In short: CGBSE Class 10 theory papers are 75 marks (25 marks internal/project). The board publishes a blueprint that fixes unit-wise marks and question types. This model paper turns that structure into a full 75-mark Maths practice paper with complete solutions. It also includes Science and Social Science practice sets with answers. Solve it in 3 hours, check against the solutions, and then solve the official CGBSE model paper from cgbse.nic.in.
How to Read the CGBSE Blueprint
A blueprint is a grid with two axes:
| Axis | What It Shows |
|---|---|
| Rows (units/chapters) | How many marks each unit carries |
| Columns (question types) | How many objective, very short, short and long questions there are |
How to use it: Write the unit marks next to each chapter in your syllabus. Study high-mark units first. Practise the exact question types listed for each unit. Blueprints can change slightly each year, so always use the 2026-27 version from cgbse.nic.in.
Maths Model Paper — 75 Marks
Time: 3 hours | Maximum Marks: 75
The layout below follows the typical CGBSE structure: objective questions first, then answers of increasing length. The exact number of questions can vary by year.
| Section | Type | Questions | Marks |
|---|---|---|---|
| A | Objective (MCQ, fill in the blanks, match, true/false) | 4 × 5 parts | 20 |
| B | Very short answer (2 marks) | 5 | 10 |
| C | Short answer (3 marks) | 5 | 15 |
| D | Long answer (4 marks) | 5 | 20 |
| E | Very long answer (5 marks) | 2 | 10 |
| Total | 75 |
Section A — Objective (20 marks)
Q1. Choose the correct option (5 × 1)
- (i) The HCF of 8 and 12 is: (a) 2 (b) 4 (c) 24 (d) 96
- (ii) If α, β are zeroes of x² − 7x + 10, then α + β = (a) 7 (b) −7 (c) 10 (d) −10
- (iii) The roots of x² − 5x + 6 = 0 are: (a) 2, 3 (b) −2, −3 (c) 1, 6 (d) −1, −6
- (iv) cos 0° + sin 90° = (a) 0 (b) 1 (c) 2 (d) √2
- (v) The mean of 2, 4, 6, 8, 10 is: (a) 5 (b) 6 (c) 7 (d) 30
Q2. Fill in the blanks (5 × 1)
- (i) The decimal expansion of a rational number is either terminating or ____.
- (ii) In an AP, if a = 5 and d = 0, then a₁₀ = ____.
- (iii) The distance between (a, 0) and (0, a) is ____.
- (iv) A tangent to a circle meets it at ____ point(s).
- (v) The probability of an event lies between ____ and ____.
Q3. Match the following (5 × 1)
| Column A | Column B |
|---|---|
| (i) Area of a circle | (a) 2πr |
| (ii) Circumference of a circle | (b) (4/3)πr³ |
| (iii) Volume of a sphere | (c) πr² |
| (iv) Curved surface area of a cone | (d) πr²h |
| (v) Volume of a cylinder | (e) πrl |
Q4. True or False (5 × 1)
- (i) Every integer is a rational number.
- (ii) A quadratic equation can have at most two roots.
- (iii) All equilateral triangles are similar.
- (iv) tan 90° is defined.
- (v) The median of grouped data can be found using an ogive.
Section B — Very Short Answer (5 × 2 = 10 marks)
- Find the LCM of 18 and 24 by prime factorisation.
- Find a quadratic polynomial whose zeroes are 3 and −2.
- Find the 25th term of the AP 7, 4, 1, −2, ...
- Find the midpoint of the segment joining (−3, 5) and (7, −1).
- If sin θ = 5/13, find cos θ and tan θ.
Section C — Short Answer (5 × 3 = 15 marks)
- Prove that 3 + 2√5 is irrational (given that √5 is irrational).
- Solve by elimination: 3x + 4y = 10 and 2x − 2y = 2.
- Find two consecutive positive integers whose squares add up to 365.
- A chord of a circle of radius 10 cm subtends a right angle at the centre. Find the area of the minor segment. (π = 3.14)
- A bag has 5 red, 8 white and 7 black balls. One ball is drawn at random. Find P(red), P(not black) and P(white or black).
Section D — Long Answer (5 × 4 = 20 marks)
- The sum of the 4th and 8th terms of an AP is 24, and the sum of the 6th and 10th terms is 44. Find the first three terms.
- Prove that the tangent at any point of a circle is perpendicular to the radius through the point of contact.
- Prove: (sin A − 2 sin³A) / (2 cos³A − cos A) = tan A.
- A 1.5 m tall boy stands some distance from a 30 m tall building. As he walks towards it, the angle of elevation of the top changes from 30° to 60°. Find how far he walked.
- A solid is a cone on top of a hemisphere, both of radius 1 cm. The height of the cone equals its radius. Find the volume of the solid in terms of π.
Section E — Very Long Answer (2 × 5 = 10 marks)
- Prove that the ratio of the areas of two similar triangles equals the square of the ratio of their corresponding sides.
- Find the mean (by step deviation) and the mode of the following data:
| Class | 10-25 | 25-40 | 40-55 | 55-70 | 70-85 | 85-100 |
|---|---|---|---|---|---|---|
| f | 2 | 3 | 7 | 6 | 6 | 6 |
Maths Solutions
Section A
| Q1 | Answer | Q2 | Answer |
|---|---|---|---|
| (i) | (b) 4 | (i) | non-terminating repeating |
| (ii) | (a) 7 | (ii) | 5 |
| (iii) | (a) 2, 3 | (iii) | a√2 |
| (iv) | (c) 2 | (iv) | one |
| (v) | (b) 6 | (v) | 0 and 1 |
Q3: (i)-(c), (ii)-(a), (iii)-(b), (iv)-(e), (v)-(d)
Q4: (i) True (ii) True (iii) True (iv) False (v) True
Section B
Q5. 18 = 2 × 3², 24 = 2³ × 3. LCM = 2³ × 3² = 72.
Q6. Sum = 1, product = −6. Polynomial = x² − x − 6.
Q7. a = 7, d = −3. a₂₅ = 7 + 24(−3) = −65.
Q8. ((−3 + 7)/2, (5 − 1)/2) = (2, 2).
Q9. Opposite = 5, hypotenuse = 13, adjacent = 12. cos θ = 12/13, tan θ = 5/12.
Section C
Q10. Assume 3 + 2√5 = r, a rational number. Then √5 = (r − 3)/2, which is rational. This contradicts the fact that √5 is irrational. So 3 + 2√5 is irrational.
Q11. Multiply the second equation by 2: 4x − 4y = 4. Add it to the first: 7x = 14, so x = 2. Then 6 + 4y = 10, so y = 1. x = 2, y = 1.
Q12. x² + (x + 1)² = 365, so 2x² + 2x − 364 = 0, which gives x² + x − 182 = 0, or (x + 14)(x − 13) = 0. Take the positive value: 13 and 14.
Q13.
- Sector area = (90/360) × 3.14 × 100 = 78.5 cm²
- Triangle area = ½ × 10 × 10 = 50 cm²
- Minor segment = 78.5 − 50 = 28.5 cm²
Q14. Total = 20.
- P(red) = 5/20 = 1/4
- P(not black) = 13/20
- P(white or black) = 15/20 = 3/4
Section D
Q15. a₄ + a₈ = 2a + 10d = 24, so a + 5d = 12. a₆ + a₁₀ = 2a + 14d = 44, so a + 7d = 22. Subtracting: 2d = 10, so d = 5 and a = −13. First three terms: −13, −8, −3.
Q16. Let OP be the radius to the point of contact P, and let Q be any other point on the tangent. Q lies outside the circle, so OQ > OP. This is true for every point on the tangent except P, so OP is the shortest distance from O to the tangent. The shortest segment from a point to a line is perpendicular to it, so OP ⊥ tangent.
Q17.
- Numerator = sin A (1 − 2 sin²A)
- Denominator = cos A (2 cos²A − 1)
- Now 1 − 2 sin²A = 2 cos²A − 1 (both equal cos 2A)
- So LHS = sin A / cos A = tan A
Q18. Effective height = 30 − 1.5 = 28.5 m.
- At 30°: distance = 28.5/tan 30° = 28.5√3
- At 60°: distance = 28.5/tan 60° = 28.5/√3
- Distance walked = 28.5(√3 − 1/√3) = 28.5 × (2/√3) = 19√3 ≈ 32.9 m
Q19.
- Cone volume = (1/3)πr²h = (1/3)π(1)(1) = π/3
- Hemisphere volume = (2/3)πr³ = 2π/3
- Total = π/3 + 2π/3 = π cm³
Section E
Q20. Take △ABC ~ △PQR. Draw altitudes AM and PN.
- ar(ABC)/ar(PQR) = (½ × BC × AM) / (½ × QR × PN)
- △ABM ~ △PQN (AA similarity), so AM/PN = AB/PQ = BC/QR
- So the ratio = (BC/QR) × (BC/QR) = (BC/QR)²
- By similarity, this also equals (AB/PQ)² = (CA/RP)². Hence proved.
Q21. Take a = 62.5 and h = 15.
| Class | xᵢ | fᵢ | uᵢ = (xᵢ − 62.5)/15 | fᵢuᵢ |
|---|---|---|---|---|
| 10-25 | 17.5 | 2 | −3 | −6 |
| 25-40 | 32.5 | 3 | −2 | −6 |
| 40-55 | 47.5 | 7 | −1 | −7 |
| 55-70 | 62.5 | 6 | 0 | 0 |
| 70-85 | 77.5 | 6 | 1 | 6 |
| 85-100 | 92.5 | 6 | 2 | 12 |
| Total | 30 | −1 |
Mean = 62.5 + (−1/30) × 15 = 62.5 − 0.5 = 62.
Mode: the modal class is 40-55 (f₁ = 7, f₀ = 3, f₂ = 6, l = 40, h = 15). Mode = 40 + [(7 − 3)/(14 − 3 − 6)] × 15 = 40 + (4/5) × 15 = 52.
Practice any chapter on Padhaao
AI tutor for CBSE, ICSE & 15 state boards · 10 learning modes · 3 free chapters daily
Science Practice Set (With Answers)
| # | Question | Marks | Answer |
|---|---|---|---|
| 1 | Name the reaction: Fe + CuSO₄ → FeSO₄ + Cu | 1 | Displacement reaction |
| 2 | Chemical name of plaster of Paris | 1 | Calcium sulphate hemihydrate (CaSO₄·½H₂O) |
| 3 | Unit of the power of a lens | 1 | Dioptre (D) |
| 4 | Which blood vessel carries blood from the lungs to the heart? | 1 | Pulmonary vein |
| 5 | Why is a respiratory pigment needed in large animals? | 2 | Diffusion alone cannot supply enough oxygen. Haemoglobin carries oxygen efficiently in the blood |
| 6 | Find the resistance of a 60 W, 220 V bulb. | 2 | R = V²/P = 48400/60 ≈ 806.7 Ω |
| 7 | A concave mirror has f = 15 cm. The object is at 10 cm. Find v. | 3 | 1/v = 1/f − 1/u = −1/15 + 1/10 = 1/30, so v = +30 cm (virtual, erect, behind the mirror) |
| 8 | Write the steps of the cleansing action of soap. | 3 | The hydrophobic tail attaches to grease and the hydrophilic head faces water. Micelles form and trap the dirt, which is rinsed away |
| 9 | Describe the structure of a neuron with a diagram. | 4 | Dendrites, cell body (cyton), axon, myelin sheath, nerve endings. The impulse travels from dendrite to axon end |
| 10 | Explain sex determination in humans. | 4 | The mother gives an X chromosome. The father gives X or Y. XX = girl, XY = boy, so the father's chromosome decides the sex |
Social Science Practice Set (With Answers)
| # | Question | Marks | Answer Points |
|---|---|---|---|
| 1 | Who founded the Swaraj Party? | 1 | C.R. Das and Motilal Nehru |
| 2 | Where is the Bhilai Steel Plant? | 1 | Durg district, Chhattisgarh |
| 3 | What is a "majoritarian" policy? | 2 | A majority community imposes its will and ignores minority needs, as in Sri Lanka after 1948 |
| 4 | Why is the per capita income used to compare countries? | 2 | Total income does not account for population. Per capita income shows the average income per person |
| 5 | Explain three features of the Indian federal system. | 3 | Three-tier government, three lists of subjects, an independent judiciary |
| 6 | Explain the importance of iron ore in Chhattisgarh's economy. | 3 | Bailadila mines produce high-grade haematite. Ore is supplied to Bhilai and exported via Visakhapatnam. It also creates jobs and revenue |
| 7 | Describe the effects of the First World War on India. | 4 | Higher defence spending and taxes, forced recruitment, crop failures and price rise, influenza epidemic, rising anger that fed the national movement |
| 8 | Map: Korba, Bhilai, Bailadila, Hirakud dam, Visakhapatnam port | 4 | Practise on a blank India map |
3-Hour Exam Plan
| Time | Section |
|---|---|
| First 15 min | Read the full paper. Mark easy questions |
| 15-45 min | Section A (objective): finish all 20 parts |
| 45-90 min | Sections B + C |
| 90-150 min | Sections D + E |
| 150-180 min | Check calculations, units, diagrams and labels |
Frequently Asked Questions
What is the CGBSE Class 10 paper pattern for 2027?
Each main subject has a 75-mark theory paper and 25 marks of internal/project assessment. The paper starts with objective questions (MCQ, fill in the blanks, match, true/false) and then moves to very short, short, long and very long answers. The official blueprint gives the exact split.
Is this model paper made by CGBSE?
No. It is a practice paper built on the CGBSE blueprint structure and the NCERT syllabus. Use it alongside the official CGBSE model papers published on cgbse.nic.in.
How does the blueprint help in scoring?
It tells you exactly how many marks each unit carries and which question types to expect. You can spend more time on high-mark units and practise the right answer lengths, without guessing.
Does CGBSE give step marks in Maths?
Yes. Correct formulas, method and intermediate steps earn marks even if the final answer is wrong. Always write each step and box the final answer.
How many model papers should I solve?
Solve at least 3 papers per main subject: this practice paper, the official CGBSE model paper, and one previous year paper. Do them under 3-hour timed conditions.
Can I write answers in Hindi?
Yes. Most CGBSE students write in Hindi medium. Choose one medium and use technical terms consistently.
Related Guides
CGBSE Class 10 Important Questions (All Subjects) | CGBSE Important Questions Hindi | CGBSE Class 10 Complete Guide | CGBSE Date Sheet 2027 | CGBSE Class 12 Preparation
Practice on Padhaao
Padhaao has NCERT-aligned Class 10 Maths, Science and SST chapters. After this paper, revise the weak chapters with Key Points mode and test yourself in Quiz Mode.
3 free chapters every day.
₹21/mo (annual plan)
Padhaao covers 17,000+ chapters across 17 boards
Your exact syllabus. 10 learning modes. 3 free daily. Pro for unlimited - ₹21/month (annual plan).
Want the full chapter with solved examples?
10 learning modes. Board-specific content. 3 free chapters every day.
Try it free