FSc Part 1 Maths formulas

Number systems

Real numbers

• Rational number: p/q with p, q integers and q ≠ 0; its decimal is terminating or recurring.
• Irrational number: non-terminating, non-recurring decimal, e.g. √2, √3, π, e.
• Additive identity is 0 and multiplicative identity is 1; the additive inverse of a is −a; the multiplicative inverse of a is 1/a (a ≠ 0).
• Commutative: a + b = b + a, ab = ba. Associative: (a + b) + c = a + (b + c), (ab)c = a(bc). Distributive: a(b + c) = ab + ac.
• Order: if a > b then a + c > b + c; if a > b and c > 0 then ac > bc; if a > b and c < 0 then ac < bc.
• Recurring decimal to p/q: let x be the decimal, multiply by powers of 10 so the recurring parts line up, then subtract.
• Closure: N is not closed under subtraction (3 − 5 is not in N); Z, Q and R are.

Complex numbers

• Powers of i repeat in fours: i¹ = i, i² = −1, i³ = −i, i⁴ = 1. Divide the power by 4 and use the remainder.
• (a + bi)(c + di) = (ac − bd) + (ad + bc)i
• Modulus: |z| = √(a² + b²); also z·z̄ = a² + b² = |z|².
• Multiplicative inverse of a + bi is (a − bi)/(a² + b²).
• |z₁z₂| = |z₁| × |z₂| and |z₁/z₂| = |z₁| ÷ |z₂|.
• Polar form: z = r(cos θ + i sin θ), where r = |z|, a = r cos θ and b = r sin θ. Choose θ from the quadrant in which the point (a, b) lies.
• If a + bi = c + di then a = c and b = d.

Sets, functions and groups

Sets and set operations

• Union A ∪ B: elements in A or in B (or both). Intersection A ∩ B: elements in both A and B.
• Difference A − B: elements of A that are not in B. Complement A′ = U − A.
• De Morgan's laws: (A ∪ B)′ = A′ ∩ B′ and (A ∩ B)′ = A′ ∪ B′.
• A set with n elements has 2ⁿ subsets, so n(P(A)) = 2ⁿ; it has 2ⁿ − 1 proper subsets.
• n(A ∪ B) = n(A) + n(B) − n(A ∩ B)
• If A ⊆ B then A ∩ B = A and A ∪ B = B.
• N = {1, 2, 3, ...}, W = {0, 1, 2, ...}, Z = the integers, Q = the rational numbers, R = the real numbers.

Logic, relations and functions

• Negation ∼p reverses the truth value. Conjunction p ∧ q is true only when both are true. Disjunction p ∨ q is false only when both are false.
• Conditional p → q is false only when p is true and q is false. Biconditional p ↔ q is true when p and q have the same truth value.
• For p → q: converse is q → p, inverse is ∼p → ∼q, contrapositive is ∼q → ∼p.
• A conditional and its contrapositive are logically equivalent; so are the converse and the inverse.
• Onto (surjective): range = B. Into: range ≠ B. One-one (injective): different elements have different images. Bijective: one-one and onto.
• Only a bijective function has an inverse function. To find f⁻¹(x), write y = f(x), solve for x, then interchange x and y.
• Domain of a real function: leave out values that make a denominator zero or put a negative number under a square root.

Binary operations and groups

• Commutative: a * b = b * a. Associative: (a * b) * c = a * (b * c), for all elements of G.
• Identity e: a * e = e * a = a for every a in G. Inverse a′ of a: a * a′ = a′ * a = e.
• Groupoid: closed under *. Semi-group: closed and associative. Monoid: a semi-group with an identity.
• Group: closed, associative, has an identity, and every element has an inverse in the set.
• Abelian group: a group in which * is also commutative.
• (Z, +) is a group with identity 0; N and W under + are not, because inverses are missing.
• {1, −1, i, −i} is an abelian group under multiplication, with identity 1.

Matrices and determinants

Matrices and matrix algebra

• If A is m × n and B is n × p, then AB is m × p. Each entry is (a row of A) × (a column of B), multiplied term by term and added.
• Symmetric: A^t = A. Skew-symmetric: A^t = −A (every diagonal entry is 0).
• Hermitian: (Ā)^t = A. Skew-Hermitian: (Ā)^t = −A. Here Ā is the matrix of conjugates.
• Diagonal matrix: square, every entry off the main diagonal is 0. Scalar matrix: a diagonal matrix with all diagonal entries equal. Identity matrix I: diagonal entries all 1.
• For the 2 × 2 matrix with rows [a b] and [c d]: |A| = ad − bc and adj A has rows [d −b] and [−c a].
• A⁻¹ = (1/|A|) × adj A, provided |A| ≠ 0; then AA⁻¹ = A⁻¹A = I.
• (AB)^t = B^tA^t and (AB)⁻¹ = B⁻¹A⁻¹ (the order reverses).

Determinants

• For rows [a b] and [c d]: |A| = ad − bc.
• Cofactor: A_ij = (−1)^i+j × M_ij, where M_ij is the minor. The signs follow the pattern + − + / − + − / + − +.
• Expansion along row 1: |A| = a₁₁A₁₁ + a₁₂A₁₂ + a₁₃A₁₃.
• |A| = 0 if a row (or column) is all zeros, or if two rows (or columns) are identical or proportional.
• Interchanging two rows (or columns) changes the sign of the determinant. |A^t| = |A|.
• Multiplying one row by k multiplies the determinant by k, so for an n × n matrix |kA| = kⁿ|A|.
• Adding a multiple of one row to another row does not change the value. For a triangular matrix, |A| = product of the diagonal entries.

Systems of linear equations

• Matrix inverse method: X = A⁻¹B (A⁻¹ on the left of B).
• Cramer's rule: x = |A₁|/|A|, y = |A₂|/|A|, where A₁ is A with its first column replaced by B, A₂ is A with its second column replaced by B, and so on.
• Echelon form: zero rows at the bottom, and each leading entry lies to the right of the leading entry in the row above. Rank = number of non-zero rows.
• Consistent: rank of A = rank of the augmented matrix. Inconsistent: the two ranks are not equal.
• Consistent with rank = number of unknowns: unique solution. Consistent with rank less than the number of unknowns: infinitely many solutions.
• Homogeneous system AX = O always has the trivial solution (all unknowns zero).
• Homogeneous system with square A: |A| ≠ 0 gives only the trivial solution; |A| = 0 gives non-trivial solutions.

Quadratic equations

Solving quadratic and reducible equations

• Quadratic formula: x = (−b ± √(b² − 4ac))/(2a).
• Factorisation: write the equation as (x − p)(x − q) = 0; then x = p or x = q.
• Completing the square (a = 1): add (half the coefficient of x)² to both sides, e.g. x² + 6x becomes (x + 3)² after adding 9.
• Exponential type: a^2x = (a^x)², so put y = a^x; reject any value y ≤ 0.
• Reciprocal type: divide by x², put y = x + 1/x and use x² + 1/x² = y² − 2.
• After a substitution y = x², each positive value of y gives two values of x: x = ±√y.
• Radical type: isolate the root, square both sides, solve, then check each root in the original equation.

Roots of unity and polynomial theorems

• Cube roots of unity: 1, ω = (−1 + √3 i)/2 and ω² = (−1 − √3 i)/2.
• 1 + ω + ω² = 0 (sum of the cube roots) and ω³ = 1 (product of the cube roots is 1).
• For a power of ω, divide the power by 3 and keep the remainder: ω³ⁿ = 1, ω³ⁿ⁺¹ = ω, ω³ⁿ⁺² = ω².
• Useful forms: 1 + ω = −ω², 1 + ω² = −ω, ω + ω² = −1.
• Fourth roots of unity: 1, −1, i, −i. Their sum is 0 and their product is −1.
• Remainder theorem: when f(x) is divided by x − a, the remainder is f(a).
• Factor theorem: x − a is a factor of f(x) if and only if f(a) = 0.

Roots, coefficients and nature of roots

• sum of roots = −(coefficient of x) ÷ (coefficient of x²): α + β = −b/a
• product of roots = constant term ÷ (coefficient of x²): αβ = c/a
• Equation with given roots: x² − (sum of roots)x + (product of roots) = 0
• α² + β² = (α + β)² − 2αβ and 1/α + 1/β = (α + β)/(αβ)
• b² − 4ac > 0: roots real and unequal. They are rational if b² − 4ac is a perfect square and irrational if it is not (a, b, c rational)
• b² − 4ac = 0: roots real (rational) and equal, each −b/(2a)
• b² − 4ac < 0: roots complex (imaginary) and unequal

Partial fractions

Resolving into partial fractions

• Proper: degree of numerator < degree of denominator. Improper: degree of numerator ≥ degree of denominator
• Improper fraction: divide first to get a polynomial + a proper fraction, then resolve the proper fraction
• Non-repeated linear factors: one constant for each factor, A/(x − a) + B/(x − b)
• Repeated linear factor (x − a)²: A/(x − a) + B/(x − a)², one term for each power
• Irreducible quadratic factor (x² + bx + c): the numerator is linear, (Ax + B)/(x² + bx + c)
• To find the constants: multiply both sides by the denominator, then put x equal to the zero of each linear factor or equate coefficients
• Identity: true for all values of x. Conditional equation: true only for particular values of x

Sequences and series

Arithmetic progressions

• common difference = any term − the term before it (d = a_n − a_n−1)
• nth term: a_n = a + (n − 1)d
• A.M. between a and b: A = (a + b)/2
• n arithmetic means between a and b: d = (b − a)/(n + 1), then the means are a + d, a + 2d, ...
• sum of n terms: S_n = (n/2)[2a + (n − 1)d] = (n/2)(a + a_n)
• 1 + 2 + 3 + ... + n = n(n + 1)/2

Geometric progressions

• common ratio = any term ÷ the term before it (r = a_n/a_n−1)
• nth term: a_n = arⁿ⁻¹
• G.M. between a and b: G² = ab, so G = ±√(ab)
• n geometric means between a and b: rⁿ⁺¹ = b/a, then the means are ar, ar², ...
• sum of n terms: S_n = a(1 − rⁿ)/(1 − r), which is the same as a(rⁿ − 1)/(r − 1), for r ≠ 1
• sum to infinity: S_∞ = a/(1 − r), only when |r| < 1

Harmonic progressions and means

• a₁, a₂, a₃, ... are in H.P. if 1/a₁, 1/a₂, 1/a₃, ... are in A.P.
• nth term of an H.P. = 1 ÷ (nth term of the matching A.P.) = 1/(a + (n − 1)d)
• H.M. between a and b: H = 2ab/(a + b)
• n harmonic means between a and b: find n arithmetic means between 1/a and 1/b, then take reciprocals
• A = (a + b)/2, G = ±√(ab) and H = 2ab/(a + b)
• G² = AH
• A > G > H for two different positive numbers, taking G = +√(ab) (A = G = H if a = b)

Permutation, combination and probability

Factorials and permutations

• n! = n(n − 1)(n − 2) ... 3 × 2 × 1, and 0! = 1
• n! = n × (n − 1)!, so for example 9!/7! = 9 × 8
• ⁿP_r = n!/(n − r)! = n(n − 1)(n − 2) ... (n − r + 1), a product of r factors
• ⁿP_n = n! (all n different objects arranged in a row)
• Objects not all different: n!/(n₁! × n₂! × ...), where n₁, n₂, ... are the numbers of alike objects
• Circular permutations of n different objects: (n − 1)!
• Necklace or key ring (can be turned over): (n − 1)!/2

Combinations

• ⁿC_r = n!/(r! × (n − r)!)
• ⁿC₀ = ⁿC_n = 1 and ⁿC₁ = n
• Complementary combinations: ⁿC_r = ⁿC_n−r, so if ⁿC_p = ⁿC_q with p ≠ q, then n = p + q
• ⁿC_r + ⁿC_r−1 = ⁿ⁺¹C_r
• ⁿP_r = r! × ⁿC_r
• Selections from two groups are multiplied, e.g. 2 from 5 and 3 from 6: ⁵C₂ × ⁶C₃
• n points, no three collinear: lines = ⁿC₂, triangles = ⁿC₃; diagonals of an n-sided polygon = ⁿC₂ − n

Probability

• P(E) = n(E)/n(S) = (number of favourable outcomes) ÷ (total number of equally likely outcomes)
• 0 ≤ P(E) ≤ 1, and P(not E) = 1 − P(E)
• Mutually exclusive events: P(A ∪ B) = P(A) + P(B)
• Events not mutually exclusive: P(A ∪ B) = P(A) + P(B) − P(A ∩ B)
• Independent events: P(A ∩ B) = P(A) × P(B)
• n(S): one die 6, two dice 36, two coins 4, a pack of cards 52

Mathematical induction and binomial theorem

Mathematical induction

• Condition 1: S(1) is true (substitute n = 1 in both sides)
• Condition 2: S(k) true implies S(k + 1) true, for any positive integer k
• S(k + 1) is found by replacing n with k + 1 everywhere in the statement
• Sum of a series: add the (k + 1)th term to both sides of S(k), then simplify the right side to the required form
• Divisibility: write the S(k) expression as a multiple, e.g. 5^k − 1 = 4m, then show the (k + 1) expression has the same factor
• Extended form: if S(i) is true and S(k) implies S(k + 1) for k ≥ i, then S(n) is true for all integers n ≥ i

Binomial theorem for positive integral index

• (a + x)ⁿ = ⁿC₀aⁿ + ⁿC₁aⁿ⁻¹x + ⁿC₂aⁿ⁻²x² + ... + ⁿC_nxⁿ
• number of terms = n + 1
• general term: T_r+1 = ⁿC_r a^n−r x^r, so the 4th term has r = 3
• n even: one middle term, T_(n/2)+1
• n odd: two middle terms, T_(n+1)/2 and T_(n+3)/2
• sum of all binomial coefficients: ⁿC₀ + ⁿC₁ + ... + ⁿC_n = 2ⁿ
• ⁿC₀ + ⁿC₂ + ⁿC₄ + ... = ⁿC₁ + ⁿC₃ + ⁿC₅ + ... = 2ⁿ⁻¹

Binomial series

• (1 + x)ⁿ = 1 + nx + [n(n − 1)/2!]x² + [n(n − 1)(n − 2)/3!]x³ + ...
• Valid only when |x| < 1. For (1 + kx)ⁿ the condition is |kx| < 1, so |x| < 1/|k|.
• General term: T_r+1 = [n(n − 1)(n − 2)...(n − r + 1)/r!]x^r
• (1 + x)⁻¹ = 1 − x + x² − x³ + ... and (1 − x)⁻¹ = 1 + x + x² + x³ + ...
• (1 + x)⁻² = 1 − 2x + 3x² − 4x³ + ... and (1 − x)⁻² = 1 + 2x + 3x² + 4x³ + ...
• If the first term is not 1, take it out first: (a + x)ⁿ = aⁿ(1 + x/a)ⁿ, valid when |x| < |a|.
• For small x, neglecting x² and higher powers: (1 + x)ⁿ ≈ 1 + nx

Fundamentals of trigonometry

Angles and their measurement

• π radians = 180°, so 1° = π/180 rad ≈ 0.0175 rad and 1 rad = 180°/π ≈ 57.296° = 57°17′45″
• 1° = 60′ (minutes) and 1′ = 60″ (seconds)
• Degrees to radians: multiply by π/180. Radians to degrees: multiply by 180/π.
• Arc length = radius × angle (l = rθ), θ in radians
• Area of a sector = ½ × radius² × angle (A = ½r²θ), θ in radians
• Coterminal angles differ by a multiple of 360° (2π): θ + 360°k, k an integer
• Quadrantal angles: 0°, 90°, 180°, 270°, 360° and their coterminal angles

Trigonometric ratios and fundamental identities

• sin θ = y/r, cos θ = x/r, tan θ = y/x; cosec θ = r/y, sec θ = r/x, cot θ = x/y
• Signs: quadrant I all positive; II only sin and cosec; III only tan and cot; IV only cos and sec
• sin 30° = 1/2, cos 30° = √3/2, tan 30° = 1/√3; sin 45° = cos 45° = 1/√2, tan 45° = 1; sin 60° = √3/2, cos 60° = 1/2, tan 60° = √3
• (cos θ, sin θ) is (1, 0) at 0°, (0, 1) at 90°, (−1, 0) at 180° and (0, −1) at 270°
• sin²θ + cos²θ = 1
• 1 + tan²θ = sec²θ
• 1 + cot²θ = cosec²θ

Trigonometric identities

Compound and allied angles

• cos(α − β) = cos α cos β + sin α sin β and cos(α + β) = cos α cos β − sin α sin β
• sin(α + β) = sin α cos β + cos α sin β and sin(α − β) = sin α cos β − cos α sin β
• tan(α + β) = (tan α + tan β)/(1 − tan α tan β) and tan(α − β) = (tan α − tan β)/(1 + tan α tan β)
• 180° ± θ and 360° ± θ: the function stays the same (sin stays sin, cos stays cos, tan stays tan)
• 90° ± θ and 270° ± θ: the function changes to its co-function (sin ↔ cos, tan ↔ cot, sec ↔ cosec)
• Sign: treat θ as acute, find the quadrant of the allied angle, and give the sign that the original function has there
• Examples: sin(90° + θ) = cos θ, cos(90° + θ) = −sin θ, cos(180° − θ) = −cos θ, tan(180° + θ) = tan θ

Double, half and triple angles; sums and products

• sin 2θ = 2 sin θ cos θ and tan 2θ = 2 tan θ/(1 − tan²θ)
• cos 2θ = cos²θ − sin²θ = 2 cos²θ − 1 = 1 − 2 sin²θ
• sin²(θ/2) = (1 − cos θ)/2 and cos²(θ/2) = (1 + cos θ)/2
• sin 3θ = 3 sin θ − 4 sin³θ and cos 3θ = 4 cos³θ − 3 cos θ
• sin P + sin Q = 2 sin((P + Q)/2) cos((P − Q)/2) and sin P − sin Q = 2 cos((P + Q)/2) sin((P − Q)/2)
• cos P + cos Q = 2 cos((P + Q)/2) cos((P − Q)/2) and cos P − cos Q = −2 sin((P + Q)/2) sin((P − Q)/2)
• 2 sin α cos β = sin(α + β) + sin(α − β); 2 cos α cos β = cos(α + β) + cos(α − β); 2 sin α sin β = cos(α − β) − cos(α + β)

Trigonometric functions and their graphs

Domain, range, period and graphs

• sin x and cos x: domain all real numbers, range −1 ≤ y ≤ 1
• tan x and sec x are undefined at odd multiples of π/2 (x = ±π/2, ±3π/2, ...); cot x and cosec x are undefined at multiples of π (x = 0, ±π, ±2π, ...)
• Range of tan x and cot x: all real numbers. Range of sec x and cosec x: y ≤ −1 or y ≥ 1
• Period of sin, cos, sec and cosec is 2π; period of tan and cot is π
• Period of sin kx or cos kx = 2π/k; period of tan kx = π/k (k > 0)
• y = a sin x and y = a cos x (a > 0) have maximum a and minimum −a; sin x = 0 at x = nπ, cos x = 0 at x = π/2 + nπ
• Even, f(−x) = f(x): cos x and sec x. Odd, f(−x) = −f(x): sin x, tan x, cot x and cosec x

Application of trigonometry

Solving triangles

• α + β + γ = 180°
• Law of cosines: a² = b² + c² − 2bc cos α, so cos α = (b² + c² − a²)/(2bc); similarly for b and c
• Law of sines: a/sin α = b/sin β = c/sin γ
• Law of tangents: (a − b)/(a + b) = tan((α − β)/2) ÷ tan((α + β)/2)
• With s = (a + b + c)/2: sin(α/2) = √((s − b)(s − c)/(bc)) and cos(α/2) = √(s(s − a)/(bc))
• tan(α/2) = √((s − b)(s − c)/(s(s − a)))
• Right-angled triangle: height = horizontal distance × tan(angle of elevation)

Area of a triangle and connected circles

• Δ = ½ab sin γ = ½bc sin α = ½ca sin β
• Semi-perimeter: s = (a + b + c)/2
• Hero's formula: Δ = √(s(s − a)(s − b)(s − c))
• Circum-radius: R = a/(2 sin α) = b/(2 sin β) = c/(2 sin γ) = abc/(4Δ)
• In-radius: r = Δ/s
• Escribed radii: r₁ = Δ/(s − a), r₂ = Δ/(s − b), r₃ = Δ/(s − c)
• In a right-angled triangle the circum-radius is half the hypotenuse

Inverse trigonometric functions

Inverse trigonometric functions

• y = sin⁻¹x: domain −1 ≤ x ≤ 1, range −π/2 ≤ y ≤ π/2
• y = cos⁻¹x: domain −1 ≤ x ≤ 1, range 0 ≤ y ≤ π
• y = tan⁻¹x: domain all real numbers, range −π/2 < y < π/2
• sin⁻¹(−x) = −sin⁻¹x, tan⁻¹(−x) = −tan⁻¹x, cos⁻¹(−x) = π − cos⁻¹x
• tan⁻¹A + tan⁻¹B = tan⁻¹((A + B)/(1 − AB)) and tan⁻¹A − tan⁻¹B = tan⁻¹((A − B)/(1 + AB))
• sin⁻¹A + sin⁻¹B = sin⁻¹(A√(1 − B²) + B√(1 − A²)) and cos⁻¹A + cos⁻¹B = cos⁻¹(AB − √((1 − A²)(1 − B²)))
• sin⁻¹x + cos⁻¹x = π/2 and 2 tan⁻¹A = tan⁻¹(2A/(1 − A²))

Solutions of trigonometric equations

Solving trigonometric equations

• sin x = 0 gives x = nπ; cos x = 0 gives x = π/2 + nπ; tan x = 0 gives x = nπ (n ∈ Z)
• sin x = k: if α is one solution, the other is π − α (add 2π if this is negative); general solution x = α + 2nπ or x = (π − α) + 2nπ
• cos x = k: if α is one solution, the other in [0, 2π] is 2π − α; general solution x = α + 2nπ or x = (2π − α) + 2nπ
• tan x = k: if α is one solution, the general solution is x = α + nπ
• sin x and cos x lie between −1 and 1, so an equation such as sin x = 2 has no solution
• A product equal to zero: put each factor equal to zero and solve each one
• Quadratic form: use sin²x + cos²x = 1 to get one function only, then factorise