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CBSE Sample Practice Paper — 9 (High 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 an advanced High Level of difficulty, with strong emphasis on HOTS (Higher Order Thinking Skills), multi-step reasoning and rigorous proof-based questions, while retaining the same CBSE marking scheme.
SECTION A (1 Mark each) — Q1 to Q20
Q1. If a = 2³ × 3² and b = 2² × 3³ × 5, then HCF(a, b) × LCM(a, b) equals: 1
Q2. The number of pairs of natural numbers whose HCF is 18 and sum is 216 is: 1
Q3. If p and q are two distinct prime numbers, then LCM(p, q) is: 1
Q4. When x⁴ − 6x³ + 16x² − 25x + 10 is divided by x² − 2x + k, the remainder is x + a. The values of k and a are: 1
Q5. The pair of equations kx + 3y = k−3 and 12x + ky = k has no solution when k equals: 1
Q6. If α and β are the zeroes of f(x) = x² − 5x + k such that α − β = 1, find k. 1
Q7. The sum of the first 15 terms of an AP equals the sum of its first 25 terms. The sum of its first 40 terms is: 1
Q8. The smallest number by which 1/13 should be multiplied so that its decimal expansion terminates after exactly one decimal place is: 1
Q9. In ΔABC, DE ∥ BC, with AD = (7x−4) cm, AE = (5x−2) cm, DB = (3x+4) cm, EC = 3x cm. Find x. 1
Q10. D, E, F are the midpoints of sides BC, CA, AB of ΔABC. The ratio of the areas of ΔDEF to ΔABC is: 1
Q11. Two tangents TP, TQ are drawn to a circle with centre O from an external point T. If ∠TPQ = 70°, find ∠POQ. 1
Q12. If tanA = n·tanB and sinA = m·sinB, then cos²A, in terms of m and n, equals: 1
Q13. If sinθ + sin²θ = 1, then the value of cos²θ + cos⁴θ is: 1
Q14. The volumes of two spheres are in the ratio 27 : 64. The ratio of their surface areas is: 1
Q15. In a single throw of a pair of dice, the probability of getting a doublet of prime numbers is: 1
Q16. If the mean of the distribution x: 2,4,6,10,(p+5) with frequencies f: 3,2,3,1,2 is 6, find p. 1
Q17. If x = 2 and x = 3 are roots of the equation 3x² − 2kx + 2m = 0, find k and m. 1
Q18. The value of p for which the polynomial x³ + 4x² − px + 8 is exactly divisible by (x−2) is: 1
Q19. Assertion (A): If the areas of two similar triangles are equal, then the triangles are congruent.
Reason (R): The ratio of the areas of two similar triangles is equal to the square of the ratio of their corresponding sides. 1
Reason (R): The ratio of the areas of two similar triangles is equal to the square of the ratio of their corresponding sides. 1
Q20. Assertion (A): The value of (sin30° + cos60°) is greater than the value of (sin60° + cos30°).
Reason (R): sin30° = cos60° = 1/2, and sin60° = cos30° = √3/2. 1
Reason (R): sin30° = cos60° = 1/2, and sin60° = cos30° = √3/2. 1
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SECTION B (2 Marks each) — Q21 to Q25
Q21. Prove that √2 is an irrational number, and hence show that 3 + 2√2 is also irrational. 2
Q22. Find the ratio in which the y-axis divides the line segment joining the points (5, −6) and (−1, −4). Also find the point of intersection. 2
— OR —
Find the value of k, if the points A(2, 3), B(4, k) and C(6, −3) are collinear. 2
Q23. If sinα = 1/2 and cosβ = 1/2, where α, β are acute angles, find the value of α + β. 2
Q24. Prove that: (cosecθ − sinθ)(secθ − cosθ)(tanθ + cotθ) = 1. 2
— OR —
If x = a·sinθ and y = b·cosθ, prove that (x/a)² + (y/b)² = 1. 2
Q25. A jar contains 24 marbles, some green and the rest blue. If a marble is drawn at random, the probability that it is green is 2/3. Find the number of blue marbles in the jar. 2
SECTION C (3 Marks each) — Q26 to Q31
Q26. If α, β are the zeroes of the quadratic polynomial f(x) = x² − 2x − 8, find a quadratic polynomial whose zeroes are 2α/β and 2β/α. 3
Q27. Solve the following pair of equations by reducing them to a pair of linear equations: 5/(x−1) + 1/(y−2) = 2 and 6/(x−1) − 3/(y−2) = 1, where x ≠ 1, y ≠ 2. 3
Q28. Which term of the AP 3, 15, 27, 39, ... will be 120 more than its 21st term? 3
Q29. ABCD is a trapezium with AB ∥ DC, and its diagonals AC and BD intersect at O. Prove that AO/OC = BO/OD. 3
Q30. In a right triangle ABC, right-angled at B, AB = 6 cm and BC = 8 cm. A circle is inscribed in the triangle, touching all three sides. Find the radius of the inscribed circle. 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 shows the daily pocket allowance of children of a locality. If the mean pocket allowance is ₹18, find the missing frequency f.
3
| Allowance (₹) | 11–13 | 13–15 | 15–17 | 17–19 | 19–21 | 21–23 | 23–25 |
|---|---|---|---|---|---|---|---|
| No. of children | 7 | 6 | 9 | 13 | f | 5 | 4 |
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SECTION D (5 Marks each) — Q32 to Q35
Q32. Places A and B are 100 km apart on a highway. One car starts from A and another from B at the same time. If the cars travel in the same direction, they meet in 5 hours. If they travel towards each other, they meet in 1 hour. Find the speeds of the two cars. 5
— OR —
A train covers a distance of 480 km at a uniform speed. If the speed had been 8 km/h less, it would have taken 3 hours more to cover the same distance. Find the original speed of the train. 5
Q33. Prove that if a perpendicular is drawn from the vertex of the right angle of a right triangle to the hypotenuse, the triangles on each side of the perpendicular are similar to the whole triangle and to each other. 5
— OR —
Prove that in a triangle, if the square of one side is equal to the sum of the squares of the other two sides, the angle opposite the first side is a right angle (Converse of the Pythagoras Theorem). 5
Q34. As observed from the top of a 75 m tall lighthouse, the angles of depression of two ships approaching it are 30° and 45°. If one ship is directly behind the other, on the same side of the lighthouse, find the distance between the two ships. 5
Q35. A solid toy is in the form of a hemisphere surmounted by a right circular cone of the same radius as the hemisphere. If the radius of the hemisphere is 4.2 cm and the total height of the toy is 10.2 cm, find the total surface area of the toy. (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 surveyor is mapping a triangular plot of land with vertices A(−3, 4), B(3, −2) and C(5, 6) on a coordinate grid (1 grid unit = 10 m).4
(i) Find the length AB (in grid units). (1)
(ii) Find the length BC (in grid units). (1)
(iii) Find the length AC, and hence classify the triangle as scalene, isosceles or right-angled. (2)
[OR for (iii): Find the coordinates of the centroid of triangle ABC.]
Q37. Case Study — Mensuration. A metallic sphere of radius 10.5 cm is melted and recast into smaller cones, each of radius 3.5 cm and height 3 cm.4
(i) Find the volume of the original sphere. (1)
(ii) Find the volume of one small cone. (1)
(iii) Find the number of cones that can be formed from the melted sphere. (2)
[OR for (iii): Find the total curved surface area of all the cones formed, if the slant height of each cone is approximately 4.6 cm.]
Q38. Case Study — Probability. In a survey of 200 people, it was found that 120 like tea, 90 like coffee, and 50 like both tea and coffee. A person is selected at random from the survey.4
(i) Find the probability that the person likes tea only (and not coffee). (1)
(ii) Find the probability that the person likes neither tea nor coffee. (1)
(iii) Find the probability that the person likes both tea and coffee. (2)
[OR for (iii): Find the probability that the person likes at least one of the two beverages.]
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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) 38880 | 6 | (B) 6 | 11 | (A) 140° | 16 | (A) 7 |
| 2 | (B) 2 | 7 | (A) 0 | 12 | (A) (m²−1)/(n²−1) | 17 | (A) k=7.5,m=9 |
| 3 | (D) pq | 8 | (A) 13/10 | 13 | (B) 1 | 18 | (A) 16 |
| 4 | (A) k=5,a=−5 | 9 | (A) 4 | 14 | (A) 9:16 | 19 | (A) |
| 5 | (B) −6 | 10 | (C) 1:4 | 15 | (A) 1/12 | 20 | (D) |
Section B — Key Answers (2 marks each)
Q21. Standard proof by contradiction for √2; then 3+2√2 must be irrational, else √2 would be expressible as a ratio of integers.
Q22. Ratio = 5 : 1; point of intersection = (0, −13/3). OR: k = 0.
Q23. α + β = 90° (α = 30°, β = 60°).
Q24. Standard identity proof — expand each factor in terms of sinθ, cosθ and simplify to 1. OR: Standard proof by substitution and simplification.
Q25. Number of blue marbles = 8 (since green = 2/3 × 24 = 16, blue = 24 − 16 = 8).
Section C — Key Answers (3 marks each)
Q26. Required polynomial: x² + 5x + 4 (zeroes of the given polynomial are 4 and −2; new zeroes are −4 and −1).
Q27. x = 4, y = 5 (using substitution u = 1/(x−1), v = 1/(y−2)).
Q28. The 31st term of the AP is 120 more than its 21st term.
Q29. Standard proof using similar triangles AOB and COD (formed since AB ∥ DC), giving AO/OC = BO/OD.
Q30. Radius of inscribed circle = 2 cm (using r = (AB+BC−AC)/2 for a right triangle). OR: Standard proof using the cyclic quadrilateral formed by the two tangents and two radii.
Q31. Missing frequency f = 20.
Section D — Key Answers (5 marks each)
Q32. Speed of car from A = 60 km/h; Speed of car from B = 40 km/h. OR: Original speed = 40 km/h.
Q33. Standard AA-similarity proof for the two sub-triangles formed by the altitude to the hypotenuse. OR: Standard converse-Pythagoras proof by constructing a congruent right triangle and using SSS congruence.
Q34. Distance between the two ships = 75(√3−1) m ≈ 54.9 m.
Q35. Total surface area of the toy ≈ 207.5 cm² (hemisphere CSA ≈ 110.88 cm² + cone CSA ≈ 96.62 cm²).
Section E — Key Answers (Case Studies, 4 marks each)
Q36. (i) AB = 6√2 units; (ii) BC = 2√17 units; (iii) AC = 2√17 units — since BC = AC, the triangle is isosceles.
Q37. (i) Volume of sphere = 4851 cm³; (ii) Volume of one cone = 38.5 cm³; (iii) Number of cones = 126.
Q38. (i) P(tea only) = 7/20; (ii) P(neither) = 1/5; (iii) P(both) = 1/4.
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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