Class 10 Math Shortcuts: Board Exam Speed Guide (CBSE 2026)

Class 10 CBSE Math board exam is one of the most important examinations of a student's academic life — and one of the most manageable once you know which chapters carry the most marks and how to solve each type of question efficiently.

The CBSE Class 10 Maths paper carries 80 marks in written exam plus 20 marks for internal assessment. It contains MCQs (20 marks), Very Short Answers (10 marks), Short Answers (18 marks), Long Answers (20 marks), and Case Study questions (12–15 marks). With the right shortcuts and chapter-wise strategy, scoring 90+ is entirely achievable for any student who practices consistently.

This guide covers all 7 units of the CBSE Class 10 Maths syllabus with shortcuts, tricks, and worked examples for every major chapter — designed for both last-minute revision and structured preparation.

Part 1: Unit-Wise Marks Distribution

CBSE Class 10 Maths 2026 — Weightage

Key takeaway: Algebra alone carries 21 marks — the single biggest unit. Algebra + Geometry + Trigonometry together = 48 marks = 60% of the paper. Master these three units and 90+ is within reach.

Part 2: Number Systems — 5 Marks, Easy Pickup

Despite low weightage, Number Systems questions are usually direct and fast to solve. Never skip them — these are the easiest marks in the paper.

Key Shortcuts

HCF and LCM using Prime Factorization:

• HCF = Product of lowest powers of common prime factors
• LCM = Product of highest powers of all prime factors

Worked Example 1:
Find HCF and LCM of 72 and 120.

72 = 2³ × 3²
120 = 2³ × 3 × 5
HCF = 2³ × 3 = 24
LCM = 2³ × 3² × 5 = 360
Verify: HCF × LCM = 24 × 360 = 8640 = 72 × 120 ✅

Worked Example 2 — Irrational Proof:
Prove that √3 is irrational.

Assume √3 = p/q (lowest terms)
→ 3q² = p² → 3 divides p → p = 3k
→ 3q² = 9k² → q² = 3k² → 3 divides q
But p and q can't both be divisible by 3 — contradiction.
∴ √3 is irrational ✅

Key Formula:
For any two positive integers a and b:
HCF(a, b) × LCM(a, b) = a × b

Part 3: Algebra — 21 Marks (Biggest Unit)

Algebra has 4 chapters — Polynomials, Linear Equations, Quadratic Equations, and Arithmetic Progressions. Together they dominate the paper.

Chapter 1 — Polynomials

Key Shortcuts:

For a quadratic ax² + bx + c:

• Sum of zeros (α + β) = −b/a
• Product of zeros (α × β) = c/a

Worked Example 3:
If α and β are zeros of 2x² − 5x + 3, find α + β and αβ.

α + β = 5/2
αβ = 3/2

Worked Example 4:
Find a polynomial whose zeros are 3 and −4.

Sum = −1, Product = −12
Polynomial = x² − (sum)x + (product) = x² + x − 12

Division Algorithm check: Dividend = Divisor × Quotient + Remainder

Chapter 2 — Linear Equations

Three methods — when to use which:

Worked Example 5:
Solve: 2x + 3y = 11, 2x − 4y = −24

Subtract: 7y = 35 → y = 5
Substituting: 2x + 15 = 11 → x = −2

Graphical shortcut for nature of solutions:

Chapter 3 — Quadratic Equations

Discriminant D = b² − 4ac — the master key:

Worked Example 6 — Factorization:
Solve: x² − 7x + 12 = 0

Find two numbers whose sum = −7 and product = 12 → −3 and −4
x² − 3x − 4x + 12 = 0
x(x − 3) − 4(x − 3) = 0
(x − 3)(x − 4) = 0
x = 3 or x = 4

Worked Example 7 — Quadratic Formula:
Solve: 2x² + x − 4 = 0

D = 1 + 32 = 33
x = (−1 ± √33) / 4
x = (−1 + √33)/4 or (−1 − √33)/4

Chapter 4 — Arithmetic Progressions (AP)

Two Core Formulas:

aₙ = a + (n − 1)d

Sₙ = n/2 × [2a + (n − 1)d] = n/2 × (a + l)

Where: a = first term, d = common difference, n = number of terms, l = last term

Worked Example 8:
Find the 15th term of AP: 3, 7, 11, 15...

a = 3, d = 4
a₁₅ = 3 + 14 × 4 = 3 + 56 = 59

Worked Example 9:
Find sum of first 20 terms of AP: 5, 8, 11...

S₂₀ = 20/2 × [2(5) + 19(3)] = 10 × [10 + 57] = 10 × 67 = 670

Shortcut — Find n when last term is given:
Last term of AP 3, 7, 11... is 79. Find n.

79 = 3 + (n − 1)4
76 = (n − 1) × 4
n − 1 = 19 → n = 20

Part 4: Coordinate Geometry — 6 Marks

Three formulas cover 90% of Coordinate Geometry questions.

Three Master Formulas

Distance Formula:
d = √[(x₂ − x₁)² + (y₂ − y₁)²]

Section Formula (internal division):
P = [(mx₂ + nx₁)/(m + n), (my₂ + ny₁)/(m + n)]

Midpoint Formula (m = n = 1):
Midpoint = [(x₁ + x₂)/2, (y₁ + y₂)/2]

Area of Triangle:
Area = ½ × |x₁(y₂ − y₃) + x₂(y₃ − y₁) + x₃(y₁ − y₂)|

Collinearity check: If Area = 0, points are collinear.

Worked Example 10:
Find midpoint of segment joining (3, −4) and (−7, 8).

Midpoint = [(3 − 7)/2, (−4 + 8)/2] = (−2, 2)

Worked Example 11:
Find area of triangle with vertices (1, 2), (4, 6), (−3, 2).

= ½ × |1(6 − 2) + 4(2 − 2) + (−3)(2 − 6)|
= ½ × |4 + 0 + 12|
= ½ × 16 = 8 sq. units

Part 5: Geometry — 15 Marks

Geometry has two chapters — Triangles and Circles. Both require theorem knowledge plus proof-writing skills.

Chapter 1 — Triangles

Must-Know Theorems:

1. Basic Proportionality Theorem (BPT): If a line is drawn parallel to one side of a triangle, it divides the other two sides in the same ratio.
2. Pythagoras Theorem: In a right-angled triangle, AC² = AB² + BC²
3. Converse of Pythagoras: If AC² = AB² + BC², then angle B = 90°

Similarity shortcuts:

Worked Example 12:
In triangle ABC, DE ∥ BC, AD = 3, DB = 5, AE = 4.5. Find EC.

By BPT: AD/DB = AE/EC
3/5 = 4.5/EC
EC = 4.5 × 5/3 = 7.5

Pythagoras Trick — Common Triples:
3-4-5, 5-12-13, 8-15-17, 7-24-25
Any multiple works: 6-8-10, 9-12-15, etc.

Chapter 2 — Circles

Tangent Properties:

1. Tangent ⊥ radius at point of contact
2. Tangents from external point are equal in length
3. Angle in a semicircle = 90°

Worked Example 13:
Two tangents PA and PB from external point P touch circle at A and B. PA = 5 cm. Find PB.

By equal tangent property: PB = 5 cm

Worked Example 14:
A tangent at point A makes angle 40° with chord AB. Find the inscribed angle in alternate segment.

By Tangent-Chord angle theorem: Inscribed angle = 40°

Part 6: Trigonometry — 12 Marks

Trigonometry is pure formula application — once you know the identities and standard values, marks are guaranteed.

Trigonometric Values Table

Memory trick for sin: √0, √1, √2, √3, √4 — all divided by 2 = 0, 1/2, 1/√2, √3/2, 1

Core Identities

sin²θ + cos²θ = 1

1 + tan²θ = sec²θ

1 + cot²θ = cosec²θ

Worked Example 15:
Prove: (sinθ + cosθ)² + (sinθ − cosθ)² = 2

LHS = sin²θ + 2sinθcosθ + cos²θ + sin²θ − 2sinθcosθ + cos²θ
= 1 + 1 = 2 = RHS ✅

Worked Example 16 — Heights and Distances:
A tower 30 m high casts a shadow 30√3 m. Find angle of elevation.

tanθ = 30 / 30√3 = 1/√3
θ = 30°

Height and Distance Formulas

• Angle of elevation: Object above horizontal → tanθ = height/distance
• Angle of depression: Object below horizontal → same tanθ applies (alternate angles)

Part 7: Mensuration — 10 Marks

Mensuration questions always involve combinations of 3D shapes — frustum, sphere + cylinder, cone + hemisphere, etc.

Complete Formula Reference

Slant height l = √(h² + r²)

Worked Example 17:
A solid is made of a cylinder (r = 5 cm, h = 10 cm) with hemisphere on top (r = 5 cm). Find volume.

Volume = πr²h + (2/3)πr³
= π × 25 × 10 + (2/3) × π × 125
= 250π + 250π/3
= π × (750 + 250)/3
= 1000π/3 ≈ 1047 cm³

Part 8: Statistics and Probability — 11 Marks

Statistics — Three Measures

Mean (Direct Method):
x̄ = Σfᵢxᵢ / Σfᵢ

Assumed Mean Method (when values are large):
x̄ = a + (Σfᵢdᵢ / Σfᵢ) — where dᵢ = xᵢ − a

Median:
Median = l + [(n/2 − cf) / f] × h

Mode:
Mode = l + [(f₁ − f₀) / (2f₁ − f₀ − f₂)] × h

Worked Example 18 — Median:
Class 20–30: l = 20, f = 25, cf = 20, h = 10, n = 50

Median = 20 + [(25 − 20)/25] × 10 = 20 + 2 = 22

Probability — Clean and Fast

P(E) = Favourable outcomes / Total outcomes

Key rules:

• P(E) + P(not E) = 1
• 0 ≤ P(E) ≤ 1
• P(certain event) = 1, P(impossible event) = 0

Worked Example 19:
A card is drawn from a deck of 52. Probability of face card?

Face cards = 12 (J, Q, K × 4 suits)
P = 12/52 = 3/13

Part 9: Answer-Writing Speed Strategy

Question-Wise Time Allocation

Board exams give 180 minutes for 80 marks — 67 minutes buffer for revision. Use it.

Board Exam Answer Tips

1. Show all steps — partial marks awarded for method even if final answer is wrong
2. Box your final answer — examiners spot it immediately
3. Write formulas first then substitute — gets method marks
4. Draw diagrams for Geometry and Trigonometry — always speeds up understanding
5. Attempt MCQs first — they are confidence builders and time efficient
6. Internal choice — always read both options before solving

30-Day Revision Plan