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Mathematics NCERT & Official PYQ Mapped

Important Math Formulas Handbook

A curated handbook of fundamental mathematical formulas essential for school examinations (Class 9-12) and competitive exams (SSC CGL, CHSL, GD, Railways, Banking, State PCS). Review key identities, equations, and solved shortcut techniques below.

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1. Essential Algebraic Identities

Algebraic identities are standard equations true for all values of variables. They are extensively used in polynomial simplification and factorization.

Square of a Binomial (Sum)

(a + b)² = a² + 2ab + b²
Variables & Units: a, b = Real numbers or algebraic terms
Exam Application: Used in expanding expressions and completing the square.

Square of a Binomial (Difference)

(a - b)² = a² - 2ab + b²
Variables & Units: a, b = Real numbers or algebraic terms
Exam Application: Useful shortcut: (a + b)² - (a - b)² = 4ab

Difference of Two Squares

a² - b² = (a - b)(a + b)
Variables & Units: a, b = Any numbers or expressions
Exam Application: Key factorization formula used frequently in competitive exams.

Cube of a Binomial

(a + b)³ = a³ + 3a²b + 3ab² + b³ = a³ + b³ + 3ab(a + b)
Variables & Units: a, b = Real numbers
Exam Application: Alternative form: a³ + b³ = (a + b)(a² - ab + b²)

Special Three-Variable Identity

a³ + b³ + c³ - 3abc = (a + b + c)(a² + b² + c² - ab - bc - ca)
Variables & Units: a, b, c = Real numbers
Exam Application: Critical Exam Rule: If a + b + c = 0, then a³ + b³ + c³ = 3abc.

Quadratic Equation Roots

x = [-b ± √(b² - 4ac)] / (2a)
Variables & Units: a, b, c = Coefficients of ax² + bx + c = 0 (a ≠ 0)
Exam Application: Discriminant D = b² - 4ac. Real & distinct if D > 0; real & equal if D = 0; imaginary if D < 0.
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2. Trigonometric Ratios & Identities

Trigonometry relationships for right-angled triangles and fundamental Pythagorean trigonometric identities.

Pythagorean Trigonometric Identities

sin²θ + cos²θ = 11 + tan²θ = sec²θ1 + cot²θ = cosec²θ
Variables & Units: θ = Angle in degrees or radians
Exam Application: Core identities used in all height-and-distance and trigonometric simplification problems.

Reciprocal & Quotient Ratios

tan θ = sin θ / cos θcot θ = cos θ / sin θsec θ = 1 / cos θcosec θ = 1 / sin θ
Variables & Units: θ = Any valid angle where denominator ≠ 0
Exam Application: Values at 0°, 30°, 45°, 60°, 90° must be thoroughly memorized.

Compound Angle Formulas

sin(A ± B) = sin A cos B ± cos A sin Bcos(A ± B) = cos A cos B ∓ sin A sin B
Variables & Units: A, B = Angles
Exam Application: Notice the sign reversal in the cosine formula.

3. Mensuration 2D & 3D Formulas

Perimeter, area, surface area, and volume formulas for geometric shapes.

Circle (Area & Circumference)

Area = πr²Circumference = 2πr
Variables & Units: r = Radius of circle (π ≈ 22/7 or 3.1416)
Exam Application: Sector Area = (θ / 360°) × πr²

Cylinder (Right Circular)

Curved Surface Area = 2πrhTotal SA = 2πr(r + h)Volume = πr²h
Variables & Units: r = Base radius, h = Height
Exam Application: Commonly asked in SSC & State police exams for ratio calculations.

Cone (Right Circular)

Slant Height l = √(r² + h²)CSA = πrlVolume = (1/3)πr²h
Variables & Units: r = Radius, h = Height, l = Slant height
Exam Application: Total Surface Area = πr(l + r)

Sphere & Hemisphere

Sphere Volume = (4/3)πr³Sphere SA = 4πr²Hemisphere Volume = (2/3)πr³
Variables & Units: r = Radius of the sphere
Exam Application: Total Surface Area of solid hemisphere = 3πr²

4. Competitive Arithmetic Formulas

Shortcut formulas for Percentage, Profit & Loss, Simple & Compound Interest, and Speed.

Percentage Increase / Decrease

Percentage Change = [(Final Value - Initial Value) / Initial Value] × 100%
Variables & Units: Initial Value = Base value before change
Exam Application: Net change for successive x% and y% changes = [x + y + (xy / 100)]%

Profit & Loss Formulas

Profit% = (Profit / CP) × 100Loss% = (Loss / CP) × 100
Variables & Units: CP = Cost Price, SP = Selling Price
Exam Application: SP = CP × [(100 + Profit%) / 100] when sold at a profit.

Time, Speed & Distance

Distance = Speed × TimeAverage Speed = (2xy) / (x + y)
Variables & Units: x, y = Speeds for equal distance trips
Exam Application: To convert km/h to m/s, multiply by (5/18). To convert m/s to km/h, multiply by (18/5).
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