CBSE Class 10 Mathematics: CBSE Class 10 Maths Question Paper 2025 with Solutions PDF

Mastering CBSE Class 10 Mathematics requires consistent problem-solving practice and a clear understanding of the board exam structure. The CBSE Class 10 Maths Question Paper 2025 with Solutions PDF serves as an indispensable benchmark for students appearing in the 2025 and 2026 board examinations. Analyzing the 2025 question paper helps you understand the balance between standard textbook problems, conceptual reasoning, and application-based competency questions across both Mathematics Standard (Subject Code: 041) and Mathematics Basic (Subject Code: 241).

Whether your goal is scoring 100/100 or securing solid foundational marks, working through verified solutions gives you insight into step-marking criteria, common algebraic traps, and time-allocation strategies for the 3-hour theoretical examination.

Key Concepts & Exam Pattern Blueprint

The CBSE Class 10 Mathematics theory examination carries 80 marks (with 20 marks allocated to internal assessment), evaluated over a duration of 3 hours. The question paper follows a standardized structure comprising 38 compulsory questions divided into five distinct sections:

• Section A (20 Marks): 20 Multiple Choice Questions (MCQs) carrying 1 mark each, including 2 Assertion-Reasoning questions (Q19 and Q20).
• Section B (10 Marks): 5 Very Short Answer (VSA) questions carrying 2 marks each, requiring crisp 2-to-3 step solutions.
• Section C (18 Marks): 6 Short Answer (SA) questions carrying 3 marks each, covering intermediate proofs and multi-step derivations.
• Section D (20 Marks): 4 Long Answer (LA) questions carrying 5 marks each, focusing on comprehensive geometric theorems, surface area problems, or algebraic modeling.
• Section E (12 Marks): 3 Case-Based / Integrated assessment units carrying 4 marks each, broken down into sub-questions of 1, 1, and 2 marks with internal choices.

Unit-Wise Weightage Breakdown

High-Yield Mathematical Formulas to Memorize

Reviewing and memorizing these foundational formulas ensures rapid recall during high-pressure sections of the board exam 10:

• Quadratic Formula: For ax² + bx + c = 0, roots are given by x = (-b ± √(b2 - 4ac)) / (2a). Discriminant D = b² - 4ac defines nature of roots: real and distinct if D > 0, real and equal if D = 0, and no real roots if D < 0.
• Arithmetic Progression: The n^th term is an = a + (n - 1)d and the sum of first n terms is Sn = (n / 2)[2a + (n - 1)d] = (n / 2)[a + l].
• Coordinate Geometry (Section Formula): The coordinates of point P(x, y) dividing the line segment joining A(x₁, y₁) and B(x₂, y₂) internally in ratio m₁ : m₂ are: x = (m1x2 + m2x1) / (m1 + m2), y = (m1y2 + m2y1) / (m1 + m2).
• Trigonometric Identity Essentials: sin2θ + cos2θ = 1, 1 + tan2θ = sec2θ, and 1 + cot2θ = cosec2θ.
• Empirical Statistical Relationship: 3 × Median = Mode + 2 × Mean.

Important CBSE Questions with Step-by-Step Solutions

Here are representative questions aligned with the latest CBSE Class 10 Mathematics paper pattern, complete with marking-scheme aligned step-by-step solutions.

Section A: Multiple Choice Question (1 Mark)

Question 1: If the quadratic equation kx² - 2√5 kx + 15 = 0 has two equal and real roots, then the non-zero value of k is:

(a) 2      (b) 3      (c) 5      (d) √5

Solution:

1. Compare the given equation with standard form ax² + bx + c = 0:
   Here, a = k, b = -2√5 k, and c = 15.
2. For real and equal roots, the discriminant D = 0:
   D = b2 - 4ac = (-2√5 k)2 - 4(k)(15) = 0
3. Simplify the algebraic expression:
   20k2 - 60k = 0
20k(k - 3) = 0
4. Since k ≠ 0 (otherwise the equation ceases to be quadratic), we obtain:
   k - 3 = 0 ⇒ k = 3

Correct Answer: (b) 3

Section B: Very Short Answer Type (2 Marks)

Question 2: Find the ratio in which the y-axis divides the line segment joining the points A(5, -6) and B(-1, -4). Also find the coordinates of the point of intersection.

Solution:

1. Let the y-axis divide the segment joining A(5, -6) and B(-1, -4) in the ratio k : 1 at point P(0, y).
2. Using the section formula for the x-coordinate:
   x = (k(-1) + 1(5)) / (k + 1)
3. Since point P lies on the y-axis, its x-coordinate is 0:
   0 = (-k + 5) / (k + 1) ⇒ -k + 5 = 0 ⇒ k = 5
   Therefore, the required ratio is 5 : 1.
4. Now, calculate the y-coordinate using k = 5:
   y = (5(-4) + 1(-6)) / (5 + 1) = (-20 - 6) / 6 = -26 / 6 = -13 / 3

Final Answer: The ratio is 5 : 1 and the point of intersection on the y-axis is (0, -13/3).

Section C: Short Answer Type (3 Marks)

Question 3: Prove that: (sin θ - 2 sin3θ) / (2 cos3θ - cos θ) = tan θ

Solution:

1. Take the Left Hand Side (LHS):
   LHS = (sin θ - 2 sin3θ) / (2 cos3θ - cos θ)
2. Factor out sin θ from the numerator and cos θ from the denominator:
   LHS = [sin θ (1 - 2 sin2θ)] / [cos θ (2 cos2θ - 1)]
3. Substitute the standard Pythagorean identity sin2θ = 1 - cos2θ into the numerator:
   1 - 2 sin2θ = 1 - 2(1 - cos2θ) = 1 - 2 + 2 cos2θ = 2 cos2θ - 1
4. Substitute this back into the expression:
   LHS = [sin θ (2 cos2θ - 1)] / [cos θ (2 cos2θ - 1)]
5. Cancel the common factor (2 cos2θ - 1):
   LHS = sin θ / cos θ = tan θ = RHS

Hence Proved.

Section D: Long Answer Type (5 Marks)

Question 4: A motor boat 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.

Solution:

1. Define variables and parameters:
   Let the speed of the stream be x km/h.
   Given speed of boat in still water = 18 km/h.
   Distance covered each way (d) = 24 km.
2. Determine effective speeds:
   Speed upstream = (18 - x) km/h (where x < 18)
   Speed downstream = (18 + x) km/h
3. Formulate the time equation:
   Time taken upstream t1 = 24 / (18 - x)
   Time taken downstream t2 = 24 / (18 + x)
   According to the problem statement: t1 - t2 = 1
[24 / (18 - x)] - [24 / (18 + x)] = 1
4. Simplify the rational equation:
   24 [ (18 + x) - (18 - x) ] / [ (18 - x)(18 + x) ] = 1
24 [ 18 + x - 18 + x ] / (324 - x2) = 1
24(2x) = 324 - x2
48x = 324 - x2
x2 + 48x - 324 = 0
5. Solve the quadratic equation by factorization:
   Find two numbers whose product is -324 and sum is 48 (54 and -6):
   x2 + 54x - 6x - 324 = 0
x(x + 54) - 6(x + 54) = 0
(x + 54)(x - 6) = 0
6. This gives x = 6 or x = -54.
7. Since the speed of a stream cannot be negative, we discard x = -54.

Final Answer: The speed of the stream is 6 km/h.

Section E: Case-Study Based Question (4 Marks)

Question 5 (Real-World Application of Arithmetic Progressions):

A manufacturing plant produces 600 electric scooters in the third year and 700 scooters in the seventh year. Assuming that the production increases uniformly by a constant number every year:

1. Find the production in the 1st year. (1 Mark)
2. Find the production in the 10th year. (1 Mark)
3. Find the total production in the first 7 years. (2 Marks)

Solution:

1. Sub-part 1: Production in the 1st year (a)
   Let the first term be a and common difference be d.
   Given a3 = a + 2d = 600   — (Equation 1)
   Given a7 = a + 6d = 700   — (Equation 2)
   Subtracting (1) from (2): 4d = 100 ⇒ d = 25.
   Substitute d = 25 into (1): a + 2(25) = 600 ⇒ a = 600 - 50 = 550.
   Production in 1st year = 550 scooters.
2. Sub-part 2: Production in the 10th year (a₁₀)
   a10 = a + 9d = 550 + 9(25) = 550 + 225 = 775.
   Production in 10th year = 775 scooters.
3. Sub-part 3: Total production in the first 7 years (S₇)
   Using the sum formula Sn = (n / 2)[a + an]:
   S7 = (7 / 2)[a + a7] = (7 / 2)[550 + 700] = (7 / 2)[1250] = 7 × 625 = 4375.
   Total production in first 7 years = 4,375 scooters.

How to Prepare for the CBSE Class 10 Mathematics Board Exam

Scoring top marks in CBSE Mathematics is not just about knowing formulas; it requires a systematic execution plan tailored to CBSE's marking scheme:

1. Master Step-Wise Solution Presentation

CBSE evaluators use detailed rubrics where every step carries dedicated weightage. Always state formulas explicitly before substituting values. For geometric problems, draw neat, labeled figures with pencil, write down the "Given", "To Prove", and "Construction" headings, and state specific geometric axioms (such as SAS, AA criterion, or Tangent Theorem) in parentheses alongside each step.

2. Eliminate Calculation and Sign Errors

Over 60% of lost marks in Class 10 Maths stem from minor arithmetic errors in Quadratic Equations, Surface Area computations, and Statistics tables. Verify your answers using reverse calculations whenever possible (e.g., substituting the obtained value of x back into the original quadratic equation or checking that probabilities sum to 1).

3. Strategic 3-Hour Time Allocation

Divide your 180 minutes of examination time systematically:

• Reading Time (First 15 minutes): Identify your choice questions in Sections B, C, and D. Highlight key numerical values and units.
• Section A (20 MCQs): 30 minutes (1.5 minutes per question).
• Section B & C (11 Questions): 50 minutes total.
• Section D & E (7 Detailed & Case Questions): 75 minutes total.
• Final Revision: 15 minutes to double-check calculation steps, missing units (cm², m³, km/h), and graph coordinates.

Where to Practice More

Consistent, timed practice with authentic test papers is the single most effective way to build exam stamina and reduce anxiety. Access official-pattern sample tests, full-length mock examinations, and chapter-wise question banks with step-by-step marking keys at QPTool by Theorify (qptool.theorify.in). Generate customized practice papers tailored to both Basic and Standard mathematics to sharpen your concepts and maximize your 2025 and 2026 board exam scores.