NDA mathematics revision sheet

The NDA maths paper is long. In recent papers it has had 120 questions for 300 marks in two and a half hours, with a third of a question's marks deducted for a wrong answer and no calculator allowed. That works out to about 75 seconds a question. In that time, you cannot derive a formula; you can only recall it. This sheet collects the results that come up again and again, with the trap attached to each, so that the last three weeks go into recall and practice rather than rereading chapters.

Use it actively. Cover everything after the bold label in each line and say the result aloud before you look. Anything you miss goes on a separate "weak list", which you read every morning until the exam.

Algebra

• Sets and relations: n(A ∪ B) = n(A) + n(B) − n(A ∩ B). A set with n elements has 2ⁿ subsets. From a set of m elements to one of n elements there are 2ᵐⁿ relations and nᵐ functions.
• Binary numbers: convert decimal to binary by repeated division by 2, reading the remainders from the bottom up. 45 = 101101 in binary (32 + 8 + 4 + 1).
• Logarithms: log(mn) = log m + log n, log(mⁿ) = n log m, and logₐ b = (log b)/(log a).
• Complex numbers: i² = −1 and i⁴ = 1; |a + bi| = √(a² + b²). The cube roots of unity satisfy 1 + ω + ω² = 0 and ω³ = 1.
• Quadratics ax² + bx + c = 0: sum of roots −b/a, product c/a, discriminant D = b² − 4ac. D > 0 gives real distinct roots, D = 0 equal roots, D < 0 complex roots.
• AP: nth term a + (n − 1)d; sum (n/2)[2a + (n − 1)d].
• GP: nth term arⁿ⁻¹; sum a(rⁿ − 1)/(r − 1); infinite sum a/(1 − r) for |r| < 1.
• Means: for positive numbers, AM ≥ GM ≥ HM. For two numbers, GM² = AM × HM.
• Sums: Σn = n(n + 1)/2, Σn² = n(n + 1)(2n + 1)/6, Σn³ = [n(n + 1)/2]².
• Permutations and combinations: nPr = n!/(n − r)!; nCr = n!/[r!(n − r)!]; nCr + nC(r − 1) = (n + 1)Cr. Circular arrangements of n objects: (n − 1)!. Diagonals of an n-gon: n(n − 3)/2.
• Binomial theorem: general term T(r + 1) = nCr aⁿ⁻ʳ bʳ; there are n + 1 terms; the sum of the coefficients is found by putting every variable equal to 1 (2ⁿ for (1 + x)ⁿ).

Matrices and determinants

For square matrices of order n:

• det(AB) = det A × det B, and det(Aᵀ) = det A.
• det(kA) = kⁿ det A, not k det A.
• A⁻¹ = adj A/det A, which exists only when det A ≠ 0.
• |adj A| = |A|ⁿ⁻¹, and (AB)⁻¹ = B⁻¹A⁻¹, (AB)ᵀ = BᵀAᵀ.
• A determinant with two equal or proportional rows (or columns) is 0.
• A skew-symmetric matrix of odd order has determinant 0.
• For a system of equations, a non-zero determinant of coefficients means a unique solution (Cramer's rule).

Trigonometry

• Identities: sin²θ + cos²θ = 1; 1 + tan²θ = sec²θ; 1 + cot²θ = cosec²θ.
• Double and triple angles: sin 2A = 2 sin A cos A; cos 2A = cos²A − sin²A = 1 − 2sin²A = 2cos²A − 1; sin 3A = 3 sin A − 4 sin³A; cos 3A = 4 cos³A − 3 cos A.
• Range: a sin θ + b cos θ lies between −√(a² + b²) and √(a² + b²).
• Inverse trig: sin⁻¹x + cos⁻¹x = π/2; tan⁻¹x + tan⁻¹y = tan⁻¹[(x + y)/(1 − xy)] when xy < 1. Principal ranges: sin⁻¹ in [−π/2, π/2], cos⁻¹ in [0, π], tan⁻¹ in (−π/2, π/2).
• Triangles: a/sin A = b/sin B = c/sin C = 2R; a² = b² + c² − 2bc cos A; area = ½bc sin A = abc/(4R) = √[s(s − a)(s − b)(s − c)].

Coordinate geometry

• Lines: perpendicular when m₁m₂ = −1; the angle between them satisfies tan θ = |(m₁ − m₂)/(1 + m₁m₂)|. The distance of (x₁, y₁) from ax + by + c = 0 is |ax₁ + by₁ + c|/√(a² + b²); between parallel lines, |c₁ − c₂|/√(a² + b²).
• Circle x² + y² + 2gx + 2fy + c = 0: centre (−g, −f), radius √(g² + f² − c).
• Parabola y² = 4ax: focus (a, 0), directrix x = −a, latus rectum 4a.
• Ellipse x²/a² + y²/b² = 1 (a > b): b² = a²(1 − e²), foci (±ae, 0), latus rectum 2b²/a.
• Hyperbola x²/a² − y²/b² = 1: b² = a²(e² − 1). A rectangular hyperbola has e = √2.
• 3D: direction cosines satisfy l² + m² + n² = 1. The angle between two lines satisfies cos θ = |l₁l₂ + m₁m₂ + n₁n₂|. The distance of a point from the plane ax + by + cz + d = 0 is |ax₁ + by₁ + cz₁ + d|/√(a² + b² + c²).

Calculus

• Standard limits (as x → 0): sin x/x → 1; (eˣ − 1)/x → 1; (aˣ − 1)/x → ln a; (1 + x) raised to the power 1/x → e. And (xⁿ − aⁿ)/(x − a) → naⁿ⁻¹ as x → a.
• Continuity: |x| is continuous everywhere but not differentiable at 0.
• Derivatives: d/dx aˣ = aˣ ln a; d/dx ln x = 1/x; d/dx tan x = sec²x; d/dx sin⁻¹x = 1/√(1 − x²); d/dx tan⁻¹x = 1/(1 + x²).
• Maxima and minima: f′ = 0 gives candidates; f″ < 0 is a maximum, f″ > 0 a minimum. On a closed interval, check the end points too.
• Integrals: ∫xⁿ dx = xⁿ⁺¹/(n + 1); ∫dx/(x² + a²) = (1/a) tan⁻¹(x/a); by parts ∫u dv = uv − ∫v du, choosing u by ILATE.
• Definite integrals: an odd function over [−a, a] gives 0; ∫[a, b] f(x) dx = ∫[a, b] f(a + b − x) dx. Areas: ellipse πab; y² = 4ax with its latus rectum 8a²/3.
• Differential equations: the order is the highest derivative; clear radicals before reading the degree. For dy/dx + Py = Q, the integrating factor is e raised to ∫P dx.

Vectors, statistics and probability

• Vectors: a · b = |a||b| cos θ; perpendicular when a · b = 0; projection of a on b = (a · b)/|b|; area of a parallelogram = |a × b|, of a triangle ½|a × b|.
• Statistics: adding c to every value shifts the mean by c and leaves the SD unchanged. Multiplying by k multiplies the mean by k, the SD by the size of k, and the variance by k². Mode ≈ 3 median − 2 mean. r² = byx × bxy.
• Probability: P(A ∪ B) = P(A) + P(B) − P(A ∩ B); for independent events, P(A ∩ B) = P(A)P(B); P(at least one) = 1 − P(none). Binomial: mean np, variance npq.

Worked mixed MCQs

Q1. The binary form of 45 is:
(a) 101011 (b) 101101 (c) 110101 (d) 100111

45 = 32 + 8 + 4 + 1, so the bits for 32, 16, 8, 4, 2, 1 are 1, 0, 1, 1, 0, 1. Answer: (b).

Q2. If ω is a complex cube root of unity, (1 + ω − ω²)(1 − ω + ω²) equals:
(a) 0 (b) 1 (c) 4 (d) −4

Since 1 + ω = −ω² and 1 + ω² = −ω, the factors are −2ω² and −2ω. Their product is 4ω³ = 4. Answer: (c).

Q3. A is a 3 × 3 matrix with det A = 5. Then |adj A| is:
(a) 5 (b) 15 (c) 25 (d) 125

|adj A| = |A|ⁿ⁻¹ = 5² = 25. Answer: (c).

Q4. The maximum value of 3 sin x + 4 cos x is:
(a) 7 (b) 5 (c) 4 (d) 1

√(3² + 4²) = 5. Option (a) simply adds the coefficients. Answer: (b).

Q5. The acute angle between the lines y = √3x and y = x/√3 is:
(a) 15° (b) 30° (c) 45° (d) 60°

The lines make angles of 60° and 30° with the x-axis, so the angle between them is 30°. Answer: (b).

Self-test (answers below)

1. The number of subsets of a set with 5 elements is: (a) 10 (b) 25 (c) 32 (d) 31
2. The sum of the coefficients in (2x − 1)⁵ is: (a) 1 (b) 32 (c) −1 (d) 243
3. 10C3 equals: (a) 720 (b) 120 (c) 30 (d) 210
4. The limit of sin 5x/x as x → 0 is: (a) 1 (b) 0 (c) 5 (d) 1/5
5. ∫[−π/2, π/2] sin x dx, the integral from −π/2 to π/2, is: (a) 2 (b) 1 (c) 0 (d) π
6. The eccentricity of x²/25 + y²/16 = 1 is: (a) 4/5 (b) 3/5 (c) 3/4 (d) 5/3
7. The distance of (1, 2, 2) from the origin is: (a) 3 (b) 5 (c) √5 (d) 9
8. log₂ 8 + log₃ 9 equals: (a) 6 (b) 5 (c) 4 (d) 17

Answers:

1. (c). 2⁵ = 32.
2. (a). Put x = 1: (2 − 1)⁵ = 1.
3. (b). (10 × 9 × 8)/(3 × 2 × 1) = 120.
4. (c). 5 × (sin 5x)/(5x) → 5.
5. (c). sin x is odd.
6. (b). e² = 1 − 16/25 = 9/25.
7. (a). √(1 + 4 + 4) = 3.
8. (b). 3 + 2.

A three-week revision plan

In every full paper, practise a two-pass method: answer what you know in the first pass, then return to the doubtful ones. A blind guess gains nothing on average, but once you can rule out even one option, a guess pays on average; the arithmetic is in NDA negative marking.

What to do next

• Cover and recall every line of this sheet once; start your weak list today.
• Rework the mixed MCQs and self-test without looking at the solutions.
• For the chapter order and time split behind this sheet, see the NDA maths plan, and for the final weeks as a whole, the last month before NDA.