Algebra, Matrices and Trigonometry

1. Chapter at a Glance

Algebra and Trigonometry together give roughly 45–55 of the 120 questions in NDA Paper I, and the same ideas appear in a simpler form in CDS Elementary Mathematics and AFCAT Numerical Ability. The level is Class XI–XII (NCERT). Most questions are direct applications of a standard result, so the winning strategy is to know every formula cold and recognise the pattern in 10–15 seconds. This chapter covers sets and relations, complex numbers, quadratic equations, sequences and series, permutations and combinations, the binomial theorem, logarithms, matrices and determinants, and trigonometry including heights and distances.

2. Sets, Relations and Functions

A set is a well-defined collection of objects. Key results:

• A set with n elements has 2ⁿ subsets and 2ⁿ − 1 proper subsets.
• n(A ∪ B) = n(A) + n(B) − n(A ∩ B); for three sets, n(A ∪ B ∪ C) = n(A) + n(B) + n(C) − n(A∩B) − n(B∩C) − n(C∩A) + n(A∩B∩C).
• De Morgan's laws: (A ∪ B)′ = A′ ∩ B′ and (A ∩ B)′ = A′ ∪ B′.
• If n(A) = m and n(B) = n, the number of relations from A to B is 2^(mn); the number of functions from A to B is nᵐ.
• A relation is an equivalence relation if it is reflexive, symmetric and transitive.
• A function is one-one (injective) if different inputs give different outputs, onto (surjective) if every element of the codomain is an image, and bijective if both. Only bijections have inverses.

3. Complex Numbers

A complex number is z = a + ib, where i² = −1. Its modulus is |z| = √(a² + b²) and its conjugate is z̄ = a − ib. In polar form, z = r(cos θ + i sin θ), where r = |z| and θ = arg z.

• Powers of i repeat in a cycle of 4: i, −1, −i, 1. So iⁿ depends only on the remainder when n is divided by 4. Example: i¹⁰³ = i³ = −i.
• z·z̄ = |z|²; |z₁z₂| = |z₁||z₂|; arg(z₁z₂) = arg z₁ + arg z₂.
• Cube roots of unity: 1, ω, ω², where ω = (−1 + i√3)/2. Key facts: 1 + ω + ω² = 0 and ω³ = 1.
• De Moivre's theorem: (cos θ + i sin θ)ⁿ = cos nθ + i sin nθ.
• |z − z₀| = r represents a circle with centre z₀ and radius r.

4. Quadratic Equations

For ax² + bx + c = 0 (a ≠ 0), the roots are x = [−b ± √(b² − 4ac)]/2a. The discriminant D = b² − 4ac decides the nature of roots:

• Sum of roots α + β = −b/a; product αβ = c/a.
• The equation with roots α, β is x² − (α + β)x + αβ = 0.
• α² + β² = (α + β)² − 2αβ; (α − β)² = (α + β)² − 4αβ; α³ + β³ = (α + β)³ − 3αβ(α + β).
• If one root is the reciprocal of the other, then c = a. If roots are equal in magnitude and opposite in sign, then b = 0.
• For a cubic ax³ + bx² + cx + d = 0: sum of roots = −b/a, sum of products taken two at a time = c/a, product = −d/a.

5. Sequences and Series

• For two positive numbers, AM ≥ GM ≥ HM, and GM² = AM × HM.
• Σn = n(n + 1)/2; Σn² = n(n + 1)(2n + 1)/6; Σn³ = [n(n + 1)/2]².
• If three numbers are in AP, take them as a − d, a, a + d; in GP, as a/r, a, ar. This makes the algebra short.

6. Permutations, Combinations and Binomial Theorem

Fundamental principle: if one job can be done in m ways and another in n ways, both together can be done in m × n ways (and either one in m + n ways when they cannot happen together).

• ⁿPᵣ = n!/(n − r)! (arrangements, order matters); ⁿCᵣ = n!/[r!(n − r)!] (selections, order does not matter).
• ⁿCᵣ = ⁿCₙ₋ᵣ; ⁿCᵣ + ⁿCᵣ₋₁ = ⁿ⁺¹Cᵣ; sum of all ⁿCᵣ = 2ⁿ.
• Arrangements of n things with p alike of one kind and q alike of another = n!/(p! q!). Example: letters of "ARMY" = 4! = 24; "NAVY" = 4! = 24; "INDIA" = 5!/2! = 60.
• Circular arrangements of n distinct objects = (n − 1)!; for a necklace or garland (clockwise = anticlockwise) = (n − 1)!/2.
• Number of diagonals of an n-sided polygon = ⁿC₂ − n = n(n − 3)/2.

Binomial theorem: (x + y)ⁿ = Σ ⁿCᵣ xⁿ⁻ʳ yʳ, r = 0 to n. The general term is T(r+1) = ⁿCᵣ xⁿ⁻ʳ yʳ. There are n + 1 terms. If n is even, the middle term is T(n/2 + 1); if n is odd, there are two middle terms.

7. Logarithms and Exponents

• log(mn) = log m + log n; log(m/n) = log m − log n; log mⁿ = n log m.
• Change of base: logₐ b = log b / log a; logₐ b × log_b a = 1; logₐ 1 = 0; logₐ a = 1.
• a^(logₐ x) = x. log₁₀ 2 ≈ 0.3010, log₁₀ 3 ≈ 0.4771 (useful to count digits: number of digits in 2³⁰ = ⌊30 × 0.3010⌋ + 1 = 10).

8. Matrices and Determinants

A matrix of order m × n has m rows and n columns. Two matrices can be multiplied only if the number of columns of the first equals the number of rows of the second; matrix multiplication is not commutative in general (AB ≠ BA).

• Transpose: (AB)ᵀ = BᵀAᵀ. A is symmetric if Aᵀ = A, skew-symmetric if Aᵀ = −A (diagonal elements of a skew-symmetric matrix are zero).
• For a 2 × 2 matrix [[a, b], [c, d]], determinant = ad − bc and inverse = (1/(ad − bc)) [[d, −b], [−c, a]].
• A⁻¹ exists only if |A| ≠ 0 (non-singular). A⁻¹ = adj A / |A|.
• For a square matrix of order n: |kA| = kⁿ|A|; |adj A| = |A|ⁿ⁻¹; |AB| = |A||B|; |Aᵀ| = |A|; |A⁻¹| = 1/|A|.
• Properties of determinants: interchanging two rows changes the sign; two identical rows make the determinant zero; adding a multiple of one row to another does not change it.
• Area of a triangle with vertices (x₁, y₁), (x₂, y₂), (x₃, y₃) = ½ |x₁(y₂ − y₃) + x₂(y₃ − y₁) + x₃(y₁ − y₂)|. Area zero ⇒ points are collinear.
• Cramer's rule (system AX = B): if |A| ≠ 0, a unique solution exists; if |A| = 0 and (adj A)B ≠ 0, no solution; if |A| = 0 and (adj A)B = 0, infinitely many or none.

9. Trigonometry

Angles are measured in degrees or radians: π radians = 180°, so 1 radian ≈ 57.3°. Length of an arc s = rθ (θ in radians).

• Identities: sin²θ + cos²θ = 1; 1 + tan²θ = sec²θ; 1 + cot²θ = cosec²θ.
• ASTC rule ("All Silver Tea Cups"): in quadrant I all ratios are positive, in II only sin (and cosec), in III only tan (and cot), in IV only cos (and sec).
• Compound angles: sin(A ± B) = sin A cos B ± cos A sin B; cos(A ± B) = cos A cos B ∓ sin A sin B; tan(A + B) = (tan A + tan B)/(1 − tan A tan B).
• Double angles: sin 2A = 2 sin A cos A; cos 2A = cos²A − sin²A = 1 − 2 sin²A = 2 cos²A − 1; tan 2A = 2 tan A/(1 − tan²A).
• Triple angles: sin 3A = 3 sin A − 4 sin³A; cos 3A = 4 cos³A − 3 cos A.
• Maximum value of a sin θ + b cos θ is √(a² + b²) and minimum is −√(a² + b²).
• In any triangle: sine rule a/sin A = b/sin B = c/sin C = 2R; cosine rule a² = b² + c² − 2bc cos A.
• Principal values of inverse functions: sin⁻¹x ∈ [−π/2, π/2], cos⁻¹x ∈ [0, π], tan⁻¹x ∈ (−π/2, π/2); sin⁻¹x + cos⁻¹x = π/2.

10. Trigonometric Equations and Multiple-Angle Identities

An identity is true for every angle, but an equation is true only for some angles. NDA papers often ask for all solutions of an equation, or for the number of solutions in a given interval. Because trigonometric functions repeat, an equation usually has infinitely many solutions, written as a general solution with an integer n.

Worked example 1. Solve 2 sin²θ − 3 sin θ + 1 = 0 for 0 ≤ θ ≤ 2π. Treat it as a quadratic in sin θ: (2 sin θ − 1)(sin θ − 1) = 0, so sin θ = 1/2 or sin θ = 1. This gives θ = π/6, 5π/6 and π/2 — three solutions in the interval.

Worked example 2. Find the value of cos 20° cos 40° cos 80°. Multiply and divide by 2 sin 20° and use 2 sin x cos x = sin 2x three times: the product becomes sin 160°/(8 sin 20°) = sin 20°/(8 sin 20°) = 1/8. The pattern cos x cos 2x cos 4x = sin 8x/(8 sin x) appears often in NDA papers.

Worked example 3. Solve sin θ + cos θ = 1 for 0 ≤ θ < 2π. Write the left side as √2 sin(θ + π/4) = 1, so sin(θ + π/4) = 1/√2. Then θ + π/4 = π/4 or 3π/4 (within the shifted range), which gives θ = 0 or θ = π/2. Squaring both sides instead gives 2 sin θ cos θ = 0, whose solutions 0, π/2, π and 3π/2 must be checked: only 0 and π/2 satisfy the original equation, which shows why substitution back is essential.

Worked example 4. If tan A = 1/2 and tan B = 1/3, find A + B for acute angles. tan(A + B) = (1/2 + 1/3)/(1 − 1/6) = (5/6)/(5/6) = 1, so A + B = π/4. This result is a favourite in NDA papers.

11. Master Formula Table

12. Master Summary Table

13. Mind Map

14. Tips and Tricks

15. Important One-Liners

• The empty set is a subset of every set; a set with n elements has 2ⁿ subsets.
• i² = −1, i³ = −i, i⁴ = 1; the sum of four consecutive powers of i is 0.
• The modulus of a complex number is never negative; the argument of a positive real number is 0.
• A quadratic with real coefficients cannot have exactly one non-real root.
• If a + b + c = 0 in ax² + bx + c = 0, then 1 is a root.
• The AM of two positive numbers is always at least their GM.
• 0! = 1 by definition; ⁿC₀ = ⁿCₙ = 1.
• The number of terms in (a + b)ⁿ is n + 1.
• log 1 = 0 in every base; logarithm of a negative number is not defined (in reals).
• A square matrix with determinant zero is called singular and has no inverse.
• The determinant of an identity matrix is 1; of a skew-symmetric matrix of odd order is 0.
• Matrix multiplication is associative but not commutative.
• 1 radian ≈ 57.3°; π radians = 180°.
• sin θ and cos θ always lie between −1 and 1; tan θ can take any real value.
• tan 45° = 1, so at 45° elevation the height equals the horizontal distance.
• sin(90° − θ) = cos θ and tan(90° − θ) = cot θ.
• The period of sin x and cos x is 2π; of tan x is π.
• In a triangle, the largest side is opposite the largest angle.

16. PYQ-Based Questions with Detailed Answers

17. Quick Revision Summary