the angle of elevation changes to 300. if the jet is flying at a speed of 720km/hr, find the constant height at which the jet is flying. (Use √3= 1.732)

Question 30

30. Find the mean and median of the following data: