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CBSE Class 10 Exam 2027 : Mathematics

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HOTS Mastery Worksheet (Master Edition) CBSE Class 10 Mathematics Standard HOTS Mastery Worksheet (Master Edition) CBSE Class 10 Mathematics Standard CBSE CLASS 10 MATHEMATICS STANDARD HOTS MASTER WORKSHEET CRITICAL GRADING GUIDE Chapters: Real Numbers, Polynomials, Linear Equations, AP, Triangles, Probability, Statistics, Mensuration TIME ALLOWED: 2 Hours 30 Minutes MAXIMUM MARKS: 70 General Instructions: 1. This question paper contains 28 questions divided into five Sections: A, B, C, D, and E. 2. Section A comprises 10 Multiple Choice Questions (MCQs) of 1 mark each. 3. Section B comprises 5 Very Short Answer (VSA) type questions of 2 marks each. 4. Section C comprises 6 Short Answer (SA) type questions of 3 marks each. 5. Section D comprises 4 Long Answer (LA) type questions of 5 marks each. 6. Section E comprises 3 Case-Based integrated units of assessment of 4 marks each with subparts. 7. All questions are compulsory. Step-by-step working must be shown in Sections B, C, D, and E. 8. Complete step-by-step solutions and marking schemes are provided at the end of the document. SECTION A (10 Marks 1 Mark Each) Q1. If p and q are two distinct prime numbers, then the HCF of p3 q 2 and p2 q 5 is: (A) pq (B) p2 q 2 (C) p3 q 5 (D) p2 q 5 Q2. If one zero of the quadratic polynomial f (x) = 4x2 8kx + 8x 9 is the negative of the other, then the value of k is: (A) 1 (B) 1 (C) 0 (D) 12 Q3. The value of c for which the pair of linear equations cx y = 2 and 6x 2y = 3 has infinitely many solutions is: (A) 3 (B) 3 (C) 12 (D) No real value Q4. The sum of the first n terms of an AP is given by Sn = 3n2 + 5n. The r-th term of this AP is: (A) 6r + 2 (B) 6r 2 (C) 5r + 3 (D) 3r + 5 Page 1 HOTS Mastery Worksheet (Master Edition) CBSE Class 10 Mathematics Standard Q5. In ABC, DE BC and AD : DB = 2 : 3. If the area of ADE is 8 cm2 , then the area of the quadrilateral BCED is: (A) 12 cm2 (B) 18 cm2 (C) 42 cm2 (D) 50 cm2 Q6. An integer is chosen at random from the first 100 positive integers. What is the probability that it is divisible by both 3 and 5, but not by 2? 3 3 3 1 (A) 50 (B) 100 (C) 20 (D) 10 Q7. If the mean of a frequency distribution is 15 and its median is 18, then its mode using the empirical relationship is: (A) 21 (B) 24 (C) 27 (D) 30 Q8. A solid metal sphere of radius R is melted and entirely recast into a solid right circular cone of height R. The radius of the base of this cone is: (A) R (B) 2R (C) 3R (D) 2 2R Q9. The decimal expansion of the rational number 14587 1250 will terminate after how many decimal places? (A) 1 decimal place (B) 2 decimal places (C) 3 decimal places (D) 4 decimal places Q10. If and are the zeroes of the polynomial f (x) = px2 2x + 3p and + = , then the value of p is: (A) 32 (B) 31 (C) 23 (D) 13 Page 2 HOTS Mastery Worksheet (Master Edition) CBSE Class 10 Mathematics Standard SECTION B (10 Marks 2 Marks Each) Q11. Prove that 3+ 5 is an irrational number. Q12. Find the zeroes of the quadratic polynomial f (x) = term. 3x2 + 10x + 7 3 by splitting the middle Q13. Solve the following pair of simultaneous linear equations for x and y: a2 x + b2 y = c2 ax + by = c (a, b = 0, a = b) Q14. In an AP, if the p-th term is 1 q and the q-th term is p1 , find the sum of its first pq terms. Q15. A card is drawn from a well-shuffled deck of 52 playing cards. Find the probability of drawing a card which is neither a spade nor a king. SECTION C (18 Marks 3 Marks Each) Q16. Prove that any positive odd integer is of the form 4q + 1 or 4q + 3, where q is some integer. Q17. If and are the zeroes of the quadratic polynomial g(s) = 3s2 6s + 4, find the exact value of: 1 1 + +2 + + 3 Q18. If the sum of the first m terms of an AP is equal to the sum of its first n terms (m = n), prove that the sum of its first (m + n) terms is zero. Q19. In ABC, a line parallel to side BC intersects AB at X and AC at Y . If AX = x, XB = x 2, AY = x + 2, and Y C = x 1, find the numerical value of x. Q20. The mean of 10 observations is 15. If one observation of value 20 is replaced by 30, and another observation of value 10 is deleted from the set, find the new mean of the remaining 9 observations. Q21. A metallic solid sphere of radius 10.5 cm is melted and recast into smaller solid cones, each of radius 3.5 cm and height 3 cm. Find the total number of cones so formed. Page 3 HOTS Mastery Worksheet (Master Edition) CBSE Class 10 Mathematics Standard SECTION D (20 Marks 5 Marks Each) Q22. State and prove the Basic Proportionality Theorem (Thales Theorem). Q23. A motor boat whose speed is 18 km/h in still water takes 1 hour more to go 24 km upstream than to return downstream to the same spot. Determine the speed of the stream. Q24. The median of the following grouped frequency distribution is 525. Find the values of the missing frequencies x and y, if the total frequency is 100: Class Interval Frequency Class Interval Frequency 0 100 100 200 200 300 300 400 400 500 2 5 x 12 17 500 600 600 700 700 800 800 900 900 1000 20 y 9 7 4 Q25. A solid toy is in the form of a hemisphere surmounted by a right circular cone of the same base radius. The height of the cone is 2 cm and the diameter of the base is 4 cm. Determine the volume of the toy. If a right circular cylinder circumscribes the toy, find the difference of the volumes of the cylinder and the toy. (Take = 3.14). SECTION E (12 Marks Case-Based Questions) Q26. Case Study 1: Polynomials (The Suspension Bridge Cable) The cable of a suspension bridge hangs in the shape of a parabola. The mathematical model for the height of the cable h(x) (in metres) above the horizontal roadbed is modeled by the quadratic polynomial: 1 h(x) = (x2 80x + 1600) + 5 100 where x represents the horizontal distance from the left-side main tower (in metres). (i) Write the polynomial in its standard quadratic form ax2 + bx + c. (1 Mark) (ii) Find the height of the suspension cable at the main left tower (x = 0). (1 Mark) (iii) Find the horizontal distance x from the tower at which the cable reaches its lowest point, and find the minimum height of the cable above the roadbed at that point. (2 Marks) Q27. Case Study 2: Arithmetic Progressions (Savings Plan) A high school student starts a dedicated savings plan for her university tuition. She deposits Rs. 1000 in the first month, and increases her monthly deposits by a constant amount of Rs. 200 every subsequent month (Rs. 1200 in the 2nd month, Rs. 1400 in the 3rd month, and so on). (i) Write down the general expression for her monthly deposit in the n-th month. State whether this sequence forms an AP and write the common difference. (1 Mark) (ii) Find her exact monthly deposit in the 15th month. (1 Mark) (iii) Calculate the cumulative total amount she will have saved in her tuition fund at the end of 2 full years (24 months). (2 Marks) Page 4 HOTS Mastery Worksheet (Master Edition) CBSE Class 10 Mathematics Standard Q28. Case Study 3: Probability Similarity (Target Shooting Game) At a school fair, a target board is designed in the shape of a right-angled triangular region P QR where Q = 90 , P Q = 6 cm, and QR = 8 cm. Within this target, there is a smaller similar triangular region P ST such that S lies on P Q, T lies on P R, and ST QR. It is given that P S = 3 cm. A dart thrown at the target board is guaranteed to land somewhere within the boundary of the larger triangle P QR. (i) Show that P ST P QR and write down the scale factor of similarity. (1 Mark) (ii) Find the hypotenuse length P R and the total area of the larger board P QR. Mark) (1 (iii) Find the probability that a dart landing randomly on the target board lands outside the smaller triangular region P ST but inside the larger region P QR. (2 Marks) Page 5

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