Maths Standard Class X Session 2024-25 Sample Question Paper PDF

SAMPLE QUESTION PAPER Class X Session 2024-25 MATHEMATICS STANDARD (Code No.041) TIME: 3 hours MAX.MARKS: 80 General Instructions: Rea...

SAMPLE QUESTION PAPER Class X Session 2024-25 MATHEMATICS STANDARD (Code No.041) TIME: 3 hours MAX.MARKS: 80 General Instructions: Read the following instructions carefully and follow them: 1. This question paper contains 38 questions. 2. This Question Paper is divided into 5 Sections A, B, C, D and E. 3. In Section A, Questions no. 1-18 are multiple choice questions (MCQs) and questions no. 19 and 20 are Assertion- Reason based questions of 1 mark each. 4. In Section B, Questions no. 21-25 are very short answer (VSA) type questions, carrying 02 marks each. 5. In Section C, Questions no. 26-31 are short answer (SA) type questions, carrying 03 marks each. 6. In Section D, Questions no. 32-35 are long answer (LA) type questions, carrying 05 marks each. 7. In Section E, Questions no. 36-38 are case study based questions carrying 4 marks each with sub parts of the values of 1, 1 and 2 marks each respectively. 8. All Questions are compulsory. However, an internal choice in 2 Question of Section B, 2 Questions of Section C and 2 Questions of Section D has been provided. An internal choice has been provided in all the 2 marks questions of Section E. 9. Draw neat and clean figures wherever required. 10. Take π =22/7 wherever required if not stated. 11. Use of calculators is not allowed. Section A Section A consists of 20 questions of 1 mark each. 1. The graph of a quadratic polynomial p(x) passes through the points (-6,0), (0, -30), 1 (4,-20) and (6,0). The zeroes of the polynomial are A) - 6,0 B) 4, 6 C) - 30,-20 D) - 6,6 2. The value of k for which the system of equations 3x-ky= 7 and 6x+ 10y =3 is 1 inconsistent, is A) -10 B) -5 C) 5 D) 7 3. Which of the following statements is not true? 1 A) A number of secants can be drawn at any point on the circle. B) Only one tangent can be drawn at any point on a circle. C) A chord is a line segment joining two points on the circle D) From a point inside a circle only two tangents can be drawn. 4. If nth term of an A.P. is 7n-4 then the common difference of the A.P. is 1 A) 7 B) 7n C) - 4 D) 4 5. The radius of the base of a right circular cone and the radius of a sphere are each 5 cm 1 in length. If the volume of the cone is equal to the volume of the sphere then the height of the cone is A) 5 cm B) 20 cm C) 10 cm D) 4 cm 6. 5 4 𝑠𝑖𝑛𝜃 + 𝑐𝑜𝑠𝜃 1 If tan𝜃 = then is equal to 2 4 𝑠𝑖𝑛𝜃 − 𝑐𝑜𝑠𝜃 11 3 9 A) B) C) D) 4 9 2 11 7. In the given figure, a tangent has been drawn at a point P on the circle centred at O. 1 Q O T P If ∠ TPQ= 110𝑂 then ∠POQ is equal to A) 110𝑂 B) 70𝑂 C) 140𝑂 D)55𝑂 8. 5 5 1 A quadratic polynomial having zeroes - √ and √ is 2 2 A) 𝑥 2 − 5√2 x +1 B) 8𝑥 2 - 20 C) 15𝑥 2 - 6 D) 𝑥 2 - 2√5 x -1 9. Consider the frequency distribution of 45 observations. 1 Class 0-10 10-20 20-30 30-40 40-50 Frequency 5 9 15 10 6 The upper limit of median class is A) 20 B) 10 C) 30 D) 40 10. O is the point of intersection of two chords AB and CD of a circle. 1 B D O A C If ∠𝐵𝑂𝐶 = 80𝑂 and OA = OD then 𝛥𝑂𝐷𝐴 𝑎𝑛𝑑 𝛥𝑂𝐵𝐶 are A) equilateral and similar B) isosceles and similar C) isosceles but not similar D) not similar 11. The roots of the quadratic equation 𝑥 2 +x-1 = 0 are 1 A) Irrational and distinct B) not real C ) rational and distinct D) real and equal 12. If 𝜃 = 30𝑜 then the value of 3tan𝜃 is 1 1 3 A)1 B) C) (D) not defined √3 √3 13. 396 1 The volume of a solid hemisphere is 𝑐𝑚 3.The total surface area of the solid 7 hemisphere (in sq.cm) is 396 594 549 604 A) B) C) D) 7 7 7 7 14. In a bag containing 24 balls, 4 are blue, 11 are green and the rest are white. One ball is 1 drawn at random. The probability that drawn ball is white in colour is 1 3 11 5 𝐴) B) C) D) 6 8 24 8 15. The point on the x- axis nearest to the point (-4,-5) is 1 A) (0, 0) B) (-4, 0) C ) (-5, 0) D) (√41, 0) 16. Which of the following gives the middle most observation of the data? 1 A) Median B) Mean C) Range D) Mode 17. A point on the x-axis divides the line segment joining the points A(2, -3) and B(5, 6) in 1 the ratio 1:2. The point is 7 3 A) (4, 0) B) ( , ) C) (3, 0) D) (0,3) 2 2 18. A card is drawn from a well shuffled deck of playing cards. The probability of getting red 1 face card is 3 1 3 3 𝐴) B) C) D) 13 2 52 26 DIRECTION: In the question number 19 and 20, a statement of Assertion (A) is followed by a statement of Reason (R). Choose the correct option A)Both assertion (A) and reason (R) are true and reason (R) is the correct explanation of assertion (A) B)Both assertion (A) and reason (R) are true and reason (R) is not the correct explanation of assertion (A) C)Assertion (A) is true but reason (R) is false. D)Assertion (A) is false but reason (R) is true. 19. Assertion (A): HCF of any two consecutive even natural numbers is always 2. 1 Reason (R): Even natural numbers are divisible by 2. 20. Assertion (A): If the radius of sector of a circle is reduced to its half and angle is 1 doubled then the perimeter of the sector remains the same.  Reason (R): The length of the arc subtending angle θ at the centre of a circle of radius r 𝛱𝑟𝜃 = 180. Section B Section B consists of 5 questions of 2 marks each. 21. (A)Find the H.C.F and L.C.M of 480 and 720 using the Prime factorisation method. 2 OR (A) The H.C.F of 85 and 238 is expressible in the form 85m -238. Find the value of m. 22. (A) Two dice are rolled together bearing numbers 4, 6, 7, 9, 11, 12. Find the 2 probability that the product of numbers obtained is an odd number OR (B) How many positive three digit integers have the hundredths digit 8 and unit’s digit 5? Find the probability of selecting one such number out of all three digit numbers. 23. 2𝑠𝑖𝑛 2 60𝑜 − 𝑡𝑎𝑛2 30𝑜 2 Evaluate: 𝑠𝑒𝑐 2 45𝑜 24. Find the point(s) on the x-axis which is at a distance of √41 units from the point (8, -5). 2 25. Show that the points A(-5,6), B(3, 0) and C( 9, 8) are the vertices of an isosceles 2 triangle. Section C Section C consists of 6 questions of 3 marks each. 26. (A) In 𝛥ABC, D, E and F are midpoints of BC,CA and AB respectively. Prove that 3 △ 𝐹𝐵𝐷 ∼ △ DEF and △ DEF ∼ △ ABC OR (B) In 𝛥ABC, P and Q are points on AB and AC respectively such that PQ is parallel to BC.  Prove that the median AD drawn from A on BC bisects PQ. A P Q R B D C 27. The sum of two numbers is 18 and the sum of their reciprocals is 9/40. Find the 3 numbers. 28. If 𝛼 and 𝛽 are zeroes of a polynomial 6𝑥 2 -5x+1 then form a quadratic 3 polynomial whose zeroes are 𝛼 2 and 𝛽 2. 29. If cosθ + sinθ = 1 , then prove that cosθ - sinθ = ±1 3 30. (A) The minute hand of a wall clock is 18 cm long. Find the area of the face 3 of the clock described by the minute hand in 35 minutes. OR (B) AB is a chord of a circle centred at O such that ∠AOB=60˚. If OA = 14 cm then find the area of the minor segment. (take √3 =1.73) O A B 31. Prove that √3 is an irrational number. 3 Section D Section D consists of 4 questions of 5 marks each 32. (A) Solve the following system of linear equations graphically: 5 x+2y = 3, 2x-3y+8 = 0 OR (B) Places A and B are 180 km apart on a highway. One car starts from A and another from B at the same time. If the car travels in the same direction at  different speeds, they meet in 9 hours. If they travel towards each other with the same speeds as before, they meet in an hour. What are the speeds of the two cars? 33. Prove that the lengths of tangents drawn from an external point to a circle are equal. 5 Using above result, find the length BC of 𝛥ABC. Given that, a circle is inscribed in 𝛥ABC touching the sides AB, BC and CA at R, P and Q respectively and AB= 10 cm, AQ= 7cm ,CQ= 5cm. A R Q B C P 34. A boy whose eye level is 1.35 m from the ground, spots a balloon moving with the wind 5 in a horizontal line at some height from the ground. The angle of elevation of the balloon from the eyes of the boy at an instant is 60𝑜. After 12 seconds, the angle of elevation reduces to 30°. If the speed of the wind is 3m/s then find the height of the balloon from the ground. (Use √3= 1.73) 35. Find the mean and median of the following data: 5 Class 85-90 90-95 95-100 100-105 105-110 110-115 frequency 15 22 20 18 20 25 OR The monthly expenditure on milk in 200 families of a Housing Society is given below Monthly 1000- 1500- 2000- 2500- 3000- 3500- 4000- 4500- Expendit 1500 2000 2500 3000 3500 4000 4500 5000 ure (in Rs.) Number 24 40 33 x 30 22 16 7 of families Find the value of x and also find the mean expenditure Section E Section E consists of 3 case study based questions of 4 marks each. 36. Ms. Sheela visited a store near her house and found that the glass jars are arranged one above the other in a specific pattern.  On the top layer there are 3 jars. In the next layer there are 6 jars. In the 3rd layer from the top there are 9 jars and so on till the 8th layer. On the basis of the above situation answer the following questions. (i) Write an A.P whose terms represent the number of jars in different layers starting from top. Also, find the common difference. 1 (ii) Is it possible to arrange 34 jars in a layer if this pattern is continued? Justify your answer. 1 (iii) (A) If there are ‘n’ number of rows in a layer then find the expression for finding the 2 total number of jars in terms of n. Hence find 𝑆8. OR (iii) (B) The shopkeeper added 3 jars in each layer. How many jars are there in the 5th 2 layer from the top?. 37. A D P Q B C E F Triangle is a very popular shape used in interior designing. The picture given above shows a cabinet designed by a famous interior designer. Here the largest triangle is represented by △ ABC and smallest one with shelf is represented by △ DEF. PQ is parallel to EF. (i) Show that △ DPQ ∼ △ DEF. 1  𝑃𝑄 (ii) If DP= 50 cm and PE = 70 cm then find. 𝐸𝐹 1 𝑝𝑒𝑟𝑖𝑚𝑒𝑡𝑒𝑟 𝑜𝑓 △𝐴𝐵𝐶 (iii) (A) If 2AB = 5DE and △ ABC ∼ △ DEF then show that is constant. 𝑝𝑒𝑟𝑖𝑚𝑒𝑡𝑒𝑟 𝑜𝑓 △𝐷𝐸𝐹 OR 2 (iii) (B) If AM and DN are medians of triangles ABC and DEF respectively then prove that △ ABM ∼ △ DEN. 2 38. Metallic silos are used by farmers for storing grains. Farmer Girdhar has decided to build a new metallic silo to store his harvested grains. It is in the shape of a cylinder mounted by a cone. Dimensions of the conical part of a silo is as follows: Radius of base = 1.5 m Height = 2 m Dimensions of the cylindrical part of a silo is as follows: Radius = 1.5 m Height = 7 m On the basis of the above information answer the following questions. (i) Calculate the slant height of the conical part of one silo. 1 (ii) Find the curved surface area of the conical part of one silo. 1 (iii)(A) Find the cost of metal sheet used to make the curved cylindrical part of 1 silo at 2 the rate of ₹2000 per 𝑚 2. OR (iii) (B) Find the total capacity of one silo to store grains. 2