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CBSE Sample Practice Paper — 6 , 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. This paper is a High-Level / HOTS practice edition — questions emphasise multi-step reasoning and application. 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. If two positive integers a and b are written as a = x³y² and b = xy³, where x, y are primes, then LCM(a, b) is: 1
Q2. n² − 1 is divisible by 8, if n is: 1
Q3. The decimal expansion of 129/(2²×5⁷×7²) is: 1
Q4. If α, β are the zeroes of x² − 6x + k such that α − β = 2, then k equals: 1
Q5. The value of k for which the pair of equations kx + 3y = k − 3 and 12x + ky = k has no solution is: 1
Q6. The value(s) of k for which the equation (k+1)x² − 2(k−1)x + 1 = 0 has equal roots is/are: 1
Q7. Which term of the AP 3, 8, 13, ... is 75 more than its 13th term? 1
Q8. The product of two positive integers is 1600 and their HCF is 5. The number of such possible pairs of integers is: 1
Q9. In ΔABC, right-angled at B, D is the midpoint of BC. Then AD² equals: 1
Q10. Two triangles are similar, and the ratio of a pair of corresponding sides is √3 : √5. The ratio of their areas is: 1
Q11. Two circles of radii 8 cm and 3 cm touch each other internally. The distance between their centres is: 1
Q12. If tanθ + cotθ = 2, then tan²θ + cot²θ equals: 1
Q13. (secθ + tanθ)(1 − sinθ) equals: 1
Q14. A solid sphere of radius r is melted and recast into a cone of the same base radius r. The height of the cone is: 1
Q15. Two dice are thrown together. The probability that the sum of the numbers on them is a prime number is: 1
Q16. For a data set, the mean is 25 and the mode is 22. Using the empirical relation Mode = 3 Median − 2 Mean, the median is: 1
Q17. If one zero of the polynomial ax² + bx + c is the reciprocal of the other, then: 1
Q18. If Sₙ = 3n² + 5n represents the sum of the first n terms of an AP, its 15th term is: 1
Q19. Assertion (A): If the sum of the first n terms of an AP is Sₙ = 2n² + 3n, the AP has a common difference of 4.
Reason (R): For an AP, if Sₙ = An² + Bn (A, B constants), then the common difference equals 2A. 1
Reason (R): For an AP, if Sₙ = An² + Bn (A, B constants), then the common difference equals 2A. 1
Q20. Assertion (A): HCF(a, b) × LCM(a, b) = a × b is true for any three positive integers a, b, c.
Reason (R): The relation HCF × LCM = product of the numbers is valid only for two positive integers, not for three. 1
Reason (R): The relation HCF × LCM = product of the numbers is valid only for two positive integers, not for three. 1
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SECTION B (2 Marks each) — Q21 to Q25
Q21. Using Euclid's Division Algorithm, find the HCF of 4052 and 12576. 2
Q22. Find the value(s) of k for which the points (k, −1), (2, 1) and (4, 5) are collinear. 2
— OR —
Find the ratio in which the y-axis divides the line segment joining (−3, 4) and (5, 6). Also find the point of division. 2
Q23. If 7sin²θ + 3cos²θ = 4, show that tanθ = 1/√3. 2
Q24. Prove that: (sinθ + cosecθ)² + (cosθ + secθ)² = 7 + tan²θ + cot²θ. 2
— OR —
If sinA + sin²A = 1, prove that cos²A + cos⁴A = 1. 2
Q25. Cards numbered 1 to 40 are put in a box and mixed thoroughly. A card is drawn at random. Find the probability that the number on it is (i) a perfect square, (ii) divisible by 5. 2
SECTION C (3 Marks each) — Q26 to Q31
Q26. Prove that √5 is an irrational number. 3
Q27. Solve for x and y: x/a + y/b = 2 and ax − by = a² − b². 3
— OR —
A fraction becomes 9/11 if 2 is added to both the numerator and denominator. If 3 is added to both, it becomes 5/6. Find the fraction. 3
Q28. The sum of the first 7 terms of an AP is 49 and the sum of its first 17 terms is 289. Find the sum of its first n terms. 3
Q29. In ΔABC, D and E are points on sides AB and AC respectively such that DE ∥ BC. If AD = 4x − 3, AE = 8x − 7, BD = 3x − 1 and CE = 5x − 3, find the value of x. 3
Q30. Prove that the tangent at any point of a circle is perpendicular to the radius through the point of contact. 3
— OR —
Two concentric circles have radii 5 cm and 3 cm. Find the length of a chord of the larger circle which touches the smaller circle. 3
Q31. The median of the following data is 28.5. Find the values of x and y, if the total frequency is 60.
3
| Class | 0–10 | 10–20 | 20–30 | 30–40 | 40–50 | 50–60 |
|---|---|---|---|---|---|---|
| Frequency | 5 | x | 20 | 15 | y | 5 |
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SECTION D (5 Marks each) — Q32 to Q35
Q32. A train travels 360 km at a uniform speed. If the speed had been 5 km/h more, it would have taken 1 hour less for the same journey. Find the speed of the train. 5
— OR —
Two pipes running together can fill a cistern in 3 1/13 hours. If one pipe takes 3 hours more than the other to fill the cistern, find the time in which each pipe alone can fill it. 5
Q33. Prove that the lengths of tangents drawn from an external point to a circle are equal. Using this result: a circle is inscribed in ΔABC, touching sides AB, BC and CA at P, Q and R respectively. If AB = 10 cm, BC = 12 cm and CA = 8 cm, find the lengths AP, BQ and CR. 5
Q34. The angle of elevation of the top of a tower from a point on the ground is 30°. On walking 100 m towards the tower, the angle of elevation becomes 60°. Find the height of the tower and the distance of the first point from the base of the tower. 5
Q35. A solid is in the form of a right circular cylinder with a hemisphere at one end and a cone at the other end. The radius of the common base is 8 cm, and the heights of the cylindrical and conical portions are 10 cm and 6 cm respectively. Find the total surface area and the volume of the solid. (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. Points A(1, 2), B(4, y), C(x, 6) and D(3, 5) are the vertices of a parallelogram ABCD, taken in order, drawn on a landscaping grid where 1 unit = 1 m.4
(i) Using the property that diagonals of a parallelogram bisect each other, find the value of x. (1)
(ii) Find the value of y. (1)
(iii) Find the length of the diagonal AC. (2)
[OR for (iii): Find the area of the parallelogram ABCD.]
Q37. Case Study — Mensuration. A bucket is in the shape of a frustum of a cone. Its height is 24 cm, and the radii of the top and bottom circular ends are 15 cm and 5 cm respectively.4
(i) Find the slant height of the bucket. (1)
(ii) Find the curved surface area of the bucket, in cm² (use π = 22/7). (1)
(iii) Find the capacity (volume) of the bucket, in cm³. (2)
[OR for (iii): Find the total surface area of the bucket, including its bottom circle (the bucket is open at the top).]
Q38. Case Study — Probability. A box contains 90 discs, numbered 1 to 90. A disc is drawn at random from the box.4
(i) Find the probability that the disc bears a perfect square number. (1)
(ii) Find the probability that the disc bears a number divisible by 5. (1)
(iii) Find the probability that the disc bears a two-digit number which is a multiple of 9. (2)
[OR for (iii): Find the probability that the disc bears a prime number less than 40.]
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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 | 3 | 2 | 3 | 0 | 0 | 8 |
| Algebra (Polynomials, Linear Eq., Quadratic Eq., AP) | 7 | 0 | 6 | 5 | 0 | 18 |
| Coordinate Geometry | 0 | 2 | 0 | 0 | 4 | 6 |
| Geometry (Triangles, Circles) | 2 | 0 | 6 | 5 | 0 | 13 |
| Trigonometry | 2 | 4 | 0 | 5 | 0 | 11 |
| Mensuration | 1 | 0 | 0 | 5 | 4 | 10 |
| Statistics & Probability | 5 | 2 | 3 | 0 | 4 | 14 |
| Total | 20 | 10 | 18 | 20 | 12 | 80 |
3. Difficulty Level Analysis (Bloom's Taxonomy) — High-Level Edition
| Cognitive Level | Description | Marks | Percentage |
|---|---|---|---|
| Remembering & Understanding | Recall of facts, definitions, direct formula-based questions | 26 | 32.5% |
| Applying | Application of concepts to solve standard problems | 26 | 32.5% |
| Analysing, Evaluating & Creating (HOTS) | Multi-step reasoning, proof-based, case-study and real-life application questions with combined concepts | 28 | 35% |
| Total | 80 | 100% |
Note: This edition intentionally weights more marks toward the HOTS band than a standard-difficulty paper, to give stronger practice for board-level application questions.
4. Learning Outcome (LO) Mapping
| Unit | Key Learning Outcomes Assessed |
|---|---|
| Number Systems | Applies Euclid's Division Algorithm to multi-digit numbers; reasons about irrationality and divisibility; analyses decimal expansions with mixed prime-factor denominators. |
| Algebra | Solves symbolic/parametric linear equations; forms and solves quadratic equations from word problems; derives AP formulae from Sₙ expressions and combined term conditions. |
| Coordinate Geometry | Uses collinearity and section-formula conditions; applies parallelogram diagonal-bisection property; computes area via the shoelace/coordinate method. |
| Geometry | Proves and applies tangent-radius and tangent-length theorems; solves incircle tangent-length problems using simultaneous relations; applies basic proportionality theorem with algebraic segment lengths. |
| Trigonometry | Manipulates and proves trigonometric identities; solves height-and-distance problems requiring two viewing points and simultaneous equations. |
| Mensuration | Computes surface area and volume of composite/frustum solids combining multiple standard shapes. |
| Statistics & Probability | Finds unknown frequencies from a given median using the grouped-data median formula; computes probability for classified/multi-condition sample spaces. |
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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 | (D) x³y³ | 6 | (A) 0 or 3 | 11 | (A) 5 cm | 16 | (C) 24 |
| 2 | (A) odd | 7 | (D) 28th | 12 | (B) 2 | 17 | (A) a=c |
| 3 | (B) non-terminating repeating | 8 | (A) 1 | 13 | (B) cosθ | 18 | (C) 92 |
| 4 | (A) 8 | 9 | (B) AB²+BC²/4 | 14 | (D) 4r | 19 | (A) |
| 5 | (B) −6 | 10 | (B) 3:5 | 15 | (B) 5/12 | 20 | (D) |
Section B — Key Answers (2 marks each)
Q21. HCF(4052, 12576) = 4 (via successive Euclid's division steps).
Q22. Area of triangle formed = 0 ⇒ k = 1. OR: Ratio = 3 : 5 (from the negative to positive x point); point of division = (0, 4.75).
Q23. 7sin²θ + 3cos²θ = 3 + 4sin²θ = 4 ⇒ sin²θ = 1/4 ⇒ θ = 30° ⇒ tanθ = 1/√3 (proved).
Q24. Standard identity expansion, both sides reduce to 7 + tan²θ + cot²θ (proved). OR: sinA = 1 − sin²A = cos²A ⇒ substituting gives cos²A + cos⁴A = 1 (proved).
Q25. Perfect squares 1–40: {1,4,9,16,25,36} ⇒ P = 6/40 = 3/20. Divisible by 5: 8 numbers ⇒ P = 8/40 = 1/5.
Section C — Key Answers (3 marks each)
Q26. Standard proof by contradiction, assuming √5 = p/q in lowest terms leads to a contradiction that p and q share a common factor 5.
Q27. Solving simultaneously: x = a, y = b. OR: Fraction = 7/9.
Q28. a = 1, d = 2 (from a+3d=7 and a+8d=17) ⇒ Sₙ = n².
Q29. Using AD/BD = AE/CE, solving 2x² − x − 1 = 0 gives x = 1 (rejecting x = −1/2 as lengths must be positive).
Q30. Standard proof by contradiction using the shortest-distance property of a perpendicular from the centre. OR: Half-chord = √(5²−3²) = 4 cm ⇒ chord = 8 cm.
Q31. Median class = 20–30; solving (30 − 5 − x)/20 × 10 = 8.5 gives x = 8, and since x + y = 15, y = 7.
Section D — Key Answers (5 marks each)
Q32. Speed = 40 km/h (solving x² + 5x − 1800 = 0). OR: Faster pipe: 5 hours; slower pipe: 8 hours.
Q33. Standard tangent-length proof using RHS congruence. Numerical part: taking AP=AR=x, BP=BQ=y, CQ=CR=z, solving x+y=10, y+z=12, z+x=8 gives AP = 3 cm, BQ = 7 cm, CR = 5 cm.
Q34. Solving √3d = (d+100)/√3 gives d = 50 m; height = 50√3 ≈ 86.6 m; distance of first point from base = 150 m.
Q35. Slant height of cone part = √(8²+6²) = 10 cm. Total surface area = CSA(cylinder) + CSA(cone) + CSA(hemisphere) = 160π + 80π + 128π = 368π ≈ 1156.6 cm². Volume = π r² h(cyl) + (1/3)π r² h(cone) + (2/3)π r³ = 640π + 128π + (1024/3)π ≈ 1109.33π ≈ 3485.9 cm³.
Section E — Key Answers (Case Studies, 4 marks each)
Q36. (i) x = 6; (ii) y = 3; (iii) AC = √41 units ≈ 6.4 units. OR: Area of parallelogram ABCD = 7 sq. units (via the shoelace formula).
Q37. (i) Slant height = √(24²+10²) = 26 cm; (ii) CSA = π(15+5)(26) = 520π ≈ 1634.3 cm²; (iii) Volume = (1/3)π(24)(15²+5²+15×5) = 2600π ≈ 8171.4 cm³. OR: Total surface area (with bottom circle) = 520π + 25π = 545π ≈ 1712.9 cm².
Q38. (i) Perfect squares 1–90 (9 numbers): P = 9/90 = 1/10; (ii) Multiples of 5 (18 numbers): P = 18/90 = 1/5; (iii) Two-digit multiples of 9 (9 numbers: 18,27,...,90): P = 9/90 = 1/10. OR: Primes less than 40 (12 numbers): P = 12/90 = 2/15.
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. Q26, 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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