General Instructions :
(i) All questions are compulsory.
(ii) The question paper consists of 31 questions divided into four sections A, B, C and D.
(iii) Section A contains 4 questions of 1 mark each. Section B contains 6 questions of 2 marks each, Section C contains 10 questions of 3 marks
each and Section D contains 11 questions of 4 marks each.
(iv) Use of calculators is not permitted.
1. PQ is a tangent drawn from an external point P to a circle with centre O, QOR is the diameter of the circle. If POR = 120, what is thenmeasure of OPQ ?
2. A solid metallic cuboid of dimensions 9 m 8 m 2 m is melted and recast into solid cubes of edge 2 m. Find the number of cubes so formed.
3. If one root of the quadratic equation 6×2– x – k = 0 is 3/ 2 , then find the value of k.
4. A ladder 15 m long makes an angle of 60 with the wall. Find the height of the point where the ladder touches the wall.
Question numbers 5 to 10 carry 2 marks each.
5. In the given figure, if AB = AC, prove that BE = EC.
6. Find the probability that in a leap year there will be 53 Tuesdays.
7. If two adjacent vertices of a parallelogram are (3, 2) and (– 1, 0) and the diagonals intersect at (2, – 5), then find the coordinates of the other two vertices.
8. If seven times the 7 th term of an A.P. is equal to eleven times the 11th term, then what will be its 18th term ?
9. Two different dice are thrown together. Find the probability that the product of the numbers appeared is less than 18.
10. Solve for x :
Question numbers 11 to 20 carry 3 marks each.
11. In the given figure, ABC is an equilateral triangle of side 3 units. Find the coordinates of the other two vertices.
12. Show that ABC with vertices A (– 2, 0), B (0, 2) and C (2, 0) is similar to DEF with vertices D (– 4, 0), F (4, 0) and E (0, 4).
13. The shadow of a tower at a time is three times as long as its shadow when the angle of elevation of the sun is 60. Find the angle of elevation of the sun at the time of the longer shadow.
14. In the given figure, ABCD is a trapezium with AB DC, AB = 18 cm, DC = 32 cm and the distance between AB and AC is 14 cm. If arcs of equal radii 7 cm taking A, B, C and D as centres, have been drawn, then find the area of the shaded region.
15. Prove that the opposite sides of a quadrilateral circumscribing a circle subtend supplementary angles at the centre of the circle.
16. Find the coordinates of the points of trisection of the line segment joining the points (3, – 2) and (– 3, – 4).
17. In the given figure, PA and PB are tangents to a circle from an external point P such that PA = 4 cm and BAC = 135. Find the length of chord AB.
18. If the roots of the quadratic equation (x – a) (x – b) + (x – b) (x – c) + (x – c) (x – a) = 0 are equal, then show that a = b = c.
19. If the sum of the first 7 terms of an A.P. is 49 and that of the first 17 terms is 289, find the sum of its first n terms.
20. A wire of diameter 3 mm is wound about a cylinder whose height is 12 cm and radius 5 cm so as to cover the curved surface of the cylinder completely. Find the length of the wire.
Question numbers 21 to 31 carry 4 marks each.
21. A child puts one five-rupee coin of her saving in the piggy bank on the first day. She increases her saving by one five-rupee coin daily. If the piggy bank can hold 190 coins of five rupees in all, find the number of days she can continue to put the five-rupee coins into it and find the total money she saved.
Write your views on the habit of saving.
22. In a rectangular park of dimensions 50 m 40 m, a rectangular pond is constructed so that the area of grass strip of uniform width surrounding the pond would be 1184 m2 . Find the length and breadth of the pond.
23. Prove that the lengths of two tangents drawn from an external point to a circle are equal.
24. A park is of the shape of a circle of diameter 7 m. It is surrounded by a path of width of 0·7 m. Find the expenditure of cementing the path, if its cost is < 110 per sq. m.
25. Two circles touch internally. The sum of their areas is 116 cm2 and the distance between their centres is 6 cm. Find the radii of the circles.
26. A well of diameter 3 m is dug 14 m deep. The soil taken out of it is spread evenly all around it to a width of 5 m to form an embankment. Find the height of the embankment.
27. A box contains 90 discs which are numbered from 1 to 90. If one disc is drawn at random from the box, find the probability that it bears (i) a two-digit number, (ii) a number divisible by 5.
28. Solve for x :
29. Draw a line segment AB of length 8 cm. Taking A as the centre, draw a circle of radius 4 cm and taking B as the centre draw another circle of radius 3 cm. Construct tangents to each circle from the centre of the other circle.
30. The angles of depression of two ships from an aeroplane flying at the height of 7500 m are 30 and 45. If both the ships are in the same line and on the same side of the aeroplane such that one ship is exactly behind the other, find the distance between the ships. [Use 3 = 1·73]
31. A hollow cone is cut by a plane parallel to the base at some height and the upper portion is removed. If the curved surface area of the remainder is 9/8 of the curved surface of the whole cone, find the ratio of the two parts into which the cone’s altitude is divided.