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KENDRIYA VIDYALAYA SANGTHAN KOLKATA REGION PRE-BOARD EXAMINATION 2020-21 CLASS: - X SUBJECT:- MATHEMATICS TIME:-3HOURS FM:- 80 This question paper consists of 11 pages General Instructions: 1. This question paper contains two parts A and B. 2. Both Part A and Part B have internal choices. Part A: 1. It consists three sections I and II 2. Section I has 16 questions of 1 mark each. Internal choice is provided in 5 questions. 3. Section II has 4 questions on case study. Each case study has 5 case-based sub-parts. An examinee is to attempt any 4 out of 5 sub-parts. Part B: 1. Question No. 21 to 26 are very short answer type questions of e marks each. 2. Questions No. 27 to 33 are Short Answer Type questions of 3 marks each. 3. Question No. 34 to 36 are Long Answer Type questions of 5 marks each. 4. Internal choice is provided in 2 questions of 2 marks, 2 questions of 3 marks and 1 question of 5 marks. Q. No. Part A Marks allocated Section 1 Section I has 16 questions of 1 mark each. Internal choice is provided in 5 questions. 1. Calculate the HCF and LCM of 33 5 and 32 52 . 1 1 OR The decimal representation of 345679 22 55 will terminate after how many decimal places? 2 2. If zeroes of the polynomial x2 + 4x + 2a are and , then find the value of . 1 3. For what values of k is the system of equations kx + 3y = k 2 , 12x + ky = k inconsistent? 1 4. 1 5. 10 students of class X took part in a Mathematics quiz. The number of girls is 4 more than the number of boys. Form the pair of linear equation to represent this situation. If k, (2k 1) and (2k + 1) are three successive terms of an A.P, find the value of k. OR The nth term of an A.P is (7 4n). Find its common difference. 6. Write the roots of the quadratic equation 2x2 x 6 = 0 1. 7. If one root of the quadratic equation 3x2 10x + 3 = 0 is 3 , then find the other root. 1 1 OR Write the nature of roots of the equation 2x2 6x + 7 = 0 . 1 8. A tangent PQ at a point P of a circle of radius 5 cm meets a line through the centre O at a point Q so that OQ = 13 cm. Find the length of PQ. 1 9. In the given fig., a circle inscribed in a triangle ABC, touches the sides AB, BC and AC at points D, E and F respectively. If AB = 12 cm, BC = 8 cm and AC = 10 cm, find the 1 lengths of AD. 2 OR In the given figure, PA and PB are two tangents to the circle with centre O. If APB = 50o then what is the measure of . 10. ABC, DE BC so that AD = 2.4 cm, AE = 3.2 cm and EC = 4.8 cm, then find the length AB. 11. To divide a line segment AB in the ratio 5:7, first a ray AX is drawn so that BAX is an 1 acute angle and then at equal distances points are marked on the ray AX . Write the minimum number of these points. 12. Evaluate:- 5 cos2 60 +4 sec2 30 tan2 45 1 1 sin2 30 +cos2 30 13. If tan A + cot A = 5, then find the value of tan2A + cot2A. 14. 1 The radii of two circles are 14 cm and 7 cm respectively. Find the radius of the circle which has circumference equal to the sum of the circumference of the two circles. A solid is in the shape of a cone standing on a hemisphere with both radii being equal to 1 1 cm and height of the cone is equal to its radius. Find the volume of the solid. A die is thrown once. Find the probability of getting a prime number. OR Gopi buys a fish from a shop for his aquarium. The shopkeeper takes out one fish at random from a tank containing 5 male fish and 8 female fish. What is the probability that the fish taken out is a male fish? 15. 16. 1 SECTION II Case study based questions are compulsory. Attempt any four sub parts of each question. Each subpart carries 1 mark. 17. Case Study 1 3 Everyday Use of Polynomials A polynomial isn't as complicated as it sounds, because it's just an algebraic expression with several terms. Usually, polynomials have more than one term, and each term can be a variable, a number or some combination of variables and numbers. Some people use polynomials in their heads every day without realizing it, while others do it more consciously. (i) A real number is called a zero of p(x), if p( ) = 0 . What will be the zeros of p(x) = x2 2x 3 are (a) (1, -3) (b) 3, - 1 (c) -3, - 1 (d) 1, 3 (ii) If the zeroes of the quadratic polynomial x2 + (a + 1) x + b are 2 and -3, then (a) a = -7, b = -1 (b) a = 5, b = -1 (c) a = 2, b = -6 (d) a = 0, b = -6 (iii) The zeroes of the quadratic polynomial x2 + 99x + 127 are (a) both positive (b) both negative (c) one positive and one negative (d) both equal (iv) Zeroes of a polynomial can be expressed graphically. Number of zeroes of polynomial is equal to the number of points where the graph of polynomial (a) intersects x-axis (b) intersect y axis (c) intersects y axis or x-axis (d) None of the above (v) If 1 is one of the zeroes of the polynomial x + x + k, then the value of k is: (a) 2 (b) -2 (c) 4 (d) -4 4 18. Case Study 2 Real Life Use of Pythagorean Theorem The Pythagorean Theorem is a statement in geometry that shows the relationship between the lengths of the sides of a right triangle a triangle with one 90-degree angle. The right triangle equation is a2 + b2 = c2. Being able to find the length of a side, given the lengths of the two other sides makes the Pythagorean Theorem a useful technique for construction and navigation. (i) If the lengths of sides of a right triangle are x cm,(x + 1) cm and (x + 2) cm, the value of x is (a) 5cm (b) 4cm (c) 3cm (d) 6cm (ii) Two buildings of height 13 m and 7 m respectively stand vertically on a plane ground at a distance of 8 m from each other. The distance between their top is (a) 9 m (b) 10 m (c) 11 m (d) 12 m (iii) (If the height of a tree is 5 m and a man stand 5 m away from its foot. The angle of elevation of the top of tree is (a) 30o (b) 45o (c) 60o (d) None of these. (iv) A ladder 25 m long just reaches the top of a building 24 m high from the ground. What is the distance of the foot of the ladder from the building? (a) 7 m (b) 14 m (c) 21 m (d) 24.5 m (v) If in the given figure, the height of tree is 20 m and its shadow on the ground is 20 3 long. The sun s altitude is (a) 300 (b) 45O (c) 60o (d) none of these 5 19. CASE STUDY - 3 USE OF DISTANCE FORMULA Do you remember what a plane is? A plane is any flat surface which can go on infinitely in both of the directions. Now, if there is a point on a plane, you can easily locate that point with the help of coordinate geometry. Using the two numbers of the coordinate geometry, a location of any point on the plane can be found. In coordinate geometry we can easily find the distance between two given points. The distance formula comes with some uses in everyday life. It can be used as a strategy for easy navigation and distance estimation. For example, if you want to estimate the distance of two places on a map, simply get the coordinate of the two places and apply the formula. (i) If the location of Bird is origin of the axis, and the location of Museum is (8, 6), the distance between Bird to Museum is (a) 8 unit (B) 2 7 unit (c) 6 unit (d) 10 unit (ii) The distance of the Museum(8,6) from x-axis is (a) 8 unit (b) 6 unit (c) 10 unit (d) 7 unit (iii) If Jones(5,6) is just between (at midpoint) Thelonious(6, 5) and Fitzgerlad(4, y), then y equals (a) 5 (b) 7 (c) 12 (d) 6 (iv) Suppose Simone lies on x-axis and it is equidistant from Miles(7, 6) and Armstrong(-3, 4), then the location of Simone is (a) (0, 4) (b) (- 4, 0) (c) (3, 0) (0, 3) 6 (v)If the distance between Spalding(4, p) and Coltrane (1, 0) is 5 then p equals to (b) 4 only (a) 4 only (c) 4 (d) 0 20. CASE STUDY - IV Lengths of Leaves A teacher gave an activity to investigate the lengths of leaves and make a frequency distribution table to class VI. In this activity a students collected some leaves from his garden and complete the given table. He just measured in a straight line as shown in the following diagram: Length (in mm) No of leaves 20 30 3 30 40 8 40 50 15 50 60 26 60 70 23 70 80 16 80 90 7 90 100 2 Total 100 (I) What is the median class of the given data? (a) 40 50 (b) 50 60 (c) 60 70 (ii) The class mark of the modal class is (a) 35 (b) 45 (c) 55 (d) 70 80 (d) 65 (iii) What is the sum of upper limit of modal class and lower limit of median class? (a) 90 (b) 110 (c) 130 (d) 150 7 (iv) If the mean length of leaf is 58.9 mm and modal length is 57.5 mm, then its median length (in mm) is (a) 58.43 (b) 38.8 (c) 54. 4 (d) 59.5 (v) Which of the following cannot be determined graphically? (a) Mean (b) Median (c) Mode (d) None of these PART B All questions are compulsory. In case on internal choices, attempt any one. 21. The HCF of 65 and 117 is expressible in the form of 65m 117. Find the value of m. Also, find its LCM. 22. Find a relation between x and y such that the point P9x, y) is equidistant from the points 2 A(- 5, 3) and B(7, 2). OR In what ratio does the line x y 2 = 0 divide the line segment joining the points A(3, -1) and B(8, 9) ? 23. 24. Form a quadratic polynomial whose zeroes are 3 + 2 and 3 2 Draw a circle of radius 6 cm with centre O and take a point P outside the circle such that OP = 10 cm. From P, draw two tangents to the circle. 2 2. 25. If 3cot A = 2, then find the value of 4 sin 3 cos 2 2 2 +6 cos OR If sin (A + B) = 1 and cos (A B) = 3 2 , find A and B 26. Prove that the lengths of tangents drawn from an external point to a circle are equal. 2 27 Prove that 2 - 3 5 is irrational, given that 3 is irrational. 3 28. Solve :- + + = + + , ( + ), , 3 OR 8 Solve :29. 30. 1 +3 + 1 2 1 = 11 7 +9 1 9 , 3, 2 , 7 In the adjoining fig. AB and CD are two diameters of a circle (with centre O) perpendicular to each other and OD is the diameter of the smaller circle. If OA = 7 cm, find the area of the shaded region. 3 The diagonals of a quadrilateral ABCD intersect each other at the point O such that 3 = . Show that ABCD is a trapezium. OR P and Q are points on the sides CA and CB of ABC, right angled at C. Prove that (AQ2 + BP2) = (AB2 + PQ2). 31 Find the mean of the following data. Class Interval 0 10 10 20 Frequency 8 12 20 30 10 30 40 11 40 50 9 If its mode is 16.7 (approx.), then find its median also. 32. 3 Two ships are there in the sea on either side of a light house in such a way that the ships and the light house are in the same straight line. The angles of depression of two ships are observed from the top of the light house are 60 and 45 respectively. If the height of the light house is 200 m, find the distance between the two ships. 33. Find the value of x and y in the following frequency distribution table, if N = 100 and its 3 median is 32. 34. Marks 0 10 10 20 20 30 30 40 40 50 50 60 Total No. of students 10 X 25 30 Y 10 100 From the top of a hill 200 m high, the angles of depression of the top and the bot tom of a pillar are 30o and 60o respectively. Find the height of the pillar and its distance 5 from the hill. [Take 3 = 1.732 ] 9 OR The angles of depression of the top and the bottom of a 7 m tall building from the top of a tower are 45o and 60o respectively. Find the height of the tower. [Take 3 = 1.732 ] 35. A cylindrical container is filled with ice-cream, whose diameter is 12 cm and height 15 cm. The whole ice-cream is distributed to 10 children in equal cones having hemispherical tops. If the height of conical portion is twice the diameter of its base, find the diameter of the ice-cream cone. 5 36. A man travels 300 km partly by train and partly by car. He takes 4 hours if the travels 60 km by train and the rest by car. If he travels 100 km by train and the remaining by car, he takes 10 minutes longer. Find the speeds of the train and the car separately. 5 10
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