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CBSE Sample Practice Paper — 2 (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 LCM of 12 and 18 is: 1
Q2. √5 is: 1
Q3. The decimal expansion of 17/8 is: 1
Q4. If α, β are the zeroes of p(x) = x² + 7x + 12, then αβ equals: 1
Q5. The pair of linear equations x + 2y = 4 and 2x + 4y = 12 represents: 1
Q6. The nature of the roots of the equation x² − 4x + 4 = 0 is: 1
Q7. The first term of an AP is 5 and the common difference is 3. The 15th term is: 1
Q8. If two positive integers a and b are written as a = x³y² and b = xy³, where x, y are prime numbers, then HCF(a, b) is: 1
Q9. If triangle ABC ~ triangle DEF and BC : EF = 1 : 3, then ar(ABC) : ar(DEF) equals: 1
Q10. In a triangle, if a line divides two sides in the same ratio, then that line is: 1
Q11. Two circles touch each other externally. The distance between their centres is 10 cm, and the radius of one circle is 6 cm. The radius of the other circle is: 1
Q12. The value of cos0° + sin90° is: 1
Q13. sec²θ − tan²θ equals: 1
Q14. The circumference of a circle whose area is 154 cm² is (use π = 22/7): 1
Q15. Two coins are tossed simultaneously. The probability of getting exactly one head is: 1
Q16. The median of the data 4, 7, 3, 9, 6 is: 1
Q17. The sum and product of the roots of the equation 3x² − 9x + 6 = 0 are respectively: 1
Q18. How many terms are there in the AP 7, 13, 19, ..., 205? 1
Q19. Assertion (A): If two triangles are similar, then their corresponding sides are proportional.
Reason (R): Equiangular triangles are always similar (AA similarity criterion). 1
Reason (R): Equiangular triangles are always similar (AA similarity criterion). 1
Q20. Assertion (A): The value of tan90° is not defined.
Reason (R): tanθ = sinθ/cosθ, and cos90° = 0. 1
Reason (R): tanθ = sinθ/cosθ, and cos90° = 0. 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 distance between the points (a, b) and (−a, −b). 2
— OR —
Find a relation between x and y such that the point (x, y) is equidistant from the points (3, 6) and (−3, 4). 2
Q23. If A and B are acute angles such that sin(A − B) = 1/2 and cos(A + B) = 1/2, find the angles A and B. 2
Q24. Evaluate: (sin45° + cos45°)². 2
— OR —
Prove that: (1 − cos²θ)(1 + cot²θ) = 1. 2
Q25. A bag contains 3 red and 5 black balls. A ball is drawn at random. Find the probability that the ball drawn is (i) red, and (ii) not black. 2
SECTION C (3 Marks each) — Q26 to Q31
Q26. Solve for x: x² − (√3 + 1)x + √3 = 0. 3
Q27. Solve the following pair of linear equations: 2x + 3y = 11 and 2x − 4y = −24. 3
— OR —
Find the value(s) of p for which the quadratic equation px² − 6x − 2 = 0 has real and equal roots. 3
Q28. Find the sum of the first 24 terms of the AP whose nth term is given by aₙ = 3 + 2n. 3
Q29. In a triangle ABC, AD is the median to side BC. Prove that AB² + AC² = 2AD² + 2BD². 3
Q30. Two tangents PA and PB are drawn from an external point P to a circle with centre O, such that ∠APB = 80°. Find ∠AOB. 3
— OR —
Prove that a parallelogram circumscribing a circle is a rhombus. 3
Q31. Find the mode of the following data:
3
| Class | 10–20 | 20–30 | 30–40 | 40–50 | 50–60 |
|---|---|---|---|---|---|
| Frequency | 5 | 8 | 12 | 6 | 3 |
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SECTION D (5 Marks each) — Q32 to Q35
Q32. The sum of the ages of a father and his son is 45 years. Five years ago, the product of their ages (in years) was 124. Find their present ages. 5
— OR —
A two-digit number is such that the product of its digits is 18. When 63 is subtracted from the number, the digits interchange their places. Find the number. 5
Q33. Prove that if the corresponding angles of two triangles are equal, then their corresponding sides are in the same ratio (AAA similarity criterion), and hence the two triangles are similar. 5
— OR —
Prove that the tangent at any point of a circle is perpendicular to the radius through the point of contact. 5
Q34. The angle of elevation of the top of a tower from a point on the ground, 30 m away from its foot, is 30°. Find the height of the tower. A flagpole is mounted on top of the tower, and the angle of elevation of the top of the flagpole from the same point is 45°. Find the height of the flagpole. 5
Q35. A bucket is in the form of a frustum of a cone with top and bottom radii of 15 cm and 10 cm respectively, and a height of 12 cm. Find the volume of the bucket, and the cost of milk that can completely fill it at ₹40 per litre. (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 drone delivery company plots its warehouse and delivery points on a coordinate grid, where each unit represents 1 km. The warehouse W is located at (0, 0), and two delivery points are P(6, 8) and Q(−6, 8).4
(i) Find the distance WP. (1)
(ii) Find the distance WQ. (1)
(iii) The company wants to place a charging station on the y-axis, equidistant from the warehouse W and delivery point P. Find its coordinates. (2)
[OR for (iii): Find the coordinates of the midpoint of PQ, and state what this point represents geometrically.]
Q37. Case Study — Mensuration. A cylindrical measuring jar of radius 7 cm contains water up to a height of 20 cm. A metallic sphere of radius 3.5 cm is dropped into the jar and gets fully submerged.4
(i) Find the volume of the sphere. (1)
(ii) Find the volume of water in the jar before the sphere was dropped. (1)
(iii) Find the rise in the water level after the sphere is dropped. (2)
[OR for (iii): Find the total volume of water plus sphere in the jar after the sphere is dropped.]
Q38. Case Study — Probability. In a class of 40 students, a survey was conducted about their favourite sport: Cricket — 16 students, Football — 10 students, Badminton — 8 students, Others — 6 students. A student is selected at random from the class.4
(i) Find the probability that the student's favourite sport is Football. (1)
(ii) Find the probability that the student's favourite sport is NOT Cricket. (1)
(iii) If two students are selected one after another, without replacement, find the probability that both prefer Badminton. (2)
[OR for (iii): Find the probability that a randomly selected student prefers either Cricket or Football.]
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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) 36 | 6 | (B) real & equal | 11 | (A) 4 cm | 16 | (B) 6 |
| 2 | (C) irrational | 7 | (A) 47 | 12 | (C) 2 | 17 | (A) 3, 2 |
| 3 | (A) terminating | 8 | (A) xy² | 13 | (B) 1 | 18 | (C) 34 |
| 4 | (C) 12 | 9 | (B) 1:9 | 14 | (B) 44 cm | 19 | (A) |
| 5 | (B) parallel | 10 | (B) parallel to 3rd side | 15 | (B) 1/2 | 20 | (A) |
Section B — Key Answers (2 marks each)
Q21. HCF(4052, 12576) = 4 (by repeated application of Euclid's division algorithm).
Q22. Distance = 2√(a² + b²). OR: Relation obtained: 3x + y − 5 = 0.
Q23. A = 45°, B = 15°.
Q24. (sin45° + cos45°)² = 2. OR: Standard identity proof — reduces to sin²θ·cosec²θ = 1.
Q25. P(red) = 3/8; P(not black) = 3/8.
Section C — Key Answers (3 marks each)
Q26. x = √3 or x = 1 (by factorisation: (x−√3)(x−1) = 0).
Q27. x = −2, y = 5. OR: p = −9/2 (for real and equal roots, D = 0).
Q28. Sum of first 24 terms = 672 (a₁ = 5, a₂₄ = 51).
Q29. Standard proof using the Apollonius theorem / median length relation, via the Pythagoras theorem in the two sub-triangles formed by the median.
Q30. ∠AOB = 100° (since ∠OAP = ∠OBP = 90°, and angle sum of quadrilateral OAPB = 360°). OR: Standard proof using tangent-length equality and congruent triangles.
Q31. Modal class = 30–40; Mode = 34 (using the mode formula for grouped data).
Section D — Key Answers (5 marks each)
Q32. Son's present age = 9 years; Father's present age = 36 years. OR: The two-digit number is 92.
Q33. Standard AAA similarity proof using construction of a triangle equal in angles, then applying BPT. OR: Standard proof using the property that the radius drawn to the point of contact is the shortest distance from the centre to the tangent line.
Q34. Height of tower = 10√3 m ≈ 17.32 m; Height of flagpole ≈ 30 − 10√3 ≈ 12.68 m.
Q35. Volume of bucket ≈ 5971.43 cm³ ≈ 5.97 litres; Cost of milk ≈ ₹238.86 (at ₹40/litre).
Section E — Key Answers (Case Studies, 4 marks each)
Q36. (i) WP = 10 km; (ii) WQ = 10 km; (iii) Charging station at (0, 6.25).
Q37. (i) Volume of sphere ≈ 179.67 cm³; (ii) Volume of water before = 3080 cm³; (iii) Rise in water level ≈ 1.17 cm.
Q38. (i) P(Football) = 1/4; (ii) P(not Cricket) = 3/5; (iii) P(both Badminton, without replacement) = 7/195.
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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