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CBSE Sample Practice Paper — 1 (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 96 and 404 is: 1
Q2. If p is a prime number, then p√2 is: 1
Q3. The decimal expansion of 129 / (2² × 5⁷ × 7) is: 1
Q4. If α, β are the zeroes of p(x) = x² − 5x + 6, then α + β equals: 1
Q5. The pair of linear equations 2x + 3y = 7 and 4x + 6y = 14 has: 1
Q6. The discriminant of the quadratic equation 2x² − 4x + 3 = 0 is: 1
Q7. The 10th term of the AP 2, 7, 12, 17, ... is: 1
Q8. The largest number which divides 70 and 125, leaving remainders 5 and 8 respectively, is: 1
Q9. If triangle ABC ~ triangle PQR and AB : PQ = 3 : 5, then the ratio of the areas of triangle ABC to triangle PQR is: 1
Q10. In triangle ABC, DE ∥ BC, with D on AB and E on AC. If AD = 2 cm and DB = 3 cm, then AD/AB equals: 1
Q11. The length of the tangent from an external point P to a circle of radius 6 cm, if the distance of P from the centre is 10 cm, is: 1
Q12. The value of sin30° · cos60° + cos30° · sin60° is: 1
Q13. If tanθ = 1, then θ equals: 1
Q14. The area of a sector of a circle of radius 7 cm with central angle 90° is (use π = 22/7): 1
Q15. The probability of an event that is certain to happen is: 1
Q16. The mean of the first five natural numbers is: 1
Q17. The roots of the quadratic equation x² − 7x + 12 = 0 are: 1
Q18. If the sum of the first n terms of an AP is Sₙ = 3n² + 2n, then the common difference of the AP is: 1
Q19. Assertion (A): The tangent to a circle is perpendicular to the radius at the point of contact.
Reason (R): A line drawn perpendicular to the radius at its outer end is a tangent to the circle. 1
Reason (R): A line drawn perpendicular to the radius at its outer end is a tangent to the circle. 1
Q20. Assertion (A): sin²θ + cos²θ = 1 for all values of θ.
Reason (R): This identity follows directly from applying the Pythagoras theorem to a right triangle with hypotenuse equal to 1. 1
Reason (R): This identity follows directly from applying the Pythagoras theorem to a right triangle with hypotenuse equal to 1. 1
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SECTION B (2 Marks each) — Q21 to Q25
Q21. Show that any positive odd integer is of the form 6q + 1, 6q + 3, or 6q + 5, where q is some integer. 2
Q22. Find the coordinates of the point which divides the line segment joining A(−1, 7) and B(4, −3) in the ratio 2 : 3. 2
— OR —
Find the ratio in which the point (−4, 6) divides the line segment joining A(−6, 10) and B(3, −8). 2
Q23. If sin A = 3/5, find the values of cos A and tan A. 2
Q24. Evaluate: cot30° × tan60° − cosec²45°. 2
— OR —
If cotθ = 7/8, evaluate [(1 + sinθ)(1 − sinθ)] / [(1 + cosθ)(1 − cosθ)]. 2
Q25. A die is thrown once. Find the probability of getting (i) a prime number, and (ii) a number greater than 4. 2
SECTION C (3 Marks each) — Q26 to Q31
Q26. Solve for x using the quadratic formula: 2x² − 5x − 3 = 0. 3
Q27. Solve the following pair of linear equations by the elimination method: 3x + 2y = 11 and 2x + 3y = 4. 3
— OR —
Determine, graphically, the coordinates of the vertices of the triangle formed by the lines x − y + 1 = 0, 3x + 2y − 12 = 0 and the x-axis. 3
Q28. Find the value of k for which the pair of equations kx + 2y = 5 and 3x + y = 1 has a unique solution. 3
— OR —
Find the value of k for which the pair of equations 2x + ky = 3 and 3x − 2y = 5 represents parallel lines. 3
Q29. Prove that in a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides (Pythagoras Theorem). 3
Q30. Prove that the lengths of tangents drawn from an external point to a circle are equal. 3
— OR —
Two concentric circles have radii 5 cm and 3 cm. Find the length of the chord of the larger circle which touches the smaller circle. 3
Q31. The following table shows the goals scored by a football team in a series of matches. Find the mean number of goals scored, using the assumed mean method.
3
| Goals scored | 0–2 | 2–4 | 4–6 | 6–8 | 8–10 |
|---|---|---|---|---|---|
| No. of matches | 4 | 6 | 5 | 3 | 2 |
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SECTION D (5 Marks each) — Q32 to Q35
Q32. A train travels a distance of 480 km at a uniform speed. If the speed had been 8 km/h more, it would have taken 3 hours less for the same journey. Find the original speed of the train. 5
— OR —
Solve the following pair of equations for x and y (x, y ≠ 0): 2/x + 3/y = 13 and 5/x − 4/y = −2. 5
Q33. State and prove the Basic Proportionality Theorem (Thales' Theorem): If a line is drawn parallel to one side of a triangle to intersect the other two sides in distinct points, prove that the other two sides are divided in the same ratio. 5
— OR —
Prove that the ratio of the areas of two similar triangles is equal to the square of the ratio of their corresponding sides. 5
Q34. From the top of a building 20 m high, the angle of elevation of the top of a tower is 60°, and the angle of depression of the foot of the tower is 30°. Find the height of the tower and the distance between the building and the tower. 5
Q35. A solid is in the shape of a cone mounted on a hemisphere of the same radius. The radius of the hemisphere is 3.5 cm, and the total height of the solid is 9.5 cm. Find 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. A city's metro map is drawn on a coordinate grid where each unit represents 1 km. Three important stations are located at A(2, 3), B(6, 7) and C(2, 7).4
(i) Find the distance between stations A and B. (1)
(ii) Find the distance between stations A and C. (1)
(iii) Show that triangle ABC is right-angled, and find its area. (2)
[OR for (iii): Find the coordinates of the point equidistant from all three stations A, B and C.]
Q37. Case Study — Mensuration. A village water tank consists of a cylindrical body with a hemispherical dome on top. The common radius is 1.4 m, and the height of the cylindrical part is 3 m.4
(i) Find the curved surface area of the hemispherical dome. (1)
(ii) Find the curved surface area of the cylindrical part. (1)
(iii) Find the total volume of water the tank can hold (cylinder + hemisphere). (2)
[OR for (iii): Find the total surface area of the tank (excluding the base).]
Q38. Case Study — Probability. At a school fete, a bag used in a game contains 5 red, 4 blue and 3 green balls. A ball is drawn at random from the bag.4
(i) Find the probability that the ball drawn is blue. (1)
(ii) Find the probability that the ball drawn is NOT green. (1)
(iii) If a ball is drawn, replaced, and a second ball is drawn, find the probability that both balls drawn are red. (2)
[OR for (iii): Find the probability that the ball drawn is either red or blue.]
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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) 4 | 6 | (A) −8 | 11 | (B) 8 cm | 16 | (B) 3 |
| 2 | (B) irrational | 7 | (A) 47 | 12 | (B) 1 | 17 | (A) 3, 4 |
| 3 | (B) non-terminating repeating | 8 | (A) 13 | 13 | (B) 45° | 18 | (B) 6 |
| 4 | (B) 5 | 9 | (B) 9:25 | 14 | (A) 38.5 cm² | 19 | (B) |
| 5 | (C) infinitely many | 10 | (A) 2/5 | 15 | (B) 1 | 20 | (A) |
Section B — Key Answers (2 marks each)
Q21. Any positive integer, by Euclid's Division Lemma, is of the form 6q, 6q+1, 6q+2, 6q+3, 6q+4 or 6q+5. Of these, 6q+1, 6q+3, 6q+5 are odd (the rest are even), proving the result.
Q22. Section formula gives point (5/3+... ) → (2/5·... ) — final point: (1, 3) [computed as ((2×4+3×−1)/5, (2×−3+3×7)/5) = (1, 3)]. OR: Ratio = 2 : 7.
Q23. cos A = 4/5, tan A = 3/4 (using the 3–4–5 right triangle).
Q24. cot30°×tan60° − cosec²45° = (√3×√3) − 2 = 3 − 2 = 1. OR: Result simplifies to cot²θ = 49/64.
Q25. P(prime) = 3/6 = 1/2; P(number > 4) = 2/6 = 1/3.
Section C — Key Answers (3 marks each)
Q26. x = 3 or x = −1/2 (using quadratic formula, D = 49).
Q27. x = 5, y = −2 (by elimination). OR: Vertices at (1,2), (4,0), (−1,0) approx. from graphical method.
Q28. k ≠ 6 for a unique solution. OR: k = −4/3 for parallel lines.
Q29. Standard proof using similar triangles formed by the altitude from the right angle to the hypotenuse.
Q30. Standard proof using congruent triangles (RHS congruence). OR: Chord length = 8 cm.
Q31. Mean ≈ 4.3 goals (using midpoints 1,3,5,7,9 and assumed mean method).
Section D — Key Answers (5 marks each)
Q32. Original speed = 32 km/h (solving 3x² + 24x − 3840 = 0). OR: x = 1/2, y = 1/3.
Q33. Standard BPT proof using parallel-line area ratios. OR: Standard proof using similar-triangle altitude ratios.
Q34. Tower height = 80 m; distance = 20√3 m ≈ 34.64 m.
Q35. Volume ≈ 166.83 cm³ (hemisphere + cone using r = 3.5 cm, cone height = 6 cm).
Section E — Key Answers (Case Studies, 4 marks each)
Q36. (i) AB = 4√2 units; (ii) AC = 4 units; (iii) BC = 4 units, AC² + BC² = AB² ⟹ right-angled at C; Area = 8 sq. units.
Q37. (i) CSA (hemisphere) = 12.32 m²; (ii) CSA (cylinder) = 26.4 m²; (iii) Total volume ≈ 24.23 m³.
Q38. (i) P(blue) = 1/3; (ii) P(not green) = 3/4; (iii) P(both red) = 25/144.
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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