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Time Allowed : 3 Hours Maximum Marks: 90
General Instructions:
(i) All questions are compulsory.
(ii) The question paper consists of 34 questions divided into four sections A, B, C and D, Section A comprises of 8 questions of 1 mark each. Section B comprises of 6 questions of 2 marks each. Section C comprises of 10 questions of 3 marks each and Section D comprises of 10 questions of 4 marks each.
(iii) Questions 1 to 8 in section A are multiple choice questions where you are to select one correct option out of the given four.
(iv) Use of calculators is not permitted.
SECTION A
Questions numbers 1 to 8 are of one mark each.
1. The common difference of the A.P., 1/p, (1-p)/p, (1-2p)/p,,.... is :
(i) p (ii) -p
(iii) -1 (iv)1
2. In figure, PA and PB are two tangents drawn from the external point P in a circle with center C and radius 4 cm. If PA is perpendicular to PB , then the length of each tangent is:
(i) 3 cm (ii) 4 cm
(iii) 5 cm (iv) 6 cm
3. In figure, a circle with centre O is inscribed in a quadrilateral ABCD such that, it touches the sides BC, AD and CD at points P, Q, R and S respectively. If AB = 29 cm, AD= 23 cm, ∠90° and DS = 5 cm, then the radius of the circle (in cm) is:
(i) 11 (ii) 18
(iii) 6 (iv) 15
4. The angle of depression of a car, standing on the ground from the top of a 75 m high tower is 30°. the distance of the car from the base of the tower (in m ) is:
(i) 25 √3 (ii) 50√3
(iii) 75 √3 (iv)150
5.The probability of getting an even number, when a dice is thrown once is :
(i) 1/2 (ii) 1/3
(iii) 1/6 (iv) 5/6
6. A box contains 90 discs, numbered from 1 to 90. If one disc is drawn at random from the box, the probability that it bears a prime-number less than 23 is:
(i) 7/90 (ii) 10/90
(iii) 4/45 (iv) 9/89
7. In figure, the area of triangle ABC (in Sq. units) is:
(a) 15 (b) 10
(c) 7.5 (d) 2.5
8. If the difference between the circumference and the radius of a circle is 37 cm, then using π = 22/7, the circumference (in cm ) of the circle is:
(a) 154 (b) 44
(c) 14 (d) 7
SECTION B
Question numbers 9 to 14 carry 2 marks each.
9. Solve the following quadratic equation for x:
4√3x² + 5x -2√3 = 0
10. How many three-digit natural numbers are divisible by 7?
11. In figure, a circle inscribed in triangle ABC touches its sides AB, BC and AC at points D, E and F respectively. If AB = 12 cm, BC = 8 cm and AC = 10 cm, then find the lengths of AD, DE and CF.
12. Prove that the parallelogram circumscribing a circle is a rhombus.
13. A card is drawn is drawn at random from a well shuffled pack of 52 playing cards. Find the probability that the drawn card is neither a king nor a queen.
14.Two circular of equal radii and maximum area, touching each other are cut out from a rectangular cardboard 14 cm X 7 cm. Find the area of the remaining cardboard.
SECTION C
Question numbers 15 to 24 carry 3 marks each.
15. For what value of k, are the roots of the quadratic equation k x (x-2) + 6 = 0 equal?
16. Find the number of terms of the A.P., 18,15(1/2), ..... - 49(1/2) and find the sum of all its terms.
17. Construct a triangle with the sides 5 cm, 4 cm and 6 cm. then construct another triangle whose sides are 2/3 times the corresponding sides of first triangle.
18. The horizontal distance between two poles is 15 m. The angle of depression of the top of the first pole as seen from the top of the second pole is 30°. If the height of the second pole is 24 cm, find the height of the first pole. (use √3 = 1.732)
19. Prove that the points (7,10), (-2,5) and (3,-4) are the vertices of an isosceles right triangle.
20. Find the ratio in which the y-axis divides the line segment joining the points (-4,-6) and (10,12). also find the coordinates of the points of the point of division.
21. In figure, AB and CD are two diameters of a circle with centre O, which are perpendicular to each other. OB is the diameter of the smaller circle. If OA = 7 cm, find the area of the shaded region.
22. A vessel is in the form a hemispherical bowl surmounted by a hollow cylinder of same diameter. The diameter of the hemispherical bowl is 14cm and the total height of the vessel is 13 cm. Find the total surface area of the vessel.
23. A wooden toy was made by scooping out a hemisphere of same radius from each end of a solid cylinder. If the height of the cylinder is 10 cm, and its base is of radius 3.5 cm , find the volume of wood in the toy. (use π = 22/7)
24. In a circle of radius 21 cm, an arc subtends an angle of 60° at the centre. Find : (i) the length of the arc (ii) area of the sector formed by the arc.
SECTION D
Question numbers 25 to 34 carry 4 marks each.
25. Solve the following for x:
1/(2a + 2 +2x) = 1/(2a) + 1/(b) + 1/(2x)
26. Sum of the areas of two squares is 400 cm². If the difference of their perimeters is 16 cm, find the sides of the two squares.
27.If the sum of first 7 terms of an A.P. is 49 and that of first 17 terms is 289, find the sum of its first n terms.
28. Prove that the tangent at any point of a circle is perpendicular to the radius through the point of contact.
29. In figure, l and m are two parallel tangents to a circle with the centre O, touching the circle at A and B respectively. Another tangent at C intersects the line l at D and m at E. Prove that ∠DOE = 90°.
30. The angle of elevation of the top of the building from the foot of the tower is 30° and the angle of elevation of the top of the tower from the foot of the building is 60°. If the tower is 60 m high, find the height of the building.
31. A group consists of 12 persons , of which 3 are extremely patient, other are extremely hones and rest are extremely kind . A person from the group is selected at random. Assuming that each person is equally likely to be selected, find the probability of selecting a person who is (i) extremely patient (ii) extremely kind or honest. Which of the above values you prefer more?
32. The three vertices of a parallelogram ABCD are A(3,-4), B(-1,-3) and C(-6,2). Find the coordinates of vertex D and find the area of ABCD.
33. Water is flowing through a cylindrical pipe, of internal diameter 2 cm, into cylindrical tank of base radius 40 cm, at the rate of 0.4 m/s. Determine the rise in level of water in the tank in half an hour.
34. A bucket at the top, and made up of a metal sheet is in the from of a frustum of a cone. The depth of the bucket is 24 cm and the diameters of its upper and lower circular ends are 30 cm and 10 cm respectively. Find the cost of metal sheet used in it at the rate of Rs. 10 per 100 cm³.
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