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CBSE Sample Practice Paper — 4 (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 two consecutive even numbers is: 1
Q2. √2 + √3 is: 1
Q3. The decimal expansion of 7/80 terminates after how many decimal places? 1
Q4. If the zeroes of the polynomial x² + px + q are double in value to the zeroes of 2x² − 5x − 3, then: 1
Q5. The value of k for which the pair of equations 3x − y + 8 = 0 and 6x − ky + 16 = 0 represent coincident lines is: 1
Q6. If the sum of the roots of the equation 3x² + (2k+1)x − (k+5) = 0 equals the product of the roots, then k equals: 1
Q7. The 4th term from the end of the AP −11, −8, −5, ..., 49 is: 1
Q8. The HCF and LCM of two numbers are 9 and 360 respectively. If one number is 45, the other number is: 1
Q9. In ΔABC, ∠A = 90° and AD ⊥ BC (D on BC). ΔABD ~ ΔCBA by which similarity criterion? 1
Q10. The diagonals of a trapezium ABCD (AB ∥ CD) intersect at O. If AB = 2CD, the ratio of the areas of ΔAOB to ΔCOD is: 1
Q11. Two circles of radii 8 cm and 3 cm have their centres 13 cm apart. The length of a direct common tangent is: 1
Q12. If sinA = cosA, then 2tanA + cos²A equals: 1
Q13. (1 + tanθ + secθ)(1 + cotθ − cosecθ) equals: 1
Q14. A cone and a cylinder have the same base radius and the same height. The ratio of their volumes (cone : cylinder) is: 1
Q15. If P(E) = 0.05, then P(not E) is: 1
Q16. The class mark of the class interval 20–30 is: 1
Q17. If the zeroes of the quadratic polynomial x² + (a+1)x + b are 2 and −3, find a and b. 1
Q18. If Sₙ = n² + 2n represents the sum of the first n terms of an AP, its 10th term is: 1
Q19. Assertion (A): The diagonals of a rhombus bisect each other at right angles.
Reason (R): A rhombus is a parallelogram in which all four sides are equal. 1
Reason (R): A rhombus is a parallelogram in which all four sides are equal. 1
Q20. Assertion (A): sin(A+B) = sinA·cosB + cosA·sinB is valid only for acute angles A and B.
Reason (R): The angle-sum formula for sine holds for all real values of A and B, not only for acute angles. 1
Reason (R): The angle-sum formula for sine holds for all real values of A and B, not only for acute angles. 1
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SECTION B (2 Marks each) — Q21 to Q25
Q21. Find the largest number that divides 245 and 1029, leaving remainder 5 in each case. 2
Q22. If the point P(x, 4) lies on the line segment joining A(−5, 8) and B(4, −10), find the ratio in which P divides AB, and hence find x. 2
— OR —
Find the coordinates of the points of trisection of the line segment joining (4, −1) and (−2, −3). 2
Q23. If sinθ = cosθ, find the value of 2tan²θ + sin²θ − 1. 2
Q24. Prove that: (1 + cotA − cosecA)(1 + tanA + secA) = 2. 2
— OR —
If tanθ = √3, find the value of sinθ · cosθ. 2
Q25. A bag contains 4 red, 5 blue and 3 green marbles. One marble is drawn at random. Find the probability that it is (i) blue, and (ii) not red. 2
SECTION C (3 Marks each) — Q26 to Q31
Q26. Solve for x: 1/(2a+b+2x) = 1/(2a) + 1/b + 1/(2x), where x ≠ 0, and a, b are constants with 2a+b ≠ 0. 3
Q27. Solve the following pair of linear equations: 44x + 55y = 110 and 55x + 44y = 88. 3
— OR —
The sum of the numerator and denominator of a fraction is 12. If the denominator is increased by 3, the fraction becomes 1/2. Find the fraction. 3
Q28. Determine the AP whose 3rd term is 5 and 7th term is 9. 3
Q29. In ΔABC, points D and E lie on AB and AC respectively such that DE ∥ BC. If AD/DB = 3/5 and AE = 3.6 cm, find EC. 3
Q30. Prove that the tangents drawn at the ends of a diameter of a circle are parallel to each other. 3
— OR —
A circle touches all four sides of a quadrilateral ABCD. If AB = 6 cm, BC = 7 cm and CD = 4 cm, find AD. 3
Q31. The following table gives the marks scored by 30 students in a test. Find the median marks.
3
| Marks | 0–10 | 10–20 | 20–30 | 30–40 | 40–50 |
|---|---|---|---|---|---|
| No. of students | 3 | 5 | 9 | 7 | 6 |
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SECTION D (5 Marks each) — Q32 to Q35
Q32. A shopkeeper buys a number of books for ₹960. If he had bought 4 more books for the same amount, each book would have cost him ₹8 less. Find the number of books he bought. 5
— OR —
Two water taps together can fill a tank in 9⅜ hours. The tap of larger diameter takes 10 hours less than the smaller one to fill the tank separately. Find the time in which each tap can separately fill the tank. 5
Q33. Prove that the lengths of tangents drawn from an external point to a circle are equal. Using this result: PQ is a chord of length 8 cm of a circle of radius 5 cm. The tangents at P and Q intersect at a point T. Find the length TP. 5
Q34. The angle of elevation of a cloud from a point 60 m above a lake is 30°, and the angle of depression of its reflection in the lake is 60°. Find the height of the cloud above the surface of the lake. 5
Q35. A tent is in the shape of a cylinder surmounted by a conical top. The height and diameter of the cylindrical part are 2.1 m and 4 m respectively, and the slant height of the conical part is 2.8 m. Find the area of canvas used for making the tent, and the cost of the canvas at ₹500 per m². (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 town-planning committee is designing a park. Three corners of a triangular flower-bed are located at P(−2, −2), Q(2, −4) and R(4, 4) on a grid where 1 unit = 1 m.4
(i) Find the length of PQ. (1)
(ii) Find the length of QR. (1)
(iii) Find the area of the triangular flower-bed PQR using the coordinate geometry area formula. (2)
[OR for (iii): Find the coordinates of the centroid of triangle PQR.]
Q37. Case Study — Mensuration. A juice glass is in the shape of a frustum of a cone, with radii of the top and bottom as 4 cm and 3 cm respectively, and a height of 10 cm.4
(i) Find the slant height of the frustum. (1)
(ii) Find the capacity (volume) of the glass, in cm³. (1)
(iii) Find the curved surface area of the glass (excluding the top and bottom circles), and state the approximate drinking capacity of the glass in millilitres. (2)
[OR for (iii): Find the total surface area of the glass, including the top and bottom circles.]
Q38. Case Study — Probability. A survey of 60 households in a colony recorded the number of smartphone users: 0 phones — 4 households, 1 phone — 10, 2 phones — 22, 3 phones — 16, 4 or more — 8. A household is selected at random.4
(i) Find the probability that the household has exactly 2 smartphones. (1)
(ii) Find the probability that the household has at least 3 smartphones. (1)
(iii) Find the probability that a randomly selected household has fewer than 2 smartphones. (2)
[OR for (iii): Find the probability that the household has more than 1 but fewer than 4 smartphones.]
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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 | (B) 2 | 6 | (A) 4 | 11 | (A) 12 cm | 16 | (B) 25 |
| 2 | (B) irrational | 7 | (A) 40 | 12 | (C) 2.5 | 17 | (A) a=0,b=−6 |
| 3 | (D) 4 | 8 | (A) 72 | 13 | (C) 2 | 18 | (A) 21 |
| 4 | (A) p=−5,q=−6 | 9 | (C) AA | 14 | (B) 1:3 | 19 | (A) |
| 5 | (B) 2 | 10 | (B) 4:1 | 15 | (B) 0.95 | 20 | (D) |
Section B — Key Answers (2 marks each)
Q21. Largest number = 16 (HCF of 240 and 1024, after subtracting the remainder 5 from each number).
Q22. Ratio = 2 : 7; x = −3. OR: Points of trisection: (2, −5/3) and (0, −7/3).
Q23. Value = 1.5 (since θ = 45° when sinθ = cosθ).
Q24. Standard identity proof, both sides reduce to 2. OR: sinθ·cosθ = √3/4.
Q25. P(blue) = 5/12; P(not red) = 8/12 = 2/3.
Section C — Key Answers (3 marks each)
Q26. x = −(2a+b)/2 (x = 0 is rejected as it makes the original expression undefined).
Q27. x = 0, y = 2. OR: Fraction = 5/7.
Q28. First term a = 3, common difference d = 1; AP: 3, 4, 5, 6, 7, ...
Q29. EC = 6 cm (using AD/DB = AE/EC).
Q30. Standard proof using alternate interior angles formed by the tangents with the diameter. OR: AD = 3 cm (using AB + CD = BC + AD).
Q31. Median ≈ 27.78 marks (median class 20–30, using the median formula for grouped data).
Section D — Key Answers (5 marks each)
Q32. Number of books = 20 (solving n² + 4n − 480 = 0). OR: Smaller tap: 25 hours; Larger tap: 15 hours.
Q33. Standard tangent-length proof using RHS congruence. Numerical part: TP = 20/3 cm ≈ 6.67 cm.
Q34. Height of cloud above the lake = 120 m (solving the two tan-equations simultaneously).
Q35. Total canvas area = 44 m² (26.4 m² cylinder + 17.6 m² cone); Cost = ₹22,000.
Section E — Key Answers (Case Studies, 4 marks each)
Q36. (i) PQ = 2√5 units; (ii) QR = 2√17 units; (iii) Area of ΔPQR = 18 sq. units.
Q37. (i) Slant height ≈ 10.05 cm; (ii) Volume ≈ 387.6 cm³; (iii) CSA ≈ 221.1 cm²; capacity ≈ 387.6 mL.
Q38. (i) P(exactly 2 phones) = 11/30; (ii) P(at least 3 phones) = 2/5; (iii) P(fewer than 2 phones) = 7/30.
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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