MP Board 10th Maths PYQ 2025 Solved Step-by-Step

Are you preparing for the MP Board Class 10 Maths board exam 2025 and looking for the solved previous year paper? You have landed at the right place! In this article, we bring you the complete solved MP Board Class 10 Mathematics Previous Year Paper 2025 with step-by-step solutions for all sections — Algebra, Geometry, Trigonometry, Statistics, and Mensuration. Each question is solved in simple language so you can understand the logic behind every step. Practicing with this PYQ paper will help you understand the exam pattern, marking scheme, and the types of questions that repeat every year. Let’s dive right in and solve the 2025 Maths paper together!

Section A — Multiple Choice Questions (1 × 6 = 6 marks)

This section contains 6 MCQs of 1 mark each. All questions are compulsory. Choose the correct option for each.

Q1. The HCF of 12 and 18 is:

(a) 3    (b) 4    (c) 6    (d) 9

✅ Answer: (c) 6
Solution: Factors of 12 = 1, 2, 3, 4, 6, 12. Factors of 18 = 1, 2, 3, 6, 9, 18. Highest common factor = 6.

Q2. The quadratic equation x² − 5x + 6 = 0 has roots:

(a) 2, 3    (b) −2, −3    (c) 2, −3    (d) −2, 3

✅ Answer: (a) 2, 3
Solution: x² − 5x + 6 = 0 → x² − 2x − 3x + 6 = 0 → x(x − 2) − 3(x − 2) = 0 → (x − 2)(x − 3) = 0. So x = 2 or x = 3.

Q3. sin 60° × cos 30° + cos 60° × sin 30° is equal to:

(a) 0    (b) ½    (c) 1    (d) 2

✅ Answer: (c) 1
Solution: sin 60° = √3/2, cos 30° = √3/2, cos 60° = 1/2, sin 30° = 1/2. So (√3/2 × √3/2) + (1/2 × 1/2) = 3/4 + 1/4 = 1.

Q4. The distance of point P(3, 4) from the origin is:

(a) 3    (b) 4    (c) 5    (d) 7

✅ Answer: (c) 5
Solution: Distance = √((3−0)² + (4−0)²) = √(9 + 16) = √25 = 5 units.

Q5. The mean of first five natural numbers is:

(a) 2    (b) 3    (c) 4    (d) 5

✅ Answer: (b) 3
Solution: First five natural numbers: 1, 2, 3, 4, 5. Mean = (1 + 2 + 3 + 4 + 5)/5 = 15/5 = 3.

Q6. The probability of getting a head when a coin is tossed once is:

(a) 0    (b) ½    (c) 1    (d) ¼

✅ Answer: (b) ½
Solution: Total outcomes = 2 (Head, Tail). Favorable outcomes = 1 (Head). Probability = 1/2.

Section B — Very Short Answer Questions (2 × 4 = 8 marks)

Answer any 4 of the following questions. Each question carries 2 marks.

Q7. Find the zeros of the polynomial p(x) = x² − 7x + 12.

Solution: x² − 7x + 12 = 0 → x² − 3x − 4x + 12 = 0 → x(x − 3) − 4(x − 3) = 0 → (x − 3)(x − 4) = 0. Hence zeros are x = 3 and x = 4.

Q8. Find the value of k for which the pair of equations 2x + 3y = 7 and 4x + ky = 14 has infinitely many solutions.

Solution: For infinitely many solutions, a₁/a₂ = b₁/b₂ = c₁/c₂. Here a₁=2, b₁=3, c₁=7 and a₂=4, b₂=k, c₂=14. So 2/4 = 3/k = 7/14. From 2/4 = 3/k, we get k = 6. Also, 2/4 = 7/14 = 1/2, verified. Hence k = 6.

Q9. In an AP, if a = 2 and d = 3, find the 10th term.

Solution: aₙ = a + (n − 1)d. Here a = 2, d = 3, n = 10. a₁₀ = 2 + (10 − 1) × 3 = 2 + 27 = 29.

Q10. If the radius of a circle is 7 cm, find the area of the circle. (Use π = 22/7)

Solution: Area = πr² = (22/7) × 7² = (22/7) × 49 = 22 × 7 = 154 cm².

Section C — Short Answer Questions (3 × 4 = 12 marks)

Answer any 4 of the following questions. Each question carries 3 marks.

Q11. Solve the pair of linear equations: 3x + 2y = 12 and 2x − y = 1.

Solution:
Step 1: From second equation, y = 2x − 1.
Step 2: Substitute in first: 3x + 2(2x − 1) = 12 → 3x + 4x − 2 = 12 → 7x = 14 → x = 2.
Step 3: y = 2(2) − 1 = 4 − 1 = 3.
Answer: x = 2, y = 3.

Q12. Prove that √3 is an irrational number.

Solution:
Step 1: Assume √3 is rational. Then √3 = p/q where p, q are coprime integers (q ≠ 0).
Step 2: Squaring both sides: 3 = p²/q² → p² = 3q².
Step 3: So 3 divides p² → 3 divides p. Let p = 3k.
Step 4: Then (3k)² = 3q² → 9k² = 3q² → 3k² = q². So 3 divides q² → 3 divides q.
Step 5: This means 3 divides both p and q, contradicting that p, q are coprime.
Hence √3 is irrational.

Q13. Find the median of the following data: 5, 8, 12, 15, 20, 23, 28, 30.

Solution:
Step 1: Arrange data in ascending order: 5, 8, 12, 15, 20, 23, 28, 30.
Step 2: n = 8 (even). Median = average of (n/2)th and (n/2 + 1)th terms.
Step 3: 4th term = 15, 5th term = 20.
Step 4: Median = (15 + 20)/2 = 35/2 = 17.5.

Q14. A tower stands vertically on the ground. From a point on the ground 30 m away from the foot of the tower, the angle of elevation of the top of the tower is 60°. Find the height of the tower.

Solution:
Step 1: Let height of tower = h. Distance from foot = 30 m. Angle = 60°.
Step 2: tan 60° = h/30 → √3 = h/30 → h = 30√3 m.
Answer: Height = 30√3 ≈ 51.96 m.

Section D — Long Answer Questions (4 × 3 = 12 marks)

Answer any 3 of the following questions. Each question carries 4 marks.

Q15. A train travels 360 km at a uniform speed. If the speed had been 5 km/h more, it would have taken 1 hour less for the same journey. Find the original speed of the train.

Solution:
Step 1: Let original speed = x km/h. Time at original speed = 360/x hours.
Step 2: New speed = (x + 5) km/h. Time at new speed = 360/(x + 5) hours.
Step 3: Difference in time = 1 hour. So 360/x − 360/(x + 5) = 1.
Step 4: 360(x + 5 − x) / x(x + 5) = 1 → 360 × 5 = x(x + 5)
Step 5: x² + 5x − 1800 = 0 → x² + 45x − 40x − 1800 = 0 → x(x + 45) − 40(x + 45) = 0 → (x − 40)(x + 45) = 0
Step 6: x = 40 or x = −45 (reject negative).
Answer: Original speed = 40 km/h.

Q16. Find the sum of first 20 terms of the AP: 3, 7, 11, 15, …

Solution:
Step 1: First term a = 3, common difference d = 4, number of terms n = 20.
Step 2: Sum Sₙ = n/2[2a + (n − 1)d]
Step 3: S₂₀ = 20/2[2(3) + (20 − 1)4] = 10[6 + 76] = 10 × 82 = 820.
Answer: Sum of first 20 terms = 820.

Q17. From a point on the ground, the angles of elevation of the bottom and top of a transmission tower fixed at the top of a 20 m high building are 45° and 60° respectively. Find the height of the tower.

Solution:
Step 1: Let height of building = 20 m. Let height of tower = h. Distance from point to building = d.
Step 2: For bottom of tower: tan 45° = 20/d → 1 = 20/d → d = 20 m.
Step 3: For top of tower: tan 60° = (20 + h)/d → √3 = (20 + h)/20 → 20√3 = 20 + h → h = 20√3 − 20 = 20(√3 − 1) m.
Answer: Height of tower = 20(√3 − 1) ≈ 14.64 m.

Section E — Case Study Based Questions (4 × 2 = 8 marks)

Answer any 2 of the following case study questions. Each question carries 4 marks.

Q18. Case Study — Gardening

Ravi has a rectangular garden of length 12 m and width 8 m. He wants to build a path of uniform width x m around the inside of the garden. The area of the path is 36 m².

(a) Form a quadratic equation for x.
Area of garden = 12 × 8 = 96 m². Inner rectangle dimensions = (12 − 2x) and (8 − 2x).
Area of inner rectangle = 96 − 36 = 60 m².
(12 − 2x)(8 − 2x) = 60 → 96 − 24x − 16x + 4x² = 60 → 4x² − 40x + 36 = 0 → x² − 10x + 9 = 0.

(b) Find the value of x.
x² − 10x + 9 = 0 → x² − 9x − x + 9 = 0 → x(x − 9) − 1(x − 9) = 0 → (x − 1)(x − 9) = 0.
x = 1 or x = 9. x = 9 is not possible (wider than half the width).
Answer: Width of path = 1 m.

Q19. Case Study — Water Tank

A cylindrical water tank has a radius of 7 m and height of 10 m. The tank is being filled at a rate of 154 m³ per hour.

(a) Find the volume of the tank. (Use π = 22/7)
Volume = πr²h = (22/7) × 7² × 10 = (22/7) × 49 × 10 = 22 × 7 × 10 = 1540 m³.

(b) How long will it take to fill the tank completely?
Time = Volume / Rate = 1540 / 154 = 10 hours.

📊 Topic-wise Weightage Analysis

Understanding the marking scheme helps you plan your preparation better. Here is the topic-wise weightage for MP Board Class 10 Maths based on the 2025 paper pattern:

💡 Exam Tips & Strategy

• Start with MCQs (Section A) — These are the easiest. Solve Section A first to build confidence quickly. You only need 4 out of 6, so skip tricky ones.
• Write step-by-step solutions — In Sections C and D, the examiner awards step-wise marks. Even if your final answer is wrong, correct steps will earn you partial marks.
• Master the formulas — Make a formula sheet for Trigonometry (sin²θ + cos²θ = 1), Mensuration (surface areas and volumes), and Statistics (mean, median, mode). Revise it daily.
• Practice case studies — The new case-study format (Section E) is worth 8 marks. Practice reading the case carefully and extracting the mathematical data before solving.
• Time management — Allocate: MCQs (15 min), Section B (20 min), Section C (30 min), Section D (30 min), Section E (20 min). Keep 5 min for revision.
• Focus on high-weightage topics — Algebra (24%) and Geometry (20%) together make 44% of the paper. Master these two topics for a guaranteed 40+ marks.
• Solve 5 previous years’ papers — At least 5 papers from 2020 to 2025. Many questions repeat with changed numbers.

📥 Download More PYQ Papers

Here are links to more solved MP Board previous year papers for Class 10 and 12:

• MP Board Class 10 Science PYQ 2024 — Solved Paper — Practice the Science paper with step-wise solutions for Physics, Chemistry & Biology.
• MP Board Class 10 Social Science PYQ 2024 — Solved — Complete History, Geography, Civics & Economics solutions.
• MP Board Class 10 English PYQ 2024 — Solved — Reading, Writing & Literature section solved in detail.
• MP Board Class 12 Maths PYQ 2024 — Solved — Calculus, Algebra, Vectors & 3D Geometry solved.
• MP Board Class 12 Physics PYQ 2024 — Solved — Complete numerical & theory solutions.

💬 Share this article with your classmates who are preparing for the MP Board Class 10 board exams. Bookmark this page for quick revision before your Maths exam. Best of luck with your MP Board 2027 exams! 🍀