NDA & NA Examination (Written Test)

Trigonometry Core

• Pythagoreansin²θ + cos²θ = 1, 1 + tan²θ = sec²θ, 1 + cot²θ = cosec²θ
• Double anglesin 2θ = 2 sinθ cosθ, cos 2θ = cos²θ − sin²θ = 1 − 2sin²θ
• Triple anglesin 3θ = 3sinθ − 4sin³θ, cos 3θ = 4cos³θ − 3cosθ
• Compoundtan(A + B) = (tan A + tan B)/(1 − tan A tan B)
• In a triangleA + B + C = π ⇒ tan A + tan B + tan C = tan A·tan B·tan C
• Sine / cosine rulea/sin A = b/sin B = c/sin C = 2R, cos A = (b² + c² − a²)/2bc

💡 Learn sin and cos of 15°, 18°, 36° and 75°: NDA uses them every paper.

Trigonometric Equations and Inverse Trig

• sin x = sin α ⇒ x = nπ + (−1)ⁿα
• cos x = cos α ⇒ x = 2nπ ± α
• tan x = tan α ⇒ x = nπ + α
• Rangessin⁻¹x ∈ [−π/2, π/2], cos⁻¹x ∈ [0, π], tan⁻¹x ∈ (−π/2, π/2)
• sin⁻¹x + cos⁻¹x = π/2, tan⁻¹x + cot⁻¹x = π/2

💡 For 'number of solutions in an interval' questions, list solutions period by period and check the end of the interval carefully.

Sequences, Quadratics, Logs

• APTₙ = a + (n − 1)d, Sₙ = n/2 [2a + (n − 1)d], Tₙ = Sₙ − Sₙ₋₁
• GPTₙ = arⁿ⁻¹, Sₙ = a(rⁿ − 1)/(r − 1), S∞ = a/(1 − r) for |r| < 1
• SumsΣn = n(n + 1)/2, Σn² = n(n + 1)(2n + 1)/6, Σn³ = [n(n + 1)/2]²
• Quadraticax² + bx + c = 0: α + β = −b/a, αβ = c/a, D = b² − 4ac
• Logslog(mn) = log m + log n, logₐb = log b / log a, logₐa = 1

💡 a, b, c in HP means 1/a, 1/b, 1/c are in AP: convert and use AP results.

Matrices and Determinants

• A⁻¹ = adj A / |A|, exists only if |A| ≠ 0
• |AB| = |A||B|, |kA| = kⁿ|A| for an n×n matrix
• |adj A| = |A|ⁿ⁻¹, (AB)⁻¹ = B⁻¹A⁻¹, (AB)ᵀ = BᵀAᵀ
• Skew-symmetricAᵀ = −A, diagonal entries 0; odd order ⇒ |A| = 0
• System AX = B|A| ≠ 0 ⇒ unique solution

💡 If AB = O and A is non-singular, multiply by A⁻¹: B must be O.

Coordinate Geometry

• Slopem = (y₂ − y₁)/(x₂ − x₁); parallel m₁ = m₂, perpendicular m₁m₂ = −1
• Distance from line|ax₁ + by₁ + c| / √(a² + b²)
• Circlex² + y² + 2gx + 2fy + c = 0: centre (−g, −f), r = √(g² + f² − c)
• Parabola y² = 4axfocus (a, 0), directrix x = −a, latus rectum 4a
• Ellipse x²/a² + y²/b² = 1 (a > b)e = √(1 − b²/a²), latus rectum 2b²/a
• Parallelogramdiagonals bisect each other: A + C = B + D (as coordinates)

💡 For 3D, direction cosines satisfy l² + m² + n² = 1; the angle between lines uses cos θ = l₁l₂ + m₁m₂ + n₁n₂.

Calculus Essentials

• Limitssin x / x → 1, (1 + x)^(1/x) → e, (aˣ − 1)/x → ln a as x → 0
• Derivativesd(xⁿ) = nxⁿ⁻¹, d(sin x) = cos x, d(ln x) = 1/x, d(eˣ) = eˣ
• Maxima / minimaf′(x) = 0 and f″(x) < 0 ⇒ local max; compare with endpoints on a closed interval
• King property∫₀ᵃ f(x) dx = ∫₀ᵃ f(a − x) dx
• Odd / even∫₋ₐᵃ f = 0 if f odd, = 2∫₀ᵃ f if f even
• Areaarea between y = f(x) and the x-axis = ∫ |f(x)| dx

💡 Order of a DE is the highest derivative; degree is that derivative's power only after clearing radicals and fractions.

Vectors

• a·b = |a||b| cos θ, |a × b| = |a||b| sin θ
• Projection of a on b = a·b / |b|
• Area of triangle = ½ |AB × AC|, of parallelogram = |a × b|
• Perpendicular ⇔ a·b = 0, parallel ⇔ a × b = 0
• Regular hexagon ABCDEFAD = 2BC, CF = 2BA, BE = 2AF (as vectors)

💡 Expressions like p sin θ + q cos θ have maximum √(p² + q²): use it for 'maximum of |a×b| + k|a·b|' questions.

Probability and Statistics

• P(A ∪ B) = P(A) + P(B) − P(A ∩ B)
• IndependentP(A ∩ B) = P(A)·P(B)
• BinomialP(X = r) = ⁿCᵣ pʳ qⁿ⁻ʳ, mean np, variance npq
• Varianceσ² = Σx²/n − (x̄)²; first n natural numbers: (n² − 1)/12
• EmpiricalMode ≈ 3 Median − 2 Mean
• Regressionr² = b_yx × b_xy, both regression coefficients have the same sign as r

💡 Adding a constant to all data changes the mean but not the variance; multiplying by k multiplies variance by k².