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CBSE Sample Practice Paper — 8 (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 exponent of 5 in the prime factorisation of 3750 is: 1
Q2. The number 0.6875 can be expressed in the form p/q as: 1
Q3. If n = 2³ × 3⁴ × 5 × 7, the number of consecutive zeros at the end of n is: 1
Q4. If the zeroes of the polynomial x³ − 3x² + x + 1 are a−b, a, a+b, find a. 1
Q5. The pair of equations x = a and y = b graphically represents lines which are: 1
Q6. If one root of the quadratic equation 2x² + kx − 6 = 0 is 2, the value of k is: 1
Q7. The sum of the first n odd natural numbers is: 1
Q8. Two numbers are in the ratio 3 : 4 and their LCM is 84. The HCF of the numbers is: 1
Q9. In ΔABC, DE ∥ BC and AD : DB = 1 : 2. Then ar(ΔADE) : ar(ΔABC) is: 1
Q10. If in ΔABC and ΔDEF, ∠A = ∠D and ∠B = ∠E, then the triangles are similar by: 1
Q11. A circle can have at most how many parallel tangents? 1
Q12. If cosecθ − cotθ = 1/3, then cosecθ + cotθ equals: 1
Q13. The angle of elevation of the sun, when the length of the shadow of a pole equals the height of the pole, is: 1
Q14. If the radius of a circle is doubled, its area becomes: 1
Q15. A number is selected at random from 1 to 100. The probability that it is a perfect square is: 1
Q16. The mean of 5 observations x, x+2, x+4, x+6, x+8 is 15. Find x. 1
Q17. If the nth term of an AP is (2n+1), the sum of its first n terms is: 1
Q18. The product of the zeroes of the polynomial 4x² − 9 is: 1
Q19. Assertion (A): The lengths of tangents drawn from an external point to a circle are equal.
Reason (R): The perpendicular from the centre of a circle to a chord bisects the chord. 1
Reason (R): The perpendicular from the centre of a circle to a chord bisects the chord. 1
Q20. Assertion (A): If cosA = 0, then sinA = 1.
Reason (R): sin²A + cos²A = 1 for all values of A. 1
Reason (R): sin²A + cos²A = 1 for all values of A. 1
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SECTION B (2 Marks each) — Q21 to Q25
Q21. Show that 12ⁿ cannot end with the digit 0 for any natural number n. 2
Q22. Find the coordinates of point A, where AB is a diameter of a circle with centre (2, −3) and B is (1, 4). 2
— OR —
If (1, 2), (4, y), (x, 6) and (3, 5) are the vertices of a parallelogram taken in order, find x and y. 2
Q23. If sinθ = 1/2, find the value of 3cosθ − 4cos³θ. 2
Q24. Evaluate: sin60°cos30° + cos60°sin30°. 2
— OR —
If secθ = 13/12, find the value of (sinθ − 2cosθ)/(3sinθ − cosθ). 2
Q25. A box of 600 electric bulbs contains 12 defective bulbs. One bulb is taken out at random. Find the probability that it is (i) defective, and (ii) not defective. 2
SECTION C (3 Marks each) — Q26 to Q31
Q26. A fraction becomes 4/5 if 1 is added to both its numerator and denominator. If 5 is subtracted from both, it becomes 1/2. Find the fraction. 3
Q27. Solve for x: √(x/(x−3)) + √((x−3)/x) = 5/2, where x ≠ 0, 3. 3
— OR —
Find the value of k for which the quadratic equation kx² + 2x + 1 = 0 has two equal real roots. 3
Q28. If the 8th term of an AP is 31, and the 15th term is 16 more than the 11th term, find the AP. 3
Q29. Two triangles ABC and DBC lie on the same base BC. If AD intersects BC at O, prove that ar(ΔABC)/ar(ΔDBC) = AO/DO. 3
Q30. Prove that the parallelogram circumscribing a circle is a rhombus. 3
— OR —
PA and PB are tangents from an external point P to a circle with centre O. If ∠PAB = 50°, find ∠AOB. 3
Q31. Calculate the median for the following distribution:
3
| Class | 0–10 | 10–20 | 20–30 | 30–40 | 40–50 |
|---|---|---|---|---|---|
| Frequency | 5 | 15 | 20 | 23 | 17 |
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SECTION D (5 Marks each) — Q32 to Q35
Q32. A peacock is sitting on top of a pillar 9 m high. From a point 27 m away from the base of the pillar, a snake is coming towards its hole at the base of the pillar. Seeing the snake, the peacock pounces on it in a straight line. If their speeds are equal, find the distance from the hole where the snake is caught. 5
— OR —
The denominator of a fraction is one more than twice its numerator. If the sum of the fraction and its reciprocal is 2 1/16, find the fraction. 5
Q33. State and prove the converse of the Pythagoras Theorem: In a triangle, if the square of one side is equal to the sum of the squares of the other two sides, then the angle opposite the first side is a right angle. 5
Q34. A statue 1.6 m tall stands on top of a pedestal. From a point on the ground, the angle of elevation of the top of the statue is 60°, and from the same point, the angle of elevation of the top of the pedestal is 45°. Find the height of the pedestal. 5
Q35. From a solid cylinder of height 20 cm and diameter 12 cm, a conical cavity of the same height and same diameter is hollowed out. Find the total surface area of the remaining solid. (Use π = 22/7, and √436 ≈ 20.88) 5
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SECTION E — Case Study Based Questions (4 Marks each) — Q36 to Q38
Q36. Case Study — Coordinate Geometry. An architect is designing a triangular garden plot with corners at A(1, 1), B(5, 1) and C(3, 4) on a blueprint grid (1 unit = 1 m).4
(i) Find the length of AB. (1)
(ii) Find the length of AC. (1)
(iii) The architect wants to place a fountain at the centroid of triangle ABC. Find its coordinates. (2)
[OR for (iii): Find the coordinates of the midpoint of BC, and hence describe how the fountain's position (centroid) relates to this midpoint and vertex A.]
Q37. Case Study — Mensuration. A company manufactures a solid toy in the shape of a hemisphere surmounted by a cone. The base radius of the cone is 3.5 cm and its height is 5 cm; the hemisphere has the same radius.4
(i) Find the slant height of the cone. (1)
(ii) Find the volume of the toy (cone + hemisphere). (1)
(iii) Find the total surface area of the toy. (2)
[OR for (iii): Find how much more volume the hemisphere contributes to the toy compared to the cone.]
Q38. Case Study — Probability. In a lottery, there are 10 prizes and 25 blanks. A lottery ticket is drawn at random from all the tickets.4
(i) Find the probability of getting a prize. (1)
(ii) Find the probability of not getting a prize. (1)
(iii) If two tickets are drawn one after another, without replacement, find the probability that both are prize-winning tickets. (2)
[OR for (iii): Find the probability that the first ticket is a prize and the second is a blank.]
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APNA CAREER PORTAL
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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 | (C) 4 | 6 | (A) −1 | 11 | (B) 2 | 16 | (A) 11 |
| 2 | (A) 11/16 | 7 | (B) n² | 12 | (B) 3 | 17 | (A) n²+2n |
| 3 | (B) 1 | 8 | (A) 7 | 13 | (B) 45° | 18 | (A) −9/4 |
| 4 | (A) 1 | 9 | (C) 1:9 | 14 | (C) 4 times | 19 | (B) |
| 5 | (C) intersect at (a,b) | 10 | (C) AA | 15 | (A) 1/10 | 20 | (A) |
Section B — Key Answers (2 marks each)
Q21. Standard proof: 12ⁿ = 2^(2n) × 3ⁿ has no factor of 5, so it can never end in 0.
Q22. A = (3, −10). OR: x = 6, y = 3.
Q23. 3cosθ − 4cos³θ = 0 (using θ = 30°, this equals cos3θ = cos90° = 0).
Q24. sin60°cos30° + cos60°sin30° = sin90° = 1. OR: Value = −19/3.
Q25. P(defective) = 12/600 = 1/50; P(not defective) = 588/600 = 49/50.
Section C — Key Answers (3 marks each)
Q26. Fraction = 7/9.
Q27. x = 4 or x = −1. OR: k = 1 (for equal roots).
Q28. First term a = 3, common difference d = 4; AP: 3, 7, 11, 15, ...
Q29. Standard proof: triangles ABC and DBC have the same base BC, so the ratio of their areas equals the ratio of their heights from A and D, which in turn equals AO/DO (using similar triangles formed with the intersection point O).
Q30. Standard proof using tangent-length equality (AP=AS, BP=BQ, etc.) and the parallelogram property. OR: ∠AOB = 100°.
Q31. Median = 30 (median class 20–30, using the median formula for grouped data).
Section D — Key Answers (5 marks each)
Q32. Distance from the hole where the snake is caught = 12 m. OR: Fraction = 3/7.
Q33. Standard proof by contradiction, constructing a right triangle with the same two legs and showing it must be congruent to the original triangle (SSS), forcing the angle to be 90°.
Q34. Height of pedestal ≈ 2.19 m (using h = 1.6/(√3−1)).
Q35. Total surface area ≈ 1261.4 cm² (curved surface of cylinder + curved surface of conical cavity + area of one flat circular base).
Section E — Key Answers (Case Studies, 4 marks each)
Q36. (i) AB = 4 units; (ii) AC = √13 units; (iii) Centroid = (3, 2).
Q37. (i) Slant height ≈ 6.10 cm; (ii) Volume = 154 cm³; (iii) Total surface area ≈ 144.1 cm².
Q38. (i) P(prize) = 2/7; (ii) P(not prize) = 5/7; (iii) P(both prizes, without replacement) = 9/119.
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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