CBSE Class 10 Mathematics: CBSE Class 10 Sample Paper 2025 Maths Standard Set 1 with Solutions

Excelling in CBSE Class 10 Mathematics demands structured practice, and the official CBSE Class 10 Sample Paper 2025 Maths Standard Set 1 with Solutions serves as the definitive benchmark for your board exam preparation. Designed strictly according to the latest 2024–25 and 2025–26 Central Board of Secondary Education guidelines, this paper reflects the exact difficulty gradient, cognitive weightage, and question typologies—from competency-based MCQs to multi-tiered case studies—that you will face in the Class 10 Mathematics Standard (Code 041) board examination.

CBSE Class 10 Maths Standard 2025 Exam Structure & Blueprint

The CBSE Class 10 Standard Mathematics paper tests conceptual clarity, algebraic manipulation, geometric reasoning, and real-world mathematical modelling. The paper carries 80 marks with a total duration of 3 hours (plus 15 minutes of initial reading time).

• Section A (20 Marks): 20 Multiple Choice Questions carrying 1 mark each (including 18 single-choice MCQs and 2 Assertion-Reason questions).
• Section B (10 Marks): 5 Very Short Answer (VSA) type questions carrying 2 marks each, requiring concise 3–4 line working.
• Section C (18 Marks): 6 Short Answer (SA) type questions carrying 3 marks each, requiring step-by-step proofs and algebraic solutions.
• Section D (20 Marks): 4 Long Answer (LA) type questions carrying 5 marks each, focusing on theorems, surface geometry, and complex word problems.
• Section E (12 Marks): 3 Case Study/Source-Based Integrated Assessment questions carrying 4 marks each, featuring sub-parts of 1, 1, and 2 marks.

Key Concepts & Core Formula Reference

Before attempting Set 1, ensure mastery over these core mathematical formulas and NCERT principles frequently assessed in CBSE Mathematics:

1. Real Numbers & Polynomials

• Fundamental Theorem of Arithmetic: Every composite number can be uniquely expressed as a product of primes:
  HCF(a, b) × LCM(a, b) = a × b
• Quadratic Polynomial: For p(x) = ax² + bx + c with zeroes α and β:
  Sum of zeroes: α + β = -b / a
  Product of zeroes: αβ = c / a
• Quadratic Formula: For ax² + bx + c = 0, x = (-b ± √(b² - 4ac)) / (2a). The discriminant D = b² - 4ac determines root nature (D > 0: two distinct real roots; D = 0: two equal real roots; D < 0: no real roots).

2. Coordinate Geometry & Trigonometry

• Distance Formula: d = √[(x&sub2; - x&sub1;)² + (y&sub2; - y&sub1;)²]
• Section Formula (Internal Division): P(x, y) = ((m&sub1;x&sub2; + m&sub2;x&sub1;)/(m&sub1; + m&sub2;), (m&sub1;y&sub2; + m&sub2;y&sub1;)/(m&sub1; + m&sub2;))
• Trigonometric Identities:
  • sin²θ + cos²θ = 1
  • 1 + tan²θ = sec²θ
  • 1 + cot²θ = cosec²θ

3. Mensuration & Statistics

• Surface Area and Volumes: Total surface area of a combined solid is the sum of exposed outer surface areas (never simply sum individual TSAs).
• Mode Formula: Mode = l + [ (f&sub1; - f&sub0;) / (2f&sub1; - f&sub0; - f&sub2;) ] × h
• Median Formula: Median = l + [ ((n/2) - cf) / f ] × h
• Empirical Relationship: 3 × Median = Mode + 2 × Mean

Important CBSE Class 10 Maths Standard Set 1 Questions with Step-by-Step Solutions

Section A: Objective & Analytical Questions (1 Mark Each)

Question 1 (MCQ - Polynomials)

Question: If one zero of the quadratic polynomial f(x) = 4x² - 8kx - 9 is negative of the other, find the value of k.

Solution:

1. Let the zeroes of the polynomial be α and -α.
2. Using the sum of zeroes formula: Sum of zeroes = -b / a
3. α + (-α) = -(-8k) / 4
4. 0 = 8k / 4 ⇒ 2k = 0 ⇒ k = 0

Answer: k = 0

Question 2 (Assertion-Reason - Coordinate Geometry)

Assertion (A): The point (0, 4) lies on the y-axis.
Reason (R): The x-coordinate of any point lying on the y-axis is always zero.

Solution: Any point on the y-axis has coordinates of the form (0, y). Since the x-coordinate of (0, 4) is 0, Assertion (A) is true, and Reason (R) is the correct mathematical explanation.

Answer: Both (A) and (R) are true, and (R) is the correct explanation of (A).

Section B: Very Short Answer Questions (2 Marks Each)

Question 3 (Real Numbers Proof)

Question: Prove that 5 + 2√3 is an irrational number, given that √3 is irrational.

Solution:

1. Let us assume to the contrary that 5 + 2√3 is a rational number.
2. Therefore, we can write 5 + 2√3 = a / b, where a and b are co-prime integers and b ≠ 0.
3. Rearranging the terms:
   2√3 = (a / b) - 5 = (a - 5b) / b
√3 = (a - 5b) / (2b)
4. Since a and b are integers, (a - 5b) / (2b) is a rational number.
5. This implies that √3 is rational. But this contradicts the given fact that √3 is irrational.
6. Hence, our assumption was false. 5 + 2√3 is irrational.

Section C: Short Answer Questions (3 Marks Each)

Question 4 (Trigonometric Identity Proof)

Question: Prove the following identity:
(sin θ - 2 sin³θ) / (2 cos³θ - cos θ) = tan θ

Solution:

1. Take the Left Hand Side (LHS):
   LHS = [sin θ (1 - 2 sin²θ)] / [cos θ (2 cos²θ - 1)]
2. Express sin²θ in terms of cos²θ using the identity sin²θ = 1 - cos²θ:
   Numerator term: 1 - 2(1 - cos²θ) = 1 - 2 + 2 cos²θ = 2 cos²θ - 1
3. Substitute back into LHS:
   LHS = [sin θ (2 cos²θ - 1)] / [cos θ (2 cos²θ - 1)]
4. Cancel the common factor (2 cos²θ - 1):
   LHS = sin θ / cos θ = tan θ = RHS
5. Hence Proved.

Section D: Long Answer Questions (5 Marks Each)

Question 5 (Applications of Quadratic Equations - Speed and Distance)

Question: 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. Let the speed of the stream be x km/h (where x < 18).
2. Speed of the boat upstream = (18 - x) km/h.
3. Speed of the boat downstream = (18 + x) km/h.
4. Distance = 24 km.
   Time taken upstream: T&sub1; = 24 / (18 - x)
   Time taken downstream: T&sub2; = 24 / (18 + x)
5. According to the problem: T&sub1; - T&sub2; = 1
[24 / (18 - x)] - [24 / (18 + x)] = 1
6. Simplify the equation:
   24 [ (18 + x) - (18 - x) ] / [(18 - x)(18 + x)] = 1
24 [ 2x ] / [ 324 - x² ] = 1
48x = 324 - x²
x² + 48x - 324 = 0
7. Factorize the quadratic equation:
   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 - 6)(x + 54) = 0
8. Therefore, x = 6 or x = -54.
9. Since speed cannot be negative, we reject x = -54.

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

Section E: Case-Study Based Real-World Application (4 Marks)

Question 6 (Case Study: Heights and Distances in Civil Surveying)

Context: A surveying team uses a theodolite stationed at point A on ground level to observe a telecommunication tower CD of height h meters erected on top of a water reservoir BC of height 20 meters. From point A, the angle of elevation of the bottom of the tower (point B) is 30°, and the angle of elevation of the top of the tower (point D) is 60°.

Sub-Questions & Solutions:

1. Find the horizontal distance AC from the observation point to the base of the reservoir. (1 Mark)
   In right-angled ΔACB:
   tan 30° = BC / AC
1 / √3 = 20 / AC ⇒ AC = 20√3 m ≈ 34.64 m
2. Find the total height AD from point A to the top of the tower. (1 Mark)
   In right-angled ΔACD:
   cos 60° = AC / AD ⇒ 1 / 2 = 20√3 / AD ⇒ AD = 40√3 m
3. Calculate the height of the tower CD. (2 Marks)
   In right-angled ΔACD:
   tan 60° = DC_total / AC = (BC + CD) / AC
√3 = (20 + CD) / (20√3)
20√3 × √3 = 20 + CD
60 = 20 + CD ⇒ CD = 40 m
   Answer: The height of the transmission tower is 40 meters.

How to Strategically Prepare for CBSE Class 10 Maths Standard 2025

Scoring 95+ in CBSE Mathematics in your board exam 10 requires combining conceptual accuracy with disciplined exam-hall time management:

• Section-Wise Time Budgeting: Allocate 30 minutes for Section A, 25 minutes for Section B, 40 minutes for Section C, 45 minutes for Section D, and 25 minutes for Section E. Reserve the remaining 15 minutes exclusively for double-checking arithmetic steps.
• Rigorous Presentation & Step Marking: CBSE marking schemes award individual marks for writing given data, stating geometric theorems, drawing neat labelled diagrams, and specifying final units (e.g., cm², km/h, rupees).
• Avoid Common Calculation Traps: Watch out for algebraic sign reversals during cross-multiplication in linear equations, verify cumulative frequencies in median tables, and always rationalize square roots in denominators.
• Solve Standard Set Papers Under Timed Conditions: Attempt at least 5 full-length standard sample sets under 3-hour exam conditions to build stamina and identify weak topics early.

Where to Practice More

To master the full spectrum of competency-based and case-study questions for the 2025 and 2026 board examinations, practice with topic-wise mock tests and curated question banks at Theorify QPTool (qptool.theorify.in). Download full-length CBSE sample papers, instant marking keys, and chapter-wise previous years' questions (PYQs) to secure a top score in your Class 10 board exams.