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

- write a rational number between √2 and √3.
- Write the number of zeroes of the polynomial y = f(x) whose graph is given in the figure.

3.Is x= -2 a solution of the equation x^{2} – 2x + 8 = 0?

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

5.D,E and F are the mid-points of the sides AB,BC and CA respectively of ΔABC. Find ar(ΔABC)/ ar ΔDEF.

6. In the figure <ATO= 40, Find < AOB.

7. If sinθ= cosθ , find the value of θ

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

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

10. Find the median class of the following data :

SECTION B

11. Find the quadratic polynomial sum of whose zeroes is 8 and their product is 12. hence, find the zeroes of the polynomial.

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

13. without using trigonometric tables, evaluate the following. ( sin² 25° sin° 65°) √3 ( tan 5° tan 15° tan 30° tan 75° tan 85°)

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

15. Cards, marked with numbers 5 to 50 are placed in a box and shuffled thoroughly. A card is shown 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

16. prove that √3 is a irrational number.

17. use Euclid’s Division Lemma to show that the square of any positive integer is either of the form 3m or 3m + 1 for some integers m

18. the sum of the 4^{th} and 8^{th} terms of an A.P is 24 and the sum of the 6^{th} and 10^{th} terms is 44. find the first three terms of the A.P.

19. solve for x and y

(a – b)x + (a + b)y = a^{2} – 2ab – b^{2}

(a + b)(x + y) = a^{2} +b^{2}

20. Prove that (sinθ cosecθ )² + (cosθ + secθ )²= 7+ tan² θ + cot ²θ

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

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 co-ordinates of R.

23. in 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

24. Draw a ΔABC with side BC= 6 cm, AB= 5cm and < ABC = 60 . Construct a ΔA’B’C’ similar to ΔABC such that the sides of the ΔA’B’C’ are 3/4 th of theΔ ABC.

^{25. }D and E are points on the sides CA and Cb respectively of ΔABC right angled at C. Prove that AE^{2} + BD^{2} = AB^{2} + DE^{2}

SECTION D

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.

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

The diagonals of a trapezium ABCD with AB parallel CD intersect at a point O. If AB= 2CD, find the ratio of ΔAOB to ΔCOD

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).

29. the angle of elevation of a jet fighter from a point A on the ground is 60^{0}. after a flight of 15 seconds, the angle of elevation changes to 30^{0}. if the jet is flying at a speed of 720km/hr,

30.find the mean, mode and the median of the following data.