SAMPLE QUESTION PAPER - I MATHEMATICS (SA - 1I) Time allowed : 3 hours Maximum Marks : 90 General Instructions 1. All questions are compulsory. 2. 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. 3. Question numbers 1 to 8 in Section A are multiple choice questions where you are to select one correct option out of the given four. 4. There is no overall choice. 5. Use of calculator is not permitted. SECTION A 1. 2. The discriminant of quadratic equation x2 – 4x + 1 = 0 is : (a) –20 (b) 20 (c) 12 (d) 12 In figure, PQ and PR are tangents to the circle with O, such that QPR = 50°, OQR is equal to : Q O P R 141 [X – Maths] 3. 4. 5. 6. (a) 25° (b) 30° (c) 40° (d) 50° How many tangent(s) to a circle can be constructed from an external point? (a) Three (b) One (c) Two (d) No The length of the shadow of a tower is 3 times that of its length. The angle of elevation of the sun is : (a) 90° (b) 30° (c) 45° (d) 60° Which of the following number cannot be the probability of an event ? (a) 1 3 (b) 0.1 (c) 3% (d) 17 16 If the probability of happening of an event is 3 , then the probability of 7 non-happening of this event is : 7. 8. (a) 0 (b) 1 (c) 2 7 (d) 4 7 The mid-point of segment AB is the point (0, 4). If the co-ordinates of B are (–2, 3), then the co-ordinates of A are : (a) (2, 5) (b) (–2, –5) (c) (2, 9) (d) (–2, 11) If the area of a circle is numerically equal to twice its circumference, then the diameter of the circle is : (a) 4 units (b) units (c) 8 units (d) 2 units 142 [X – Maths] SECTION B 9. If the equation kx2 – 2kx + 6 = 0 has equal roots, then find the value of k. 10. Which term of the arithmetic progression 3, 10, 17, .... will be 84 more than its 13th term ? 11. Prove that in two concentric circles, the chord of the larger circle, which touches the smaller circle is bisected at the point of contact. 12. In the given figure, BOA is a diameter of a circle and the tangent at a point P meets BA produced at T. If PBO = 30°, what is the measure of PTA? P B 13. 14. 30° O A T An integer is chosen between 0 and 100. Find the probability that the number is dividible by 3 and 5 both. 22 If the perimeter of a protractor is 72 cm, calculate its area. Take 7 SECTION C 15. Solve for x : 1 1 1 1 ; where a 0, b 0, x 0 and a + b + x 0. ab x a b x 16. The first and the last terms of an A.P. are 8 and 350 respectively. If its common difference is 9, how many terms are there and what is their sum? 17. Draw a triangle ABC in which BC = 6.5 cm, AB = 4.5 cm and ABC = 3 60°. Also construct a triangle similar to this triangle whose sides are 4 of the corresponding sides of the triangle ABC. 18. A 1.2 m tall girl spots a balloon moving with the wind in a horizontal line at a height of 88.2 m from the ground. The angle of elevation of the balloon from the eye of the girl at any instant is 60°. After sometimes the 143 [X – Maths] angle of elevation reduces at 30°. Find the distance travelled by balloon during the interval (Take 3 = 1.7). 19. Find a point on the y-axis which is equidistant from the points A(6, 5) and B(–4, 3). 20. In what ratio does the point (½, 6) divide the line segment joint the point (3, 5) and (–7, 9). 21. A chord 10 cm long is drawn in a circle whose radius is 50 cm. Find the area of segments. 22. Find the area of the shaded region in the given figure, if PQ = 24 cm, PR = 7 cm and O is the centre of the circle. Q O R P 23. The largest sphere is curved out of a cube of side 7 cm. Find the volume of sphere. 24. A glass cylinder with diameter 20 cm has water to a height of 9 cm. A metal cube of 8 cm edge is immersed in it completely. Calculate the 22 height by which water will rise in the cylinder. Use . 7 SECTION D 25. 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. 26. In a school, students thought of planting trees in and around the school to reduce air pollution. It was decided that the number of trees that each section of each class will plant, will be the same as the class in which they are studying, e.g., a section of class I will plant one tree, a section of class II will plant two trees and so on till class XII. There are three sections of each class. Find : (a) How many trees will be planted by the students? 144 [X – Maths] (b) Which mathematical concept is used in above problem? (c) Which value is depicted in this problem? 27. The sum of 5th and 9th terms of A.P. is 72, and the sum of 7th and 12th terms is 97. Find the A.P. 28. Prove that the length of tangents drawn from an external point to a circle are equal. 29. Prove that the intercept of a tangent between a pair of parallel tangents to a circle subtend a right angle at the centre of the circle. (Figure is given). O 30. There is a small island in the middle of a 100 meter wide river and a tall tree stands on the island. P and Q are points directly opposite to each other on two banks and in line with the tree. If the angles of elevation of the top of the tree from P and Q are respectively 30° and 45°, find the height of the tree. 31. All the three face cards of spades are removed from a well shuffled pack of 52 cards. A card is drawn at random from the remaining pack. Find the probability of getting : (a) a black-faced card (b) a queen. (c) a black card (d) a spade 32. Find the value of K, if the points A (2, 3), B (4, K) and C (6, –3) are collinear. 33. A cylinderical bucket, 32 cm high and with radius of base 18cm, is filled with sand. This bucket is emptied on the gound and a conical heap of sand is formed. If the height of the conical heap is 24 cm, find the radius and slant height of the heap. 34. A cap is shaped like the frustum of a cone. If its radius on the open side is 10 cm, and radius at the upper base is 4 cm and its slant height is 15 cm, find the area of cloth used for making it. 145 [X – Maths]

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