2021 Mathematics (Basic) Past Paper PDF

## CENTRE FOR ADVANCEMENT OF STANDARDS IN EXAMINATIONS (GEMS ASIAN SCHOOLS) COMMON REHEARSAL EXAMINATIONS – JANUARY 2021 (ALL INDIA SECONDARY SCHOOL EXAMINATION) Subject: Mathematics (Basic) Subject Code: 241 Time: 3 Hours Max. Marks: 80 ### General Instructions: 1. This question paper contains tw...

## CENTRE FOR ADVANCEMENT OF STANDARDS IN EXAMINATIONS (GEMS ASIAN SCHOOLS) COMMON REHEARSAL EXAMINATIONS – JANUARY 2021 (ALL INDIA SECONDARY SCHOOL EXAMINATION) Subject: Mathematics (Basic) Subject Code: 241 Time: 3 Hours Max. 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. Internal choice is provided in 5 questions. 3. Section II has four case study-based questions. 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 2 marks 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. ### Part-A ### Section-I Section I has 16 questions of 1 mark each. Internal choice is provided in 5 questions. 1. Without actual division, determine whether 202/625 is terminating or non terminating. 2. AB is a tangent to the circle with centre O. If ∠BPQ = 39°, then find ∠POQ. [Diagram is of a circle with centre O, point P on the circumference, AB as the tangent to the circle at point B, point Q on the circumference, and line PQ connecting point P and Q.] 3. The HCF of two numbers is 145 and their LCM is 2175. If one number is 725, then find the other number. **OR** If a = x6y3 and b = x²y5, where x and y are prime numbers, then find the LCM (a,b). 4. Is the system of linear equations 2x+3y-9=0 and 4x+6y-18=0 consistent? Justify your answer. 5. What is the H.C.F. of two consecutive odd natural numbers? 6. In the given figure, AB is the diameter where AP = 12 cm and PB =16 cm. Find the perimeter of the shaded region in terms of π. [Diagram is of a circle with centre O, point P on the circumference, point A on the circumference, AB as the diameter, and line PB connecting point P and B.] **OR** The length of the minute hand of a clock is 14 cm. Find the area swept by the minute hand in 15 minutes. 7. In the figure, P and Q are points on AC and AB respectively such that PQ || CB, PC/AC = 3/4 and BQ = 4.5cm. Find AB. [Diagram is of a triangle ABC, point P on AC, point Q on AB, and line PQ parallel to BC.] **OR** In the given figure, T and B are right angles. If the lengths of TS, BC and AS are 8 cm, 16 cm and 17 cm respectively, then find the length of AC. [Diagram is of a triangle ABC, point S on AC, point T on BC, and line AS connecting point A and S.] 8. To draw a pair of tangents to a circle which are inclined to each other at an angle of 60°, it is required to draw tangents at the end points of two radii of the circle. Find the angle between the two radii of the circle. 9. If the angle of depression of a car from the top of a 75 m high tower is 30º, find the distance of the car from the foot of the tower. 10 In a circle of radius 14 cm, an arc subtends an angle of 450 at the centre. Find the length of the arc. 11 For the following distribution, find the median class. | Class | Frequency | |---|---| | 0-5 | 10 | | 5-10 | 15 | | 10-15 | 12 | | 15-20 | 20 | | 20-25 | 9 | 12. If the quadratic equation 2x2 – 10x + k = 0 has two real and equal roots, then what is the value of k? 13. Find the values of k for which the pair of equations given below has unique solution? kx + 2y = 5 3x - y = 1 **OR** If x = a, y = b, is the solution of the equations x - y = 2 and x + y = 4, then the find the values of a and b. 14. One ticket is drawn at random from a bag containing tickets numbered 1 to 40. What is the probability that the selected ticket has a number, which is a multiple of 5? **OR** One card is drawn from a well-shuffled deck of 52 cards. Calculate the probability that the card drawn is an ace. 15. If tan 20 = √3, then find the value of θ. 16. The probability of getting a non-defective pen from a lot of 1000 of pens is 24/25. Find the number of defective pens in the lot. ### Section-II Case study-based questions are compulsory. Attempt any 4 sub parts from each question. Each question carries 1 mark. 17. Students of a school are standing in rows and columns in their playground for a drill practice. A, B, C and D are the positions of four students as shown in figure. [Diagram is of a grid with x and y axis. Each mark is numbered from 1 to 13 on both axes. Points A, B ,C and D are marked on the grid. A is in the 3rd column and 4th row . B is in the 9th column and 7th row, C is the 12th column and 2nd row, and D is in the 7th column and 1st row.] Considering O as the origin, answer the following questions: i) The coordinates of A and B are (a) A (3, 5), B (7,8) (b) A (3, 4), B (9, 7) (c) A (3, 4), B (7, 9) (d) A (3, 5), B (7, 9) ii) In the drill if Jaspal stands in such a way that he is equidistant from all of the four students A, B, C and D. What would be the position (coordinates) of Jaspal? (a) (7,5) (b) (5,7) (c) (0,5) (d) (7,0) iii) What is the distance between B and D? (a) 4√2 units (b) 1 unit (c) 8 units (d) 3 units iv) What is the distance between A and C? (a)) 4√2 units (b) unit (c) units (d) 8 units v) What is the most appropriate name of the figure obtained by joining the points A, B, C and D? (a) Rectangle (b) Parallelogram (c) Rhombus (d) Square 18. A pen stand made of wood is in the shape of a cuboid with four conical depressions to hold pens. The dimensions of the cuboid are 15 cm by 10 cm by 3.5 cm. The radius of each of the depressions is 0.5 cm and the depth is 1.4 cm. (see the figure). [Diagram is of a cuboid, with a rectangle on the top and 4 conical depressions inside the cuboid. ] i) What is the formula to find the volume of a cone? a) (1/2).πr²h b) (1/3).πr²h c) πr²h d) πr1 ii) Volume of the wooden cuboid is: a) 5250 cm³ b) 150cm³ c) 350 cm³ d) 525 cm³ iii) Volume of wood removed in order to make the conical depressions is: a) 1.47cm³ b) 14.7 cm³ c) 147cm³ d) 35.7 cm³ iv) What is the volume of wood remaining in the entire stand? a) 523.53cm³ b) 378cm³ c) 539.7cm³ d) 335.3cm³ v) What is the ratio of the volume of a cone and a cylinder having the same radius and height? a) 1:3 b) 1:2 c) 3:1 d) 2:1 19. The reflector of dish antenna follows a mathematical shape as shown in the figure. Answer the following questions below. [Diagram is of a parabola. ] i) The polynomial which forms the above curve is a) Linear b) quadratic c) cubic d) None of these ii) How many zeroes are there for the polynomial (shape of the reflector)? a) 2 b) 3 c) 1 d) 0 iii) The zeroes of the polynomial are a) 1, 1 b) 2, 2 c) 1, -1 d) 0,0 iv) What will be the expression of the polynomial? a) x2 b) x2 + x -1 c) x²+x d) x² -1 v) What is the value of the polynomial if x = 2? a) 6 b) 4 c) 3 d) 5 20. A dilation stretches or shrinks a figure. The image created by a dilation is similar to the original figure. The scale factor (k) of a dilation is the ratio of corresponding side lengths. The center of a dilation is a fixed point in the plane about which all points are expanded or contracted. [Diagram is of a triangle ABC. A'B'C' is image of a triangle ABC after dilation. The centre of dilation is marked as C.P.] A' B i) From the above given information, triangles ABC and A' B' C ' are similar. Which of the following options will hold correct? (a) ∆ ВАС ~ AB'C' A' (c) Д СВА ~ AB'C' A' C B ii) 6 m D A (b) ДВСА ~ AC' B' Α' (d) A CAB AC' A' B' ~ A vertical pole of length 6 m casts a shadow 4 m long on the ground and at the same time a tower casts a shadow 28 m long. Find the height of the tower. E (a) 42 m iii) B 4 m c (b) 24 m F In the given figure, ADEF is a dilation of ДАВС. OD = 2 OA OE = 2 OB OF = 2 OC Find by which criteria ΔΟΑΒ-ΔODE 1 1 (c) 7 m (d) 12 m 1 center of O dilation B 6 A 8 12 (a) RHS (b) SAS (c) AAS (d) SSS iv) Find the ratio, area (∆ABC): area (ADEF) (a) 1:4 (b) 4:1 (c) 2:1 (d) 1:2 E 8 F 16 1 v) In the figure given, the ratio of PR to PT is equal to the ratio of PQ to PS. If PR = 25 cm, RT = 50 cm and RQ, is 4 cm, find the length of TS. [Diagram is of a figure having points P, Q, R, S and T. Line PQ is parallel to line RS] (a) 14 cm (b) 12 cm (c) 8 cm (d) 11 cm ## Part -B All questions are compulsory. In case of internal choices, attempt anyone. 21. A line intersects the y-axis and the x-axis at points P and Q respectively. If (2, -5) is the 2 mid-point of PQ, then find the co-ordinates of P and Q. **OR** If the distance between the point (4, p) and (1, 0) is 5, then find the value of p. 22. Prove that the area of the equilateral triangle drawn on the hypotenuse of a right-angled triangle is equal to the sum of the areas of the equilateral triangles drawn on the other two sides of the triangle. 23. A The circle of touches the sides BC, CA and AB at D, E and F respectively. If AB= AC, prove that BD = CD [Diagram is of a triangle ABC, with a circle inside, touching all its sides. Points D, E and F are marked on the sides of the triangle.] E F B D C 24. Find the 9th term from the end of the A.P: 5, 9, 13, . . . . . . 185 25. Evaluate: (cos 30° + sin 60°)/(1 + cos 60° + sin 30°) **OR** In A PQR, right-angled at Q, PR + QR = 25 cm and PQ = 5 cm. Determine the value of sin P. 26. If a large sized pizza of diameter 35 cm is kept on a rectangular base of dimensions 50 cm x 60 cm. Find the area of the base excluding the pizza. [Diagram is of a pizza. ] ## Part -B All questions are compulsory. In case of internal choices, attempt anyone. 27. Prove that 3 +2√5 is irrational, given that √5 is irrational. 28. Draw a circle of diameter 8 cm. From a point 10 cm away from its centre, construct a pair of tangents to the circle. Measure the lengths of the tangents. 29. Four years hence the age of Roney is 8 more than 3 times his son's age. Four years ago, Roney's age was 9 times that of his son. Find their present ages. 30. Two dice are thrown simultaneously. What is the probability i) that the sum of the two numbers appearing on the top of the dice is a prime number? ii) that the sum of the two numbers appearing on the top of the dice is 1? iii) of getting a doublet? **OR** A box contains cards bearing numbers from 5 and 70. If one card is drawn at random from the box, find the probability that it bears (i) a one-digit number. (ii) an odd number less than 30. (iii) a composite number between 50 and 70. 31. A solid cylinder of diameter 12cm and height 15cm is melted and recast into toys in the form of a right circular cone mounted on a hemisphere of radius 3cm. If the height of the toy is 12cm, find the number of toys so formed. 32. Prove that: (sin30+cos30)/(sine+cose) + sin cos 0 = 1. 33. Solve the quadratic equation for x: 4x2-4ax + (a²-b²) = 0 **OR** 16/(x) - 1/(x+1) = 15, x≠0.-1 ## Part B All questions are compulsory. In case of internal choices, attempt any one. 34. Two pillars of equal height are standing opposite cach other and on either side of a road 5 which is 100 m wide. The angles of elevation of the top of the pillars are 600 and 300 from a point on the road between the pillars. Find the height of each pillar and the distance of the point from the pillars. (Take √3 = 1.732) **OR** As observed from the top of a 75 m high light house from the sea level, the angles of depression of two ships are 300 and 45º. If one ship is exactly behind the other on the same side of the light house, find the distance between the two ships. [Take √3 =1.732] 35. The 2nd term of an AP is 10 and the 5th term is 19. i) Find the AP. ii) Find the sum of first 25 terms. 36. Find the mean and mode for the following data: | Marks obtained | Number of students | |---|---| | 25-35 | 7 | | 35-45 | 31 | | 45-55 | 33 | | 55-65 | 17 | | 65-75 | 11 | | 75-85 | 1 | 37. The International Kite Festival (Uttarayan) is one of the biggest festivals celebrated in Gujarat. Months beforehand, homes in Gujarat begin to manufacture kites for the festival. [Diagram is of a kite flying in the sky.] **OR** A boy is standing on the ground and flying a kite with 150 m of string at an elevation of 30°. At the same time another boy is standing on the roof of a 25 m high building and is flying his kite at an elevation of 45°. Both the boys are on opposite sides of the kites and the length of kite's string of the boy on the roof is such that the two kites meet. The following figure represents this situation diagrammatically. [Diagram is of :A boy standing on the ground and flying a kite attached to a string of 150 m at an elevation of 30°. Another boy is standing on the roof of a 25 m high building and flying a kite at an elevation of 45°. ] Based on the above information answer the following question I. What is the distance of the kites from the ground when they meet? II. If the building casts a shadow of 25/√3 m on the ground, then find the angle of elevation of Sun at that time. III. Find the length of the string that the boy standing on the roof must have so that the two kites meet. (Use √2 = 1.41) **OR** Find the distance between the boy standing on the ground and the building. (Use √√3 = 1.73) 38: The best use of Geometry in daily life is the construction of buildings, dams, rivers, roads, etc. The applications of coordinate geometry in daily life can also be found in interior design. Setting new items in an open space is done perfectly using the concepts of coordinate geometry. Sara is considering two different layouts for her new garden. The following diagram shows both layouts on a coordinate grid. [Diagram is of a coordinate grid with points O,M,L,P,Q,A,S,R,B, and N marked. Line LS is parallel to line MR. Point O is at the origin.] I. Find the co-ordinates of the points O and M. II. What is the length of diagonal OM? III. Find the ratio in which x-axis divides the line segment joining the points P and S. **OR** Find a relation between x and y such that the point (x, y) is equidistant from the points Land M.