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Section - D 29. . 6500 were divided equally among a certain number of persons. Had there been 15 more persons, each would have got . 30 less. Find the original number of persons. Sample Paper OR (Based on CBSE CCE SA - 2) One fourth of a herd of camels was seen in the forest. Twice the square root of the herd had gone to the mountains and 15 camels were on the bank of the river. Find the number of camels. (X) MATHEMATICS [Time allowed: 3 hours] [Maximum marks: 80] 30. Find the sum of all natural numbers between 200 and 1502 which are exactly divisible by 3. 31. Prove that the lengths of tangents drawn from an external point to a circle are equal. General Instructions: 1. All questions are compulsory. 32. A hemispherical depression is cut out from one face of a cubical wooden block such that the diameter ‘a’ 2. The question paper consists of 34 questions divided into 4 sections, section A, B, C, and D. of the hemisphere is equal to the edge of the cube. Determine the surface area of the remaining solid. 3. Section A contains 10 multiple choice type questions of 1 mark each. Section B contains 8 questions of 2 marks each, section C contains 10 questions of 3 marks each and section D contains 6 questions of 4 a marks each. 4. There is no overall choice. However, internal choice has been provided in 1 question of two marks, 3 questions of three marks each and 2 questions of four marks each. Attempt only one of the alternatives in all such questions. a Section - A 1. A quadratic equation whose one root is 3 and the sum of the roots is 0, is OR (a) x2 + 3 = 0 (b) x2 + 9 = 0 A shuttle cock used for playing badminton has the shape of a frustum of a cone mounted on a hemisphere (c) x2 - 9 = 0 (d) x2 - 3 = 0 as shown in the figure. The external diameters of the frustum are 5 cm and 2 cm and the height of the entire shuttle cock is 7 cm. Find its external surface area. 2. The next term of the AP 8 , 18 , 32 ... is (a) 65 (b) 60 5 cm (c) 50 (d) 55 3. The distance of the point (3, -4) from the y-axis is 7 cm (a) 7 units (b) 6 units (c) 3 units (d) 4 units 4. To draw a pair of tangents to a circle which are inclined to each other at an angle x°, it is required to 2 cm draw tangents at the end points of those two radii of the circle, the angle between which is 33. The middle portion of a wooden rolling pin (belan) is a cylinder of diameter 6 cm with hemispherical (a) 90° - x° (b) 90° + x° ends of same diameter. Two cylindrical rods, each of length 9 cm and diameter 2 cm are attached on (c) 180° - x° (d) 180° + x° either side of the hemispherical ends. If the total length of the rolling pin is 42.5 cm, find its volume. 5. The sum of n terms of the series 3 + 12 + 27 + 18 + ... is Also, compute its mass if 1 cubic cm of wood has a mass of 0.9 grams. 2n(n + 1) 3n(n - 1) 9 cm (a) (b) 3 2 6 cm 2 cm 3n (n + 1) 2n(n - 1) (c) (d) 42.5 cm 2 3 6. The famous mathematician associated with finding the sum of first 100 natural numbers is 34. The angle of elevation of the top of an unfinished tower at a point 120 m from its base is 45°. How much (a) Euclid (b) Newton must the tower be raised so that the angle of elevation at the same point be 60°? (c) Gauss (d) Pythagoras AVTE/X/M/10-11/Feb./SA-2 4 AVTE/X/M/10-11/Feb./SA-2 1 7. A cylindrical pencil sharpened at one edge is the combination of Section - C (a) two cylinders (b) frustum of a cone and a cylinder 19. Solve: 36x2 - 12ax + (a2 - b2) = 0 (c) a hemisphere and a cylinder (d) a cone and a cylinder OR 8. If the radius of the wheel is 49 cm, then the number of times the wheel will rotate in a journey of 2002 m x x + 1 34 is Solve: + = , x ¹ - 1and x ¹ 0 (a) 1300 (b) 650 x+1 x 15 1 15 (c) 1000 (d) 1500 20. Find the sum of two middle most terms of the A.P - 1, - , 0,....... . 2 2 9. If the diameter of a protractor is 14 cm, then its perimeter is 21. From a point P, two tangents PA and PB are drawn to a circle with centre O. If OP is equal to the (a) 24 cm (b) 36 cm diameter of the circle, show that ∆APB is equilateral. (c) 48 cm (d) 60 cm 22. From an external point P, tangents PA and PB are drawn to a circle with centre O. If CD is the tangent to 10. The given figure shows the observation of point C from point A. The angle of depression from A is the circle at a point E and PA = 14 cm, find the perimeter of ∆PCD. A A C 2m O E P C B D 2 3m B 23. AOB is a quadrant of a circle of radius 14 cm. From it, an equilateral ∆OPQ is cut off as shown in the (a) 60° (b) 30° é 22 ù (c) 45° (d) 75° figure. Calculate the area of the shaded part. ê Take 3 = 1.732, p= ú ëê 7 úû Section - B A Q 11. Find the roots of the quadratic equation: 21x2 - x - 2 = 0. 12. AP is a tangent to the circle with centre O such that OP = 4 cm and ∠OPA = 30°. Find AP P O O B 24. The rain water from a roof 22 m × 20 m drains into a cylindrical vessel having diameter of base 2 m and 4 cm height 3.5 m. If the vessel is just full, find the height of rainfall in cm. 30° A P OR 13. The minute hand of a clock is 21 cm long. Find the area described by the minute hand on the face of A well of diameter 3 m is dug 21 m deep. The earth taken out of it has been spread evenly all around it the clock between 7 : 00 am and 7 : 05 am. to a width of 4 m to form an embankment. Find the height of the embankment. 25. From an aeroplane vertically above a straight horizontal plane, the angles of depression of two consecutive 14. A cuboidal box of dimensions 16 cm × 8 cm × 8 cm contains glass spheres of diameter 6 cm. If 119.68 cm3 kilometre stones on the opposite sides of the aeroplane are found to α and β. Show that the height of the of space remains vacant, find the number of glass spheres. [Take π = 3.14] 15. Line segment PQ is divided into five equal parts at A, B, C and D. If A is (-3, -7) and C is (1, 1), find the tan a . tan b aeroplane, in kilometres is . tan a + tan b coordinates of P, B, D and Q. 26. Find a relation between x and y if the points (x, y), (1, 2) and (7, 0) are collinear. 16. If A(1, 2), B(4, y), C(x, 6) and D(3, 5) be the vertices of a parallelogram taken in order, find x and y. 27. The end points of the diameter AOB of a circle C(O, r) are A(-3, 4) and B(3, -4). Find; 17. Two dice are thrown at the same time. Find the probability of getting a doublet of odd. (i) coordinates of its centre (ii) area of the circle OR 28. A class has 15 girls and 10 boys. The teacher calls on a student at random to answer a question. Express A coin is tossed two times. Find the probability of getting at least one head. in decimal form, the probability that a student called upon is 18. The first term of an A.P is 4 and the last term is 61. The sum of all these terms is 650. Find the common (i) a girl (ii) a boy difference. (iii) a student of the class (iv) the teacher of the class. AVTE/X/M/10-11/Feb./SA-2 2 AVTE/X/M/10-11/Feb./SA-2 3

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