QP_MATHS

The Star Global School MATHEMATICS GRADE-10 Class 10 - Mathematics Time Allowed: 3 hours Maximum Marks: 80 General Instructions: 1. This Question Paper has 5 Sections A, B, C, D and E. 2. Section A has 20 MCQs carrying 1 mark each The Star Global School3. Section B has 5 questions carrying 02 marks each. 4. Section C has 6 questions carrying 03 marks each. 5. Section D has 4 questions carrying 05 marks each. 6. Section E has 3 case based integrated units of assessment (04 marks each) with sub- parts of the values of 1, 1 and 2 marks each respectively. 7. All Questions are compulsory. However, an internal choice in 2 Qs of 5 marks, 2 Qs of 3 marks and 2 Questions of 2 marks has been provided. An internal choice has been provided in the 2marks questions of Section E 8. Draw neat figures wherever required. Take π = 22  wherever required if not stated. [1] 7 Section A 1. The exponent of 3 in the prime factorization of 864 is: a) 2 b) 3 c) 4 d) 8 [1] 2. The graph of a polynomial is shown in Figure, then the number of its zeroes is: a) 4 b) 3 [1] c) 1 d) 2 3. The pair of equations x + 2y + 5 = 0 and –3x – 6y + 1 = 0 have a) a unique solution b) infinitely many solutions 1/7

c) no solution d) exactly two solutions 4. The quadratic equation 2x2 – – + 1 = 0 has [1] √5x [1] [1] a) two equal real roots b) no real root [1] [1] c) two distinct real roots d) more than 2 real roots 5. In an AP if a = –7.2, d = 3.6, an = 7.2, then n is a) 3 b) 1 c) 5 d) 4 6. AOBC is a rectangle whose three vertices are A(0, 3), O(0, 0) and B(5, 0). The length of its diagonal is a) 5 b) 3 The Star Global School c) −− d) 4 √34 7. The centroid of a triangle divides the median in the ratio a) 2 : 1 b) 1 : 2 c) 1 : 3 d) 3 : 1 8. In the adjoining figure∠P QR = ∠P RS. If PR = 8cm, PS = 4 cm, then PQ is equal to a) 16 cm. b) 12 cm. c) 24 cm. d) 32 cm. 9. In Figure, PQ and PR are two tangents to a circle with centre O. If ∠QPR = 46°, then ∠QOR equals [1] a) 67o b) 46o c) 134o d) 44o 10. From an external point Q, the length of the tangent to a circle is 5 cm and the distance of Q from the centre is 8 [1] cm. The radius of the circle is: a) 3 cm b) 7 cm c) 39 cm d) −− cm √39 11. If x = r sin θ cos ϕ, y = r sin θ sin ϕ and z = r cos θ, then [1] a) x2 + y2 + z2 = r2 b) x2 - y2 + z2 = r2 2/7

c) z2 + y2 - x2 = r2 d) x2 + y2 - z2 = r2 12. If x cos A = 1 and tan A = y, then the value of x2 - y2 is [1] a) -1 b) 0 c) 1 d) 2 13. From the top of a building 60m high, the angles of depression of the top and the bottom of a tower are observed [1] to be 30° and 60°. The height of the tower is a) 40 m b) 60 m c) 45 m d) 50 m 14. A piece of paper in the shape of a sector of a circle (see figure 1) is rolled up to form a right-circular cone (see [1] figure 2). The value of angle θ is : The Star Global School a) 5π b) 10π 13 13 c) 9π d) 6π 13 13 15. If the area of a sector of a circle is 5 of the area of the circle, then the sector angle is equal to [1] 18 [1] [1] a) 100° b) 120° [1] [1] c) 90° d) 60° 16. Which of the following cannot be the probability of an event? a) - 1.5 b) 0.7 c) 2 d) 15% 3 17. If a two digit number is chosen at random, then the probability that the number chosen is a multiple of 3, is a) 3 b) 29 10 100 c) 7 d) 1 25 3 18. The median and mode of a frequency distribution are 26 and 29 respectively. Then, the mean is a) 28.4 b) 22.5 c) 25.8 d) 24.5 19. Assertion (A): In the given figure, a sphere circumscribes a right cylinder whose height is 8 cm and radius of the base is 3 cm. The ratio of the volumes of the sphere and the cylinder is 125 : 54 Reason (R): Ratio of their volume = V olume of sphere V olume of cylinder a) Both A and R are true and R is the correct b) Both A and R are true but R is not the explanation of A. correct explanation of A. 3/7

c) A is true but R is false. d) A is false but R is true. 20. Assertion (A): Sum of first n terms in an A.P. is given by the formula: Sn = 2n ×[2a + (n - 1)d] [1] Reason (R): Sum of first 15 terms of 2 , 5 , 8 ... is 345. a) Both A and R are true and R is the correct b) Both A and R are true but R is not the explanation of A. correct explanation of A. c) A is true but R is false. d) A is false but R is true. Section B 21. Prove that 1 is irrational. [2] √2 22. In Fig. check whether AD is the bisector of ∠A of ΔABC if AB =6 cm, AC = 8 cm, BD = 1.5 cm and CD = 2 [2] cm The Star Global School 23. In the given figure, common tangents AB and CD to the two circles with centres O1 and O2 intersect at E. Prove [2] that AB = CD. 24. Prove 3 = tan θ, where the angles involved are acute angles for which the expressions are defined. [2] sin θ−2 sin θ 2 cos2 θ−cos θ OR If sin (A + B) = 1 and sin (A -B ) = 1 , 0≤A + B = 90° and A > B, then find A and B. 2 25. In a circle of radius 10 cm, an arc subtends an angle of 108° at the centre. What is the area of the sector in terms [2] of π? OR The circumference of a circle is 8 cm. Find the area of the sector whose central angle is 72°. Section C 26. Renu has collected 8 U.S. stamps and 12 international stamps. She wants to display them in identical groups of [3] U.S. and international stamps, with no stamps left over. What is the greatest number of groups Renu can display them in? 27. Read the following statement carefully and deduce about the sign of the constants p, q, and r. [3] “The zeroes of a quadratic polynomial 2 + qx + r are both negatives.” px 28. Solve the system of equations by using the method of substitution: [3] 2x + 3y=9 3x + 4y=5 OR The angles of a cyclic quadrilateral ABCD are ∠A = (6x + 10)°, ∠B = (5x)°, ∠C = (x + y)°, ∠D = (3y – 10)° Find x and y, and hence the values of the four angles. 4/7

29. In an isosceles triangle ABC in which AB = AC = 6 cm is inscribed in a circle of radius 9 cm, find the area of [3] the triangle. OR In the given figure, a circle is inscribed in a triangle PQR. If PQ = 10 cm, QR = 8 cm and PR = 12 cm, find the lengths of QM, RN and PL. 30. Prove the following identity: sin θ + tan θ =​​​​secθ ⋅ cos ecθ + cotθ [3] 1−cos θ 1+cos θ [3] 31. Find the mean of the following data, using step-deviation method: The Star Global School Class 5 - 15 15 - 25 25 - 35 35 - 45 45 - 55 55 - 65 65 - 75 Frequency 6 10 16 15 24 8 7 Section D 32. A journey of 192 km from a town A to town B takes 2 hours more by an ordinary passenger train than a super [5] fast train. If the speed of the faster train is 16 km/h more, find the speed of the faster and the passenger train. OR Swati can row her boat at a speed of 5 km/hr in still water. If it takes her 1 hour more to row the boat 5.25 km upstream than to return downstream, find the speed of the stream. 33. In an isosceles ΔABC, the base AB is produced both the ways to P and Q such that AP × BQ = AC 2. Prove [5] that ΔAP C ∼ ΔBC Q. 34. A spherical glass vessel has a cylindrical neck 8 cm long and 1 cm in radius. The radius of the spherical part is 9 [5] cm. Find the amount of water (in litres) it can hold, when filled completely. OR A toy is in the form of a cone mounted on a hemisphere. The diameter of the base of the cone is 7 cm and its height is 15.5 cm. Find the volume of the toy. (Use π = 3.14). 35. The median of the following data is 525. Find the values of x and y, if the total frequency is 100. [5] Class interval Frequency 0-100 2 100-200 5 200-300 x 300-400 12 400-500 17 500-600 20 5/7

600-700 y 700-800 9 800-900 7 900-1000 4 Section E 36. Read the text carefully and answer the questions: [4] Kamla and her husband were working in a factory in Seelampur, New Delhi. During the pandemic, they were asked to leave the job. As they have very limited resources to survive in a metro city, they decided to go back to their hometown in Himachal Pradesh. After a few months of struggle, they thought to grow roses in their fields and sell them to local vendors as roses have been always in demand. Their business started growing up and they The Star Global School hired many workers to manage their garden and do packaging of the flowers. In their garden bed, there are 23 rose plants in the first row, 21 are in the 2nd, 19 in 3rd row and so on. There are 5 plants in the last row. (i) How many rows are there of rose plants? (ii) Also, find the total number of rose plants in the garden. OR If total number of plants are 80 in the garden, then find number of rows? (iii) How many plants are there in 6th row. 37. Read the text carefully and answer the questions: [4] The design of Christmas tree is shown in the following graph: (i) What is the distance of point A from x-axis? (ii) What is the Length of BC? OR What is the perimeter of its trunk LMPN?​ (iii) What is the Length of FG? 38. Read the text carefully and answer the questions: [4] Basant Kumar is a farmer in a remote village of Rajasthan. He has a small square farm land. He wants to do fencing of the land so that stray animals may not enter his farmland. For this, he wants to get the perimeter of the land. There is a pole at one corner of this field. He wants to hang an effigy on the top of it to keep birds away. He 6/7

The Star Global Schoolstanding in one corner of his square field and observes that the angle subtended by the pole in the corner just diagonally opposite to this corner is 60o. When he retires 80 m from the corner, along the same straight line, he finds the angle to be 30o. (i) Find the height of the pole too so that he can arrange a ladder accordingly to put an effigy on the pole. (ii) Find the length of his square field so that he can buy material to do the fencing work accordingly. OR Find the Distance from Farmer at position D and top of the pole? (iii) Find the Distance from Farmer at position C and top of the pole? 7/7