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K.R. MANGALAM WORLD SCHOOL, GK-II Class- X 2020-21 Subject- Mathematics Pre-Board Examination Time Allowed: 3 Hours Maximum Marks: 80 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 of two 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 casebased 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 2 mark each, 2. Question 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. Page 1 of 11 PART-A Section I has 16 questions of 1 mark each. Internal choice is provided in 5 questions. 1. Write down the decimal expansion of 6 3125 without actual division. OR HCF(144,198) = 18, find LCM(144,198) 2. If one of the zeroes of the quadratic polynomial 2 2 8 is , find the value of m. 3. For what values of m and n will the following pair of linear equations have infinitely many solutions? 2 + 3 = 11 and ( + ) + (2 ) = 33. 4. Express the given statements in the form of a linear equation in two variables. When one is added to numerator and 4 is subtracted from denominator, the fraction becomes 1. 5. For what value of k will k+9, 2k-1 and 2k+7 are the consecutive terms of an A.P.? OR How many multiples of 4 lie between 10 and 260? 6. Find the value of p for which the quadratic equation x2 + px + 3 = 0 has equal roots. 7. Find the roots of 2 2 + 7 + 5 2 = 0 Solve for x: OR 2 ( 2 + 1) + 2 = 0 8. If in a , = 90 , = 5 = 12 . , then find AD. Page 2 of 11 9. Two concentric circles are of radii a and b(a>b). Find the length of the chord of the larger circle which touches the smaller circle. OR In figure 2 below, AB is a chord of the circle and AOC is its diameter, such that ACB = 50 . If AT is the tangent to the circle at the point A, find BAT. Figure-1 Figure - 2 10.In figure 1 above, two circles touch each other externally at C and AB is common tangent of circles, then find ACB. 11.To divide a line segment AB in the ratio 3:4, first, a ray AX is drawn so that BAX is an acute angle and then at equal distances points are marked on the ray AX. What is the minimum number of these points on the ray AX. 12. If = , then find the value of + 13. If = and = , then find the value of ( ) 14. What is the area of the circle that can be inscribed in a square of side 6cm? 15. A cone, a hemisphere and a cylinder stand on equal bases and have the same height. What is the ratio of their volumes? 16. Find the probability that a non- leap year has 53 Sundays. OR Find the probability of not getting a doublet in a throw of a pair of dice. Page 3 of 11 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: LAVANYA S BALL Lavanya throws a ball upwards, from a rooftop, which is 20 m above from ground. It will reach a maximum height and then fall back to the ground. The height of the ball from the ground at time t is h , which is given by h = 4t2 + 16t + 20 On the basis of the above information, answer any four of the following questions: (i) What is the height reached by the ball after 1 second? (a) 64 m (b) 128 m (c) 32 m (d) (ii) 20 m Name the shape of the curve formed by the trajectory of the ball. (a)Circle (b) Parabola (c) Ellipse (d) none of these (iii) How long will the ball take to hit the ground? (a) 4 seconds (b) 3 seconds (c) 5 seconds (d) 6 seconds (iv) What are the two possible times to reach the ball at the same height of 32 m? (a)1 and 3 seconds (b) 1.5 and 2.5 seconds (c) 0.5 and 2.5 seconds (d) 1.6 and 2.6 seconds (v) Where is the ball after 5 seconds ? (a) at the ground (b) rebounds (c) at highest point (d) fall back Page 4 of 11 18. Case Study -2- PRE BOARD MARKS OF STUDENTS The graph given below represent the marks obtained by students of class X in preboard examination. On the basis of the above information, answer any four of the following questions: i. Find the mean marks obtained by the students of class X. a. 51.2 b. 61.2 c. 50.56 d. 62 ii. What will be the upper limit of the modal class? a. 60 b. 100 c. 80 d. 70 iii. The sum of lower limit and the upper limit of the median class is: a. 100 b. 140 c. 180 d. 60 iv. The measure of central tendency that cannot be read from the above graph directly without using any formula: a. Mean b. Mode c. Median d. All of these v. In the formula for finding the mean of grouped data using assumed mean method, are the deviations from a of : a. Lower limits of the classes b. upper limits of the classes c. mid-points of the classes d. Frequencies of the class marks. Page 5 of 11 19.Case study 3 AN AQUARIUM An aquarium is a transparent tank of water in which fish and other water creatures and plants are kept. The diagrams below show the plans for an aquarium. It will be built in hexagonal shape. It will be made using six rectangular shaped clear glasses. One regular hexagon clear glass for roof. On the basis of the above information, answer any four of the following questions: Refer to top view (i) The mid-point of the line segment joining the points E(7,12) and B(12, 14) is (a) (6,8) (b) (19/2, 13) (c) (5,10) (d) (19/2, 15) (ii) The value of x for which the distance between the points F(2,-3) and C(x,5) is 10, is (a) 5 or 10 (b) 4 or 8 (c) -5,-10 (d) 8 or -4 Refer to Front view (iii) Find the distance of the point ( 6, 8) from the origin. (a) 8 (b) 11 (c) 10 (d) 9 (iv) Find the ratio in which the line joining the points (6, 4) and (1, 7) is divided by x-axis. (a) 1 : 3 (b) 2:7 (c) 4 : 7 (d) 6:7 (v) Find the coordinates of the point equidistant from the points A(1, 2), B (3, 4) and C(5, 6). (a) (2, 3) (b) (11, 2) (c) ( 1 , 2) (d) (1, 3) Page 6 of 11 20.Case study 4 The picture given below represent the formation of shadow of two different objects at the same time. On the basis of the above information, answer any four of the following questions: i. By which similarity criteria the two triangles can be proven similar? a. AA b. AAS c. SSS d. SSA ii. Find x in the figure. a. 14 m b. 12 m c. 8 m d. 24 m iii. If the ratio of the corresponding sides of the triangles shown above is 1:8, what is the ratio of their corresponding medians? a. 1:64 b. 1:2 c. 8:1 d. 1:8 iv. What will affect the similarity of any two polygons? a. Translated down b. Enlarge by scale factor c. Flipped horizontally d. None of these v. If value of x is 12 m what would be the length of the stick? a. 1.5 m b. 2 m c. 0.75 m d. 1 m Page 7 of 11 PART- B Section III 21.A circular field has a circumference of 360 km. Three cyclists start together and can cycle 48, 60 and 72 km a day, round the field. When will they meet again?. 22.If the points A(4, 3) and B(x, 5) are on the circle with the centre O(2, 3), find the value of x. OR A (-2, 5) and B (3, 2) are two points. Find the co-ordinates of the point R on AB such that AR=3BR 23. Find a quadratic polynomial whose zeroes are 7 3 5 and 7+3 5. 24. Draw a line segment PQ = 9cm. With P as centre draw a circle of radius 3cm. Construct two tangents from the point Q to the circle drawn. 25. Prove that = 2 2 . OR If A = 60 and B = 30 , verify that sin (A B) = sin A cos B cos A sin B. 26. AB is a diameter and AC is a chord of a circle with centre O such that BAC = 30 . The tangent at C intersects extended AB at a point D. Prove that BC = BD. Section IV 27. Prove that 1 3 is irrational. 28. and are zeroes of the quadratic polynomial x2 6x + y. Find the value of y if 3 + 2 = 20. OR If and are zeroes of the quadratic polynomial 4x2 + 4x + 1, then form a quadratic polynomial whose zeroes are 2 and 2 . Page 8 of 11 29. The following table shows the ages of the patients admitted in a hospital during a year. Find the mode and the median of the above data if the mean is 35.4 years. 30. In the figure, ABC is right angled at C and DE AB. Prove that ABC ~ ADE and hence find the lengths of AE and DE. OR In the figure, M is the mid-point of side CD of a parallelogram ABCD. The line BM is drawn intersecting AC at L and AD produced at E. Prove that EL = 2BL. Page 9 of 11 31. Find the value of p, if the mean of the following distribution is 18: 32. A man rowing a boat away from a lighthouse 150 m high takes 2 minutes to change the angle of elevation of the top of lighthouse from 45 to 30 . Find the speed of the boat. (Use 3 = 1.732) 33. In Figure, APB and CCD are semi-circles of diameter 7 cm each, while ARC and a BSD are semi-circles of diameter 14 cm each. Find the perimeter of the region. [Use = 22/7]. Section V 34. The angle of elevation of a cloud from a point 60 m above a lake is 30 and the angle of depression of the reflection of the cloud in the lake is 60 . Find the height of the cloud from the surface of the lake. Page 10 of 11 OR A bird is sitting on the top of a tree, which is 80 m high. The angle of elevation of the bird, from a point on the ground is 45 . The bird flies away from the point of observation horizontally and remains at a constant height. After 2 seconds, the angle of elevation of the bird from the point of observation becomes 30 . Find the speed of the bird. 35. Water is flowing through a cylindrical pipe into a cylindrical tank of base radius 40 cm at the rate of 2.52 km/hr. If the increase in the level of the water in the tank, in half an hour is 3.15 m, find the internal diameter of the pipe. 36. Nine times the side of one square exceeds the perimeter of a second square by one metre and six times the area of the second square exceeds twenty-nine times the area of the first by one square metre. Find the side of each square. Page 11 of 11
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