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CBSE Sample Practice Paper — 7 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. The HCF of 2³×3²×5 and 2²×3³×5² is: 1
Q2. The sum of a rational number and an irrational number is always: 1
Q3. The number of decimal places after which the decimal expansion of 17/8 terminates is: 1
Q4. If the zeroes of x² − (k+6)x + 2(2k−1) are equal in magnitude but opposite in sign, then k equals: 1
Q5. The value of k for which 2x + 3y = 7 and (k−1)x + (k+2)y = 3k have infinitely many solutions is: 1
Q6. If the roots of the equation (a−b)x² + (b−c)x + (c−a) = 0 are equal, then: 1
Q7. If 7 times the 7th term of an AP equals 11 times its 11th term, its 18th term is: 1
Q8. Three bells toll together at 9-minute, 12-minute and 15-minute intervals. If they toll together at 8:00 a.m., after how many minutes will they toll together again? 1
Q9. In ΔABC, DE ∥ BC with AD = x, DB = x−2, AE = x+2, EC = x−1. The value of x is: 1
Q10. ΔABC ~ ΔPQR with ar(ABC) : ar(PQR) = 25 : 16. If BC = 10 cm, then QR equals: 1
Q11. Tangents PA and PB are drawn to a circle with centre O from an external point P such that ∠APB = 60°. Then ∠AOB equals: 1
Q12. If sinθ + cosθ = √2, then tanθ + cotθ equals: 1
Q13. (1 + tan²A)(1 − sinA)(1 + sinA) equals: 1
Q14. A right circular cylinder and a right circular cone have equal base radii and equal heights. The ratio of the volume of the cylinder to that of the cone is: 1
Q15. A card is drawn from a well-shuffled deck of 52 cards. The probability that it is neither a king nor a queen is: 1
Q16. In the formula for mean, Mean = a + (Σfᵢdᵢ)/(Σfᵢ), the term 'a' denotes: 1
Q17. A quadratic polynomial whose zeroes are 3+√2 and 3−√2 is: 1
Q18. If the nth term of an AP is 7 − 4n, its common difference is: 1
Q19. Assertion (A): The tangent to a circle at a point is perpendicular to the radius through that point.
Reason (R): A line perpendicular to the radius at its outer end is a tangent to the circle. 1
Reason (R): A line perpendicular to the radius at its outer end is a tangent to the circle. 1
Q20. Assertion (A): The solution of the pair of equations x − 2y = 0 and 3x + 4y = 20 is x = 4, y = 2.
Reason (R): A pair of linear equations in two variables whose graphs intersect at exactly one point has a unique solution. 1
Reason (R): A pair of linear equations in two variables whose graphs intersect at exactly one point has a unique solution. 1
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SECTION B (2 Marks each) — Q21 to Q25
Q21. Find the largest number which divides 615 and 963, leaving remainder 6 in each case. 2
Q22. The points A(4, 3) and B(x, 5) lie on a circle with centre O(2, 3). Find the value of x. 2
— OR —
Find the distance between the points (a cosθ, a sinθ) and (a cosφ, a sinφ). 2
Q23. Prove that: cosA/(1+sinA) + (1+sinA)/cosA = 2secA. 2
Q24. If sec4A = cosec(A − 20°), where 4A is an acute angle, find the value of A. 2
— OR —
Evaluate: (2cos²60° + 3sec²30° − 2tan²45°) / (sin²30° + cos²30°). 2
Q25. Two dice are rolled together. Find the probability that the product of the numbers appearing on them is a perfect square. 2
SECTION C (3 Marks each) — Q26 to Q31
Q26. Given that √5 is irrational, prove that 3 + 2√5 is an irrational number. 3
Q27. Solve the pair of equations: 2/x + 3/y = 13 and 5/x − 4/y = −2, where x, y ≠ 0. 3
— OR —
Places A and B are 100 km apart. A car starts from A and another from B at the same time. If they travel in the same direction, they meet in 5 hours; if they travel towards each other, they meet in 1 hour. Find their speeds. 3
Q28. Find the sum of all three-digit natural numbers which are divisible by 7. 3
Q29. In ΔPQR, N is a point on side PR such that QN ⊥ PR. If PN × NR = QN², prove that ∠PQR = 90°. 3
Q30. Two tangents TP and TQ are drawn to a circle with centre O from an external point T. Prove that ∠PTQ = 2∠OPQ. 3
— OR —
Prove that the parallelogram circumscribing a circle is a rhombus. 3
Q31. If the mean of the following distribution is 54, find the value of p.
3
| Class | 0–20 | 20–40 | 40–60 | 60–80 | 80–100 |
|---|---|---|---|---|---|
| Frequency | 7 | p | 10 | 9 | 13 |
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SECTION D (5 Marks each) — Q32 to Q35
Q32. A motorboat whose speed is 18 km/h in still water 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 —
A passenger train takes 3 hours less for a journey of 360 km if its speed is increased by 10 km/h from its usual speed. Find the usual speed of the train. 5
Q33. State and prove the Basic Proportionality Theorem (Thales' Theorem). Using it: In trapezium ABCD with AB ∥ CD, diagonals AC and BD intersect at O such that AO/OC = BO/OD. If AO = 3x−1, OC = 5x−3, BO = 2x+1 and OD = 6x−5, find the value of x. 5
Q34. From the top of a 60 m high building, the angles of depression of the top and bottom of a tower are 30° and 60° respectively. Find the height of the tower. 5
Q35. A solid toy is in the form of a hemisphere surmounted by a right circular cone. The height of the cone is 4 cm and the diameter of the base is 6 cm. A right circular cylinder circumscribes the solid (same base radius, height = hemisphere radius + cone height). Find the volume of air space between the solid toy and the cylinder. 5
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SECTION E — Case Study Based Questions (4 Marks each) — Q36 to Q38
Q36. Case Study — Coordinate Geometry. A surveyor marks three points on a map representing the corners of a triangular plot: A(2, 3), B(6, −2) and C(−4, −8), where 1 unit = 1 km.4
(i) Find the length of AB. (1)
(ii) Find the length of BC. (1)
(iii) Find the coordinates of the point which divides AB internally in the ratio 2 : 3. (2)
[OR for (iii): Find the area of triangle ABC.]
Q37. Case Study — Mensuration. A cylindrical vessel of radius 12 cm and height 6 cm is full of ice cream. This ice cream is to be filled into cones of height 12 cm and radius 3 cm, each having a hemispherical top.4
(i) Find the volume of the cylindrical vessel, in cm³. (1)
(ii) Find the volume of ice cream in one cone (cone + hemisphere), in cm³. (1)
(iii) Find the number of such cones required to empty the vessel. (2)
[OR for (iii): Find the curved surface area of one cone together with its hemispherical top (excluding the flat base).]
Q38. Case Study — Probability. In a survey of 200 students, their mode of transport to school was recorded: Bus — 70, Bicycle — 50, Walking — 40, Car — 30, Other — 10. A student is chosen at random.4
(i) Find the probability that the student travels by bicycle. (1)
(ii) Find the probability that the student does not travel by car. (1)
(iii) Find the probability that the student travels by bus or by walking. (2)
[OR for (iii): Find the probability that the student travels by bicycle or by car.]
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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) | 3 | 0 | 6 | 5 | 0 | 14 |
| Trigonometry | 2 | 4 | 0 | 5 | 0 | 11 |
| Mensuration | 1 | 0 | 0 | 5 | 4 | 10 |
| Statistics & Probability | 4 | 2 | 3 | 0 | 4 | 13 |
| 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 | 25 | 31% |
| Applying | Application of concepts to solve standard problems | 27 | 34% |
| 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 remainder-based problems; reasons about the sum of rational and irrational numbers; uses LCM to solve periodic real-life (bell-tolling) problems. |
| Algebra | Analyses parametric conditions for infinite/no solutions; solves reciprocal-substitution equation systems; derives AP terms from combined-term conditions; sums arithmetic sequences with a divisibility constraint. |
| Coordinate Geometry | Uses the distance formula with circle-radius conditions and parametric points; applies the section formula and the coordinate area formula. |
| Geometry | Proves and applies converses of circle theorems; uses similar-triangle reasoning to establish right-angle conditions; applies the Basic Proportionality Theorem to algebraic segment ratios. |
| Trigonometry | Manipulates compound trigonometric identities; solves complementary-angle equations for an unknown angle; evaluates mixed trigonometric-ratio expressions. |
| Mensuration | Computes volumes of composite solids (hemisphere + cone) and the air space within a circumscribing cylinder; solves capacity-transfer problems between vessels. |
| Statistics & Probability | Finds an unknown frequency from a given mean using the assumed-mean/direct method; computes probability for combined and complementary events from real-life survey data. |
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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) 180 | 6 | (A) 2b=a+c | 11 | (C) 120° | 16 | (A) assumed mean |
| 2 | (B) irrational | 7 | (A) 0 | 12 | (B) 2 | 17 | (A) x²−6x+7 |
| 3 | (C) 3 | 8 | (C) 180 | 13 | (B) 1 | 18 | (A) −4 |
| 4 | (A) −6 | 9 | (C) 4 | 14 | (B) 3:1 | 19 | (B) |
| 5 | (C) 7 | 10 | (C) 8 cm | 15 | (A) 11/13 | 20 | (A) |
Section B — Key Answers (2 marks each)
Q21. Subtracting the remainder 6: HCF(609, 957) = 87.
Q22. Since O is the centre, OA = OB (both radii) ⇒ (x−2)² + 4 = 4 ⇒ x = 2. OR: Distance = 2a·sin[(θ−φ)/2].
Q23. Combining over a common denominator, the LHS simplifies to 2(1+sinA)/[cosA(1+sinA)] = 2secA (proved).
Q24. sec4A = cosec(90°−4A); equating angles: 90−4A = A−20 ⇒ A = 22°. OR: Value = 2.5.
Q25. Favourable outcomes (product a perfect square): 8 out of 36 ⇒ P = 8/36 = 2/9.
Section C — Key Answers (3 marks each)
Q26. Standard proof by contradiction: assuming 3+2√5 is rational forces √5 to be rational, a contradiction.
Q27. Substituting u=1/x, v=1/y and solving gives u=2, v=3 ⇒ x = 1/2, y = 1/3. OR: Speeds = 60 km/h and 40 km/h.
Q28. First term 105, last term 994, n = 128 ⇒ Sum = 64 × 1099 = 70,336.
Q29. Using PN×NR=QN² to show ΔPNQ ~ ΔQNR, corresponding angles give ∠PQR = 90° (proved).
Q30. Standard proof using the tangent-radius perpendicularity and the isosceles triangle OPQ. OR: Standard proof that all sides of the parallelogram are equal using equal tangent lengths.
Q31. Solving (2370+30p)/(39+p) = 54 gives 264 = 24p ⇒ p = 11.
Section D — Key Answers (5 marks each)
Q32. Speed of stream = 6 km/h (solving x² + 48x − 324 = 0). OR: Usual speed = 30 km/h (solving x² + 10x − 1200 = 0).
Q33. Standard BPT proof using areas of triangles with a common vertex. Numerical part: solving 2x² − 5x + 2 = 0 gives x = 2 (x = 1/2 rejected as it makes OC negative).
Q34. Height of tower = 40 m (solving tan60° and tan30° equations simultaneously; d = 20√3 m).
Q35. Cylinder: r = 3 cm, H = 7 cm ⇒ Volume = 63π cm³. Toy volume = hemisphere (18π) + cone (12π) = 30π cm³. Air space = 33π ≈ 103.71 cm³.
Section E — Key Answers (Case Studies, 4 marks each)
Q36. (i) AB = √41 km; (ii) BC = 2√34 km; (iii) Point dividing AB in ratio 2:3 = (3.6, 1). OR: Area of ΔABC = 37 sq. km.
Q37. (i) Volume of cylinder = 864π cm³; (ii) Volume of one cone (with hemisphere) = 54π cm³; (iii) Number of cones = 864π/54π = 16. OR: CSA of cone + hemisphere ≈ (3√153 + 18)π ≈ 173.1 cm².
Q38. (i) P(bicycle) = 50/200 = 1/4; (ii) P(not car) = 170/200 = 17/20; (iii) P(bus or walking) = 110/200 = 11/20. OR: P(bicycle or car) = 80/200 = 2/5.
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, 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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