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CBSE Sample Practice Paper — 10 (Session 2026–27)
| Student Name: ______________________ | Roll No.: __________ | Section: __________ |
General Instructions:
1. This Question Paper has 5 Sections A, B, C, D and E.
2. Section A comprises 20 MCQs (including 2 Assertion-Reason questions) of 1 mark each.
3. Section B comprises 5 Very Short Answer (VSA) questions of 2 marks each.
4. Section C comprises 6 Short Answer (SA) questions of 3 marks each.
5. Section D comprises 4 Long Answer (LA) questions of 5 marks each.
6. Section E comprises 3 Case-Study based questions of 4 marks each, with sub-parts of 1, 1 and 2 marks.
7. All questions are compulsory. Internal choices have been provided in 2 questions of Section B, 3 questions of Section C, 2 questions of Section D, and in one sub-part of each Case-Study question of Section E.
8. Draw neat figures wherever required. Take π = 22/7 wherever not stated otherwise.
9. Use of calculators is not permitted.
SECTION A (1 Mark each) — Q1 to Q20
Q1. The HCF of 336 and 54 is: 1
Q2. Which of the following is an irrational number? 1
Q3. If two positive integers p and q can be expressed as p = ab² and q = a³b, where a, b are prime numbers, then LCM(p, q) is: 1
Q4. A quadratic polynomial whose zeroes are −3 and 4 is: 1
Q5. The pair of equations y = 0 and y = −7 has: 1
Q6. If one root of the equation x² + px + 12 = 0 is 4, and the equation x² + px + q = 0 has equal roots, find p and q. 1
Q7. If the 9th term of an AP is zero, then the ratio of its 29th term to its 19th term is: 1
Q8. The largest number that divides 2053 and 967, leaving remainders 5 and 7 respectively, is: 1
Q9. If, in ΔABC and ΔPQR, AB/QR = BC/PR = CA/PQ, then: 1
Q10. In ΔABC, D is a point on BC such that ∠ADC = ∠BAC. Then: 1
Q11. If two tangents inclined at an angle of 60° are drawn to a circle of radius 3 cm, the length of each tangent is: 1
Q12. If sinA + sinB = √3(cosB − cosA), the value of sin3A + sin3B is: 1
Q13. If cosθ = 8/17, find the value of sinθ. 1
Q14. If the height of a cone is 24 cm and the radius of its base is 7 cm, the slant height of the cone is: 1
Q15. A number is chosen at random from −3, −2, −1, 0, 1, 2, 3. The probability that the square of this number is less than or equal to 1 is: 1
Q16. The mode of the data 15, 18, 15, 20, 25, 18, 15, 22 is: 1
Q17. The value of k for which the polynomial 2x³ + kx² + 11x + (k+3) has (x+1) as a factor is: 1
Q18. The 11th term of the AP −5, −5/2, 0, 5/2, ... is: 1
Q19. Assertion (A): If a line is drawn parallel to one side of a triangle, intersecting the other two sides in distinct points, the other two sides are divided in the same ratio.
Reason (R): This is the Basic Proportionality Theorem (Thales' Theorem). 1
Reason (R): This is the Basic Proportionality Theorem (Thales' Theorem). 1
Q20. Assertion (A): sin(A−B) = sinA·cosB − cosA·sinB.
Reason (R): This formula can be derived using the sine-addition formula applied to A + (−B). 1
Reason (R): This formula can be derived using the sine-addition formula applied to A + (−B). 1
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SECTION B (2 Marks each) — Q21 to Q25
Q21. Prove that √7 is an irrational number. 2
Q22. Find the coordinates of the point equidistant from the vertices O(0,0), A(5,0) and B(0,5). 2
— OR —
Find the value of a, if the distance between the points (3, a) and (4, 1) is √10. 2
Q23. If tanθ = 8/15, find the value of [(1+sinθ)(1−sinθ)] / [(1+cosθ)(1−cosθ)]. 2
Q24. Evaluate: (cos²45° − sin²45°) + tan²60°. 2
— OR —
If sinA = cosA, find the value of 3tan²A + 2sin²A. 2
Q25. Two dice are thrown once. Find the probability of getting an even number on one die and a multiple of 3 on the other. 2
SECTION C (3 Marks each) — Q26 to Q31
Q26. Find the zeroes of the quadratic polynomial 6x² − 7x − 3, and verify the relationship between the zeroes and the coefficients. 3
Q27. Solve for x and y: 0.4x + 0.3y = 1.7 and 0.7x − 0.2y = 0.8. 3
— OR —
The coach of a cricket team buys 7 bats and 6 balls for ₹3800. Later, she buys 3 bats and 5 balls for ₹1750. Find the cost of each bat and each ball. 3
Q28. How many terms of the AP 24, 21, 18, ... must be taken so that their sum is 78? Explain the double answer. 3
Q29. In an equilateral triangle ABC, D is a point on side BC such that BD = (1/3)BC. Prove that 9AD² = 7AB². 3
Q30. A tangent PQ at a point P of a circle of radius 5 cm meets a line through the centre O at a point Q such that OQ = 13 cm. Find the length PQ. 3
— OR —
Prove that the tangent at any point of a circle is perpendicular to the radius through the point of contact. 3
Q31. Find the mean of the following frequency distribution using the direct method:
3
| Class | 0–10 | 10–20 | 20–30 | 30–40 | 40–50 |
|---|---|---|---|---|---|
| Frequency | 7 | 10 | 15 | 8 | 10 |
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SECTION D (5 Marks each) — Q32 to Q35
Q32. A takes 6 days less than the time taken by B to finish a piece of work. If both A and B together can finish the work in 4 days, find the time taken by B to finish the work alone. 5
Q33. Prove that the ratio of the areas of two similar triangles is equal to the square of the ratio of their corresponding sides. Using this result, ΔABC ~ ΔDEF, and their areas are 64 cm² and 121 cm² respectively. If EF = 15.4 cm, find BC. 5
Q34. Two poles of equal heights stand opposite each other on either side of a road 80 m wide. From a point between them on the road, the angles of elevation of the tops of the poles are 60° and 30° respectively. Find the height of the poles and the distances of the point from each pole. 5
Q35. Water is flowing at 7 m/s through a circular pipe of internal diameter 2 cm, into a cylindrical tank of radius 40 cm. Find the rise in the water level of the tank in half an hour. (Use π = 22/7) 5
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SECTION E — Case Study Based Questions (4 Marks each) — Q36 to Q38
Q36. Case Study — Coordinate Geometry. A robotics team programs a robot to move on a coordinate grid (1 unit = 1 m), starting at S(2, 3), passing through waypoint W(6, 7), and ending at destination D(10, 3).4
(i) Find the distance SW. (1)
(ii) Find the distance WD. (1)
(iii) Find the distance SD, verify that ΔSWD is isosceles, and calculate its area. (2)
[OR for (iii): Find the coordinates of the midpoint of SD, and verify that W, this midpoint, and the triangle's shape are consistent with an isosceles triangle.]
Q37. Case Study — Mensuration. A company packages ice cream in cone-shaped cups. Each cone has a radius of 3 cm and height 12 cm, topped with a hemispherical scoop of the same radius.4
(i) Find the volume of the conical part. (1)
(ii) Find the volume of the hemispherical scoop. (1)
(iii) Find the total volume of ice cream in one cup, and state how many complete cups can be filled from 500 cm³ of ice cream mixture. (2)
[OR for (iii): Find the total surface area of one filled cup (curved surface of cone + curved surface of hemisphere).]
Q38. Case Study — Probability. A weather app recorded whether it rained on each of 30 days in a month: it rained on 12 days. A day is selected at random from this month.4
(i) Find the probability that it rained on the selected day. (1)
(ii) Find the probability that it did not rain on the selected day. (1)
(iii) If two different days are selected at random from the month, without replacement, find the probability that it rained on both selected days. (2)
[OR for (iii): Find the probability that it rained on the first selected day but not on the second.]
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Design of Question Paper — Blueprint & Analysis
1. Blueprint — Section-wise Design
| Section | Question Type | No. of Questions | Marks per Question | Total Marks |
|---|---|---|---|---|
| A | MCQ (incl. 2 Assertion–Reason) | 20 | 1 | 20 |
| B | Very Short Answer (VSA) | 5 | 2 | 10 |
| C | Short Answer (SA) | 6 | 3 | 18 |
| D | Long Answer (LA) | 4 | 5 | 20 |
| E | Case Study Based (1+1+2) | 3 | 4 | 12 |
| Total | 80 | |||
2. Chapter-wise / Unit-wise Marks Distribution
| Unit / Chapter Group | Sec A | Sec B | Sec C | Sec D | Sec E | Total |
|---|---|---|---|---|---|---|
| Number Systems | 4 | 2 | 0 | 0 | 0 | 6 |
| Algebra (Polynomials, Linear Eq., Quadratic Eq., AP) | 6 | 0 | 9 | 5 | 0 | 20 |
| Coordinate Geometry | 0 | 2 | 0 | 0 | 4 | 6 |
| Geometry (Triangles, Circles) | 4 | 0 | 6 | 5 | 0 | 15 |
| Trigonometry | 3 | 4 | 0 | 5 | 0 | 12 |
| Mensuration | 1 | 0 | 0 | 5 | 4 | 10 |
| Statistics & Probability | 2 | 2 | 3 | 0 | 4 | 11 |
| Total | 20 | 10 | 18 | 20 | 12 | 80 |
3. Difficulty Level Analysis (Bloom's Taxonomy)
| Cognitive Level | Description | Marks | Percentage |
|---|---|---|---|
| Remembering & Understanding | Recall of facts, definitions, direct formula-based questions | 43 | 54% |
| Applying | Application of concepts to solve standard/routine problems | 19 | 24% |
| Analysing, Evaluating & Creating (HOTS) | Multi-step reasoning, proof-based, case-study and real-life application questions | 18 | 22% |
| Total | 80 | 100% |
4. Learning Outcome (LO) Mapping
| Unit | Key Learning Outcomes Assessed |
|---|---|
| Number Systems | Applies Euclid's Division Lemma; identifies rational/irrational numbers; determines nature of decimal expansions. |
| Algebra | Finds zeroes and relationships of polynomials; solves pairs of linear equations by multiple methods; solves quadratic equations; applies AP formulae to real-life contexts. |
| Coordinate Geometry | Applies distance and section formulae; determines area of a triangle using coordinates; interprets geometric figures on a coordinate plane. |
| Geometry | Proves and applies theorems on similar triangles, Pythagoras theorem, and tangents to a circle. |
| Trigonometry | Evaluates trigonometric ratios and identities; applies trigonometry to height-and-distance real-life problems. |
| Mensuration | Computes areas of circles/sectors; finds surface areas and volumes of combined solids. |
| Statistics & Probability | Computes mean of grouped data; determines theoretical probability of simple and compound events. |
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Complete Answer Key & Step-wise Marking Scheme
Section A — Answer Key (1 mark each)
| Q | Ans | Q | Ans | Q | Ans | Q | Ans |
|---|---|---|---|---|---|---|---|
| 1 | (A) 6 | 6 | (A) p=−7,q=49/4 | 11 | (A) 3√3 cm | 16 | (A) 15 |
| 2 | (C) 0.1010010001... | 7 | (B) 2:1 | 12 | (A) 0 | 17 | (A) 5 |
| 3 | (B) a³b² | 8 | (A) 64 | 13 | (A) 15/17 | 18 | (A) 20 |
| 4 | (A) x²−x−12 | 9 | (C) ΔCBA~ΔPQR | 14 | (A) 25 cm | 19 | (A) |
| 5 | (B) no solution | 10 | (A) CA²=CB.CD | 15 | (A) 3/7 | 20 | (A) |
Section B — Key Answers (2 marks each)
Q21. Standard proof by contradiction for the irrationality of √7.
Q22. Point equidistant = (2.5, 2.5), the midpoint of the hypotenuse AB (circumcentre of right ΔOAB). OR: a = −2 or a = 4.
Q23. Value = cot²θ = 225/64 (using tanθ = 8/15).
Q24. Value = 3 (since cos²45°−sin²45° = 0, and tan²60° = 3). OR: Value = 4.
Q25. P(even on one die and multiple of 3 on other) = 20/36 = 5/9 (counting all valid ordered outcomes).
Section C — Key Answers (3 marks each)
Q26. Zeroes: x = −1/3 and x = 3/2 (by factorisation: (3x+1)(2x−3) = 0); relationship verified via sum = 7/6 and product = −1/2.
Q27. x = 2, y = 3. OR: Cost of one bat = ₹500; cost of one ball = ₹50.
Q28. n = 4 or n = 13 (both values give sum = 78, since later terms become negative).
Q29. Standard proof: drop a perpendicular from A to BC, then apply the Pythagoras theorem in the resulting right triangle to establish 9AD² = 7AB².
Q30. PQ = 12 cm (using PQ² = OQ² − OP²). OR: Standard proof using the radius-tangent perpendicularity property.
Q31. Mean = 25.8 (using the direct method with midpoints 5,15,25,35,45).
Section D — Key Answers (5 marks each)
Q32. B alone takes 12 days to finish the work; A alone takes 6 days (solving x² − 14x + 24 = 0).
Q33. Standard proof using the altitude-ratio / area-ratio argument for similar triangles. Numerical part: BC = 11.2 cm.
Q34. Height of each pole = 20√3 m ≈ 34.64 m; distances from the point are 20 m and 60 m.
Q35. Rise in water level ≈ 787.5 cm ≈ 7.875 m.
Section E — Key Answers (Case Studies, 4 marks each)
Q36. (i) SW = 4√2 units; (ii) WD = 4√2 units; (iii) SD = 8 units, so ΔSWD is isosceles (SW=WD); Area = 16 sq. units.
Q37. (i) Volume of cone ≈ 113.14 cm³; (ii) Volume of hemisphere ≈ 56.57 cm³; (iii) Total volume ≈ 169.71 cm³; only 2 complete cups can be filled from 500 cm³.
Q38. (i) P(rain) = 2/5; (ii) P(no rain) = 3/5; (iii) P(rain on both days, without replacement) = 22/145.
Step-wise Marking Scheme — General Guidelines
• Section A (MCQ/AR): 1 mark for the correct option only; no partial credit.
• Section B/C/D (VSA/SA/LA): Marks are awarded step-wise — correct method/formula (partial marks even if the final numeric answer is wrong due to a minor computational slip), correct substitution of values, and correct final answer with unit. Alternative correct methods must be given full credit.
• Proof-based questions (e.g. Q29, Q30, Q33): Marks are distributed across statement of the theorem/given-to-prove, construction (if any), logical steps of the proof, and the concluding statement.
• Section E (Case Study): Each sub-part is marked independently as per its allotted marks (1, 1, 2); no negative marking for an incorrect sub-part affecting others.
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