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CBSE Sample Practice Paper — 3 (Difficult Level) (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.
10. This paper is set at a higher difficulty level, with increased emphasis on multi-step, HOTS (Higher Order Thinking Skills) and application-based questions, while retaining the same CBSE marking scheme.
SECTION A (1 Mark each) — Q1 to Q20
Q1. The total number of prime factors (counted with multiplicity) in the prime factorisation of 2² × 3³ × 5² × 7 is: 1
Q2. For any natural number n, the number 6ⁿ − 5ⁿ always ends with the digit: 1
Q3. The least number divisible by all natural numbers from 1 to 10 is: 1
Q4. If one zero of the polynomial (a² + 9)x² + 13x + 6a is the reciprocal of the other, then a equals: 1
Q5. For what value of k does the pair of equations 2x + 3y = 7 and kx + 9y = 21 have infinitely many solutions? 1
Q6. If the equation (k+1)x² − 2(k−1)x + 1 = 0 has equal roots, then k equals: 1
Q7. If the sum of first n terms of an AP is Sₙ = 2n² + 3n, then the first term and common difference of the AP are: 1
Q8. If HCF(a, b) = 12 and a × b = 1800, then LCM(a, b) is: 1
Q9. D and E are points on sides AB and AC of ΔABC such that AD = 2 cm, DB = 3 cm, AE = 3 cm, AC = 7.5 cm. Is DE ∥ BC? 1
Q10. In ΔPQR, PQ = PR, and S is a point on QR such that PS ⊥ QR. Then: 1
Q11. From an external point, two tangents are drawn to a circle of radius 5 cm, and the length of each tangent is 12 cm. The distance of the external point from the centre is: 1
Q12. If sinθ + cosθ = √2, then θ equals: 1
Q13. The value of (tan1° · tan2° · tan3° ... tan89°) is: 1
Q14. The radius of a sphere is increased by 100%. The percentage increase in its volume is: 1
Q15. Cards numbered 1 to 25 are placed in a box and mixed thoroughly. One card is drawn at random. The probability that the number on the card is a multiple of both 2 and 3 is: 1
Q16. The mean of 20 observations is 15. If each observation is increased by 3 and then multiplied by 2, the new mean is: 1
Q17. If α, β are the zeroes of 2x² − 5x + 7, then 1/α + 1/β equals: 1
Q18. The value of k for which the system kx − y = 2, 6x − 2y = 3 has no solution is: 1
Q19. Assertion (A): If, in two triangles, one angle of a triangle is equal to one angle of another triangle and the sides including these angles are proportional, then the triangles are similar.
Reason (R): This is the SAS (Side-Angle-Side) similarity criterion for triangles. 1
Reason (R): This is the SAS (Side-Angle-Side) similarity criterion for triangles. 1
Q20. Assertion (A): The value of sinθ cannot exceed 1 for any real angle θ.
Reason (R): sinθ represents the ratio of the perpendicular to the hypotenuse in a right triangle, and the hypotenuse is always the longest side. 1
Reason (R): sinθ represents the ratio of the perpendicular to the hypotenuse in a right triangle, and the hypotenuse is always the longest side. 1
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SECTION B (2 Marks each) — Q21 to Q25
Q21. Prove that 5 + 2√3 is an irrational number. 2
Q22. Find the value of k such that the points (7, −2), (5, 1) and (3, k) are collinear. 2
— OR —
Find the area of the triangle formed by the points (0, 0), (6, 0) and (4, 3). 2
Q23. If tanA + cotA = 2, find the value of tan²A + cot²A. 2
Q24. Prove that: (secθ − tanθ)(secθ + tanθ) = 1. 2
— OR —
If 3cotA = 4, find the value of (1 − tan²A) / (1 + tan²A). 2
Q25. Two dice are thrown simultaneously. Find the probability of getting a sum of 9. 2
SECTION C (3 Marks each) — Q26 to Q31
Q26. If the roots of the equation (a − b)x² + (b − c)x + (c − a) = 0 are equal, prove that 2a = b + c. 3
Q27. Solve for x: 1/(x+1) + 2/(x+2) = 4/(x+4), where x ≠ −1, −2, −4. 3
— OR —
Find the values of a and b for which the system of equations 2x + 3y = 7 and (a+b)x + (2a−b)y = 21 has infinitely many solutions. 3
Q28. The sums of the first n terms of two APs are in the ratio (7n+1) : (4n+27). Find the ratio of their 9th terms. 3
Q29. Two poles of heights 6 m and 11 m stand vertically on a plane ground. If the distance between their feet is 12 m, find the distance between their tops. 3
Q30. PA and PB are tangents drawn from an external point P to a circle with centre O. If OP = 13 cm and the radius OA = 5 cm, find the length of the chord of contact AB. 3
— OR —
Prove that the angle between the two tangents drawn from an external point to a circle is supplementary to the angle subtended by the line segment joining the points of contact at the centre. 3
Q31. The following distribution gives the daily income of 50 workers of a factory. Find the mean daily income using the step-deviation method.
3
| Daily income (₹) | 100–120 | 120–140 | 140–160 | 160–180 | 180–200 |
|---|---|---|---|---|---|
| No. of workers | 12 | 14 | 8 | 6 | 10 |
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SECTION D (5 Marks each) — Q32 to Q35
Q32. A motorboat, whose speed in still water is 18 km/h, takes 1 hour more to go 24 km upstream than to return downstream to the same spot. Find the speed of the stream. 5
— OR —
If the sum of the first p terms of an AP is q, and the sum of the first q terms is p (p ≠ q), prove that the sum of the first (p+q) terms is −(p+q). 5
Q33. Prove that if a line divides any two sides of a triangle in the same ratio, then the line is parallel to the third side (Converse of BPT). Using this result, in a trapezium ABCD with AB ∥ CD, whose diagonals intersect at O, prove that OA/OC = OB/OD. 5
— OR —
Prove that the areas of two similar triangles are in the ratio of the squares of their corresponding medians. 5
Q34. The angles of elevation of the top of a tower from two points at distances a and b (a > b) from its base, on the same straight line and on the same side, are complementary. Prove that the height of the tower is √(ab). 5
Q35. A container, open at the top, is in the form of a frustum of a cone of height 24 cm, with radii of its lower and upper circular ends as 8 cm and 20 cm respectively. Find the cost of milk which can completely fill the container, at ₹50 per litre, and the cost of the metal sheet used to make the container, at ₹10 per 100 cm² (ignore the thickness of the sheet). (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 game designer places three checkpoints on a coordinate grid (1 unit = 1 m): A(1, 2), B(7, 4) and C(4, 8).4
(i) Find the distance AB. (1)
(ii) Find the distance AC. (1)
(iii) Find the coordinates of the centroid of triangle ABC formed by these checkpoints, and explain its significance as the "balance point" where the designer places a power-up item. (2)
[OR for (iii): Find the coordinates of the point which divides BC in the ratio 2:1, and describe where this places a checkpoint relative to B and C.]
Q37. Case Study — Mensuration. A factory manufactures a metal component in the shape of a cylinder of radius 6 cm and height 14 cm, with a conical cavity of the same radius and height 4 cm drilled out from one end.4
(i) Find the volume of the cylinder before drilling. (1)
(ii) Find the volume of the conical cavity removed. (1)
(iii) Find the volume of the remaining metal component. (2)
[OR for (iii): Find the total exposed surface area of the component (curved surface of the cylinder + curved surface of the conical cavity + one flat circular end).]
Q38. Case Study — Probability. A quality control manager checks a sample of 100 LED bulbs for defects: 8 are found defective and 92 are non-defective. Two bulbs are selected at random from the sample, one after another, without replacement.4
(i) Find the probability that the first bulb selected is defective. (1)
(ii) Find the probability that the first bulb selected is non-defective. (1)
(iii) Find the probability that both bulbs selected are defective. (2)
[OR for (iii): Find the probability that the first bulb is defective and the second is non-defective.]
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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 | (B) 8 | 6 | (C) 0 or 3 | 11 | (A) 13 cm | 16 | (C) 36 |
| 2 | (A) 1 | 7 | (A) a=5,d=4 | 12 | (B) 45° | 17 | (A) 5/7 |
| 3 | (A) 2520 | 8 | (A) 150 | 13 | (B) 1 | 18 | (A) 3 |
| 4 | (A) 3 | 9 | (A) Yes | 14 | (C) 700% | 19 | (A) |
| 5 | (B) 6 | 10 | (A) QS=SR | 15 | (A) 4/25 | 20 | (A) |
Section B — Key Answers (2 marks each)
Q21. Standard proof by contradiction: assume 5+2√3 is rational, derive that √3 must then be rational, contradicting the known irrationality of √3.
Q22. k = 4 (using the collinearity/slope condition). OR: Area = 9 sq. units.
Q23. tan²A + cot²A = 2 (using (tanA+cotA)² = tan²A + cot²A + 2).
Q24. Standard identity proof using sec²θ − tan²θ = 1. OR: Result = 7/25.
Q25. P(sum = 9) = 4/36 = 1/9.
Section C — Key Answers (3 marks each)
Q26. Standard proof: expand the discriminant condition (b−c)² = 4(a−b)(c−a) and simplify algebraically to obtain 2a = b + c.
Q27. x = 2 + 2√3 or x = 2 − 2√3. OR: a = 5, b = 1.
Q28. Ratio of 9th terms = 24 : 19 (using n = 17 in the given ratio expression).
Q29. Distance between tops = 13 m (using Pythagoras: √(5² + 12²)).
Q30. AB = 120/13 cm ≈ 9.23 cm. OR: Standard proof using the cyclic quadrilateral OAPB.
Q31. Mean daily income = ₹145.20 (using step-deviation method with assumed mean A = 150, h = 20).
Section D — Key Answers (5 marks each)
Q32. Speed of stream = 6 km/h (solving x² + 48x − 324 = 0). OR: Standard AP proof using Sₚ and S_q expressions.
Q33. Standard converse-BPT proof by contradiction, followed by the trapezium diagonal-ratio proof using similar triangles AOB and COD. OR: Standard proof using the property that the median divides the opposite side in the same ratio as corresponding sides.
Q34. Standard proof: tanθ = h/a, tan(90°−θ) = cotθ = h/b; multiplying gives h² = ab, so h = √(ab).
Q35. Volume ≈ 15,689.14 cm³ ≈ 15.69 litres; Cost of milk ≈ ₹784.50; Total surface area ≈ 2561.8 cm²; Cost of metal sheet ≈ ₹256.18.
Section E — Key Answers (Case Studies, 4 marks each)
Q36. (i) AB = 2√10 units; (ii) AC = 3√5 units; (iii) Centroid = (4, 14/3), the point where the medians of the triangle meet, used as the balanced "power-up" location.
Q37. (i) Volume of cylinder = 1584 cm³; (ii) Volume of conical cavity ≈ 150.86 cm³; (iii) Remaining volume ≈ 1433.14 cm³.
Q38. (i) P(defective) = 2/25; (ii) P(non-defective) = 23/25; (iii) P(both defective) = 14/2475.
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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