MSCE Additional Mathematics cheat sheet

Quadratic formula: for ax² + bx + c = 0, x = (−b ± √(b² − 4ac)) / 2a; the discriminant b² − 4ac gives two real roots if positive, one repeated root if zero, no real roots if negative.

Remainder Theorem: when polynomial P(x) is divided by (x − a), the remainder equals P(a); dividing by (ax − b) gives remainder P(b/a).

Factor Theorem: (x − a) is a factor of P(x) if and only if P(a) = 0; equate coefficients to find unknown constants in identities.

Laws of indices: a^m × a^n = a^(m+n), a^m ÷ a^n = a^(m−n), (a^m)^n = a^mn, a^0 = 1, a^(−n) = 1/a^n, a^(1/n) = ⁿ√a.

Laws of logarithms: log(ab) = log a + log b, log(a/b) = log a − log b, log(a^n) = n log a, log_a a = 1, log_a 1 = 0, log_a b = log c b / log c a.

Solving a^x = b: take logarithms of both sides to get x = log b / log a; convert y = a^x to x = log_a y.

Partial fractions: for distinct linear factors use A/(x + a) + B/(x + b); for a repeated factor include terms up to (x + a)²; for an irreducible quadratic include (Cx + D)/(ax² + bx + c).

Absolute value and inequality rules: |x| < k means −k < x < k; |x| > k means x < −k or x > k; solve quadratic inequalities by finding critical roots and testing sign intervals.

Simultaneous equations: one linear and one quadratic - substitute the linear expression into the quadratic, solve the resulting quadratic, then back-substitute for the paired values.

A function is one-to-one if each output corresponds to exactly one input; only one-to-one functions have inverses, and the graph of f⁻¹ is the reflection of f in the line y = x.

Domain of f⁻¹ equals range of f, and range of f⁻¹ equals domain of f; find f⁻¹ by writing y = f(x) and solving for x in terms of y.

Composite function: (fg)(x) means f(g(x)); the domain of fg is the set of x in the domain of g for which g(x) lies in the domain of f.

Transformations: y = f(x) + a shifts up by a; y = f(x + a) shifts left by a; y = af(x) stretches vertically by factor a; y = f(ax) stretches horizontally by factor 1/a.

Reflections: y = −f(x) reflects in the x-axis; y = f(−x) reflects in the y-axis; y = |f(x)| reflects any part below the x-axis upward.

A graph y = f(x) has an asymptote where the function tends to infinity or a finite limit as x tends to a value or ±∞; vertical asymptotes occur where the denominator is zero.

Even functions satisfy f(−x) = f(x) and are symmetric about the y-axis; odd functions satisfy f(−x) = −f(x) and have rotational symmetry about the origin.

Exponential graph y = a^x (a > 1) passes through (0, 1) and has the x-axis as a horizontal asymptote; y = log_a x is its reflection in y = x and passes through (1, 0).

Distance between (x₁, y₁) and (x₂, y₂): √((x₂ − x₁)² + (y₂ − y₁)²); midpoint is ((x₁ + x₂)/2, (y₁ + y₂)/2).

Gradient of a line through two points: m = (y₂ − y₁)/(x₂ − x₁); equation forms y = mx + c and y − y₁ = m(x − x₁).

Parallel lines have equal gradients; perpendicular lines satisfy m₁ × m₂ = −1.

A circle with centre (a, b) and radius r has equation (x − a)² + (y − b)² = r²; expanded form x² + y² + 2gx + 2fy + c = 0 has centre (−g, −f) and radius √(g² + f² − c).

The perpendicular from the centre of a circle to a chord bisects the chord; the tangent at a point is perpendicular to the radius at that point.

Area of a triangle with vertices (x₁, y₁), (x₂, y₂), (x₃, y₃): ½|x₁(y₂ − y₃) + x₂(y₃ − y₁) + x₃(y₁ − y₂)|.

A point dividing a segment in ratio m : n has coordinates ((mx₂ + nx₁)/(m + n), (my₂ + ny₁)/(m + n)).

To find intersections of a line and a curve, substitute the line equation into the curve equation and solve the resulting quadratic; the discriminant indicates 0, 1 (tangent) or 2 intersections.

Arithmetic progression: nth term a + (n − 1)d; sum of n terms Sₙ = n/2 [2a + (n − 1)d] = n/2 (a + l) where l is the last term.

Geometric progression: nth term ar^(n−1); sum of n terms Sₙ = a(1 − rⁿ)/(1 − r) for r ≠ 1; sum to infinity S∞ = a/(1 − r) valid only when |r| < 1.

For a GP with |r| ≥ 1 the series diverges and has no sum to infinity; for r = 1 the sum of n terms is na.

The sum of the first n natural numbers is n(n + 1)/2; the sum of the first n squares is n(n + 1)(2n + 1)/6.

Binomial expansion: (a + b)ⁿ = Σ from r = 0 to n of ⁿCᵣ a^(n−r) b^r; the general term is ⁿCᵣ a^(n−r) b^r.

ⁿCᵣ = n!/(r!(n − r)!); the term independent of x is found by setting the power of x in the general term to zero.

Sigma notation: Σ from r = 1 to n of (Ar + B) splits into A Σr + B Σ1; use the standard sums of r and r² for polynomial terms.

Convergence of an arithmetic series: it diverges unless d = 0; a geometric series converges only when |r| < 1.

Recurrence relations: a term like uₙ₊₁ = kuₙ + c defines a sequence; iterate to find terms, and solve a fixed point u = ku + c to find any limit.

Reciprocal identities: cosec θ = 1/sin θ, sec θ = 1/cos θ, cot θ = 1/tan θ; Pythagorean identities sin²θ + cos²θ = 1, 1 + tan²θ = sec²θ, 1 + cot²θ = cosec²θ.

Exact values: sin 30° = ½, cos 30° = √3/2, tan 30° = 1/√3; sin 45° = cos 45° = 1/√2; sin 60° = √3/2, cos 60° = ½, tan 60° = √3.

Sine rule: a/sin A = b/sin B = c/sin C; cosine rule: a² = b² + c² − 2bc cos A; area of triangle = ½ab sin C.

Identities: 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 angle: sin 2A = 2 sin A cos A; cos 2A = cos²A − sin²A = 2cos²A − 1 = 1 − 2sin²A; tan 2A = 2tan A/(1 − tan²A).

Radians: π radians = 180°; arc length s = rθ and sector area = ½r²θ for θ in radians.

General solutions: sin θ = k gives θ = α + 360°n or 180° − α + 360°n; cos θ = k gives θ = ±α + 360°n; tan θ = k gives θ = α + 180°n.

The equation a sin θ + b cos θ = c is solved by writing it as R sin(θ + α) with R = √(a² + b²) and tan α = b/a.

Graphs: y = sin x and y = cos x have period 360° and range [−1, 1]; y = tan x has period 180° and asymptotes at 90° + 180°n.

Derivative from first principles: f′(x) = limit as h → 0 of [f(x + h) − f(x)]/h; the derivative is the gradient of the tangent to the curve.

Power rule: d/dx (axⁿ) = anx^(n−1); the derivative of a constant is zero; d/dx (ln x) = 1/x and d/dx (eˣ) = eˣ.

Chain rule: dy/dx = dy/du × du/dx; product rule: d/dx (uv) = u dv/dx + v du/dx; quotient rule: d/dx (u/v) = (v du/dx − u dv/dx)/v².

Trig derivatives: d/dx (sin x) = cos x, d/dx (cos x) = −sin x, d/dx (tan x) = sec²x, with x in radians.

Stationary points occur where dy/dx = 0; use the second derivative to classify: d²y/dx² > 0 gives a minimum, < 0 gives a maximum, = 0 requires further investigation.

Tangents and normals: tangent gradient is dy/dx at the point; normal gradient is −1/(dy/dx); use y − y₁ = m(x − x₁).

Integration reverses differentiation: ∫ axⁿ dx = ax^(n+1)/(n + 1) + c for n ≠ −1; ∫ x⁻¹ dx = ln|x| + c; ∫ eˣ dx = eˣ + c.

Definite integrals: ∫ from a to b of f(x) dx = F(b) − F(a); the area between two curves is ∫ (upper − lower) dx between intersection points.

Volumes of revolution: rotating y = f(x) about the x-axis gives V = π ∫ y² dx; about the y-axis use V = π ∫ x² dy.

A vector has magnitude and direction; a scalar has magnitude only; vectors are equal if they have the same magnitude and direction.

Magnitude of a column vector (x, y): √(x² + y²); in three dimensions |(x, y, z)| = √(x² + y² + z²).

Unit vector in the direction of a is a/|a|; the position vector of a point relative to the origin is written using i, j, k components.

Vector addition is head-to-tail; subtraction a − b gives the vector from b to a; scalar multiplication k a changes magnitude by |k| and reverses direction if k < 0.

Scalar (dot) product: a · b = |a||b| cos θ = a₁b₁ + a₂b₂ + a₃b₃; perpendicular vectors have a · b = 0.

Angle between vectors: cos θ = (a · b)/(|a||b|); this is used to find angles in triangles and between lines.

A line has equation r = a + t b, where a is a point on the line and b is a direction vector; parallel lines have parallel direction vectors.

Velocity and displacement: if r = a + t v, the velocity is v and speed is |v|; the position at time t is found by substituting t into r.

A matrix of order m × n has m rows and n columns; addition and subtraction require the same order, and multiplication requires the inner dimensions to match.

Matrix multiplication is not commutative: AB ≠ BA in general; the identity matrix I satisfies AI = IA = A.

Determinant of a 2 × 2 matrix [[a, b], [c, d]] is ad − bc; the matrix is singular (no inverse) when the determinant is zero.

Inverse of a 2 × 2 matrix: (1/(ad − bc)) [[d, −b], [−c, a]]; the inverse satisfies AA⁻¹ = A⁻¹A = I.

Transformation matrices: reflection in the x-axis [[1, 0], [0, −1]], in the y-axis [[−1, 0], [0, 1]], in y = x [[0, 1], [1, 0]]; rotation by θ anticlockwise [[cos θ, −sin θ], [sin θ, cos θ]].

Enlargement with scale factor k has matrix [[k, 0], [0, k]]; a one-way stretch parallel to the x-axis has matrix [[k, 0], [0, 1]].

To find the matrix of a transformation, apply it to the unit vectors (1, 0) and (0, 1); the images form the columns of the matrix.

Combined transformations: if transformation A is followed by transformation B, the combined matrix is BA; order matters.

Solving simultaneous equations with matrices: write AX = B and solve X = A⁻¹B, provided A is non-singular.

Mean of a data set: x̄ = Σx/n; for grouped data use class midpoints, x̄ = Σfx/Σf; median is the middle value when data are ordered.

Variance and standard deviation: σ² = Σ(x − x̄)²/n = Σx²/n − x̄²; standard deviation is the positive square root of the variance.

For grouped data, the mean is Σfx/Σf and variance is Σfx²/Σf − x̄²; use midpoints of classes.

Probability: P(A ∪ B) = P(A) + P(B) − P(A ∩ B); for mutually exclusive events P(A ∩ B) = 0; for independent events P(A ∩ B) = P(A)P(B).

Conditional probability: P(A|B) = P(A ∩ B)/P(B); events are independent if P(A|B) = P(A).

Permutations: the number of arrangements of n distinct objects is n!; choosing r from n and ordering them is ⁿPᵣ = n!/(n − r)!.

Combinations: the number of ways of choosing r objects from n without regard to order is ⁿCᵣ = n!/(r!(n − r)!).

Binomial distribution: if X ~ B(n, p), then P(X = r) = ⁿCᵣ pʳ (1 − p)^(n−r); mean is np and variance is np(1 − p).

Tree diagrams: multiply probabilities along branches and add probabilities for mutually exclusive paths; the sum of all branch probabilities from one node is 1.

Quadratic formula: for ax² + bx + c = 0, x = (−b ± √(b² − 4ac)) / 2a; the discriminant b² − 4ac gives two real roots if positive, one repeated root if zero, no real roots if negative.

Remainder Theorem: when polynomial P(x) is divided by (x − a), the remainder equals P(a); dividing by (ax − b) gives remainder P(b/a).

Factor Theorem: (x − a) is a factor of P(x) if and only if P(a) = 0; equate coefficients to find unknown constants in identities.

Laws of indices: a^m × a^n = a^(m+n), a^m ÷ a^n = a^(m−n), (a^m)^n = a^mn, a^0 = 1, a^(−n) = 1/a^n, a^(1/n) = ⁿ√a.

Laws of logarithms: log(ab) = log a + log b, log(a/b) = log a − log b, log(a^n) = n log a, log_a a = 1, log_a 1 = 0, log_a b = log c b / log c a.

Solving a^x = b: take logarithms of both sides to get x = log b / log a; convert y = a^x to x = log_a y.

Partial fractions: for distinct linear factors use A/(x + a) + B/(x + b); for a repeated factor include terms up to (x + a)²; for an irreducible quadratic include (Cx + D)/(ax² + bx + c).

Absolute value and inequality rules: |x| < k means −k < x < k; |x| > k means x < −k or x > k; solve quadratic inequalities by finding critical roots and testing sign intervals.

Simultaneous equations: one linear and one quadratic - substitute the linear expression into the quadratic, solve the resulting quadratic, then back-substitute for the paired values.

A function is one-to-one if each output corresponds to exactly one input; only one-to-one functions have inverses, and the graph of f⁻¹ is the reflection of f in the line y = x.

Domain of f⁻¹ equals range of f, and range of f⁻¹ equals domain of f; find f⁻¹ by writing y = f(x) and solving for x in terms of y.

Composite function: (fg)(x) means f(g(x)); the domain of fg is the set of x in the domain of g for which g(x) lies in the domain of f.

Transformations: y = f(x) + a shifts up by a; y = f(x + a) shifts left by a; y = af(x) stretches vertically by factor a; y = f(ax) stretches horizontally by factor 1/a.

Reflections: y = −f(x) reflects in the x-axis; y = f(−x) reflects in the y-axis; y = |f(x)| reflects any part below the x-axis upward.

A graph y = f(x) has an asymptote where the function tends to infinity or a finite limit as x tends to a value or ±∞; vertical asymptotes occur where the denominator is zero.

Even functions satisfy f(−x) = f(x) and are symmetric about the y-axis; odd functions satisfy f(−x) = −f(x) and have rotational symmetry about the origin.

Exponential graph y = a^x (a > 1) passes through (0, 1) and has the x-axis as a horizontal asymptote; y = log_a x is its reflection in y = x and passes through (1, 0).

Distance between (x₁, y₁) and (x₂, y₂): √((x₂ − x₁)² + (y₂ − y₁)²); midpoint is ((x₁ + x₂)/2, (y₁ + y₂)/2).

Gradient of a line through two points: m = (y₂ − y₁)/(x₂ − x₁); equation forms y = mx + c and y − y₁ = m(x − x₁).

Parallel lines have equal gradients; perpendicular lines satisfy m₁ × m₂ = −1.

A circle with centre (a, b) and radius r has equation (x − a)² + (y − b)² = r²; expanded form x² + y² + 2gx + 2fy + c = 0 has centre (−g, −f) and radius √(g² + f² − c).

The perpendicular from the centre of a circle to a chord bisects the chord; the tangent at a point is perpendicular to the radius at that point.

Area of a triangle with vertices (x₁, y₁), (x₂, y₂), (x₃, y₃): ½|x₁(y₂ − y₃) + x₂(y₃ − y₁) + x₃(y₁ − y₂)|.

A point dividing a segment in ratio m : n has coordinates ((mx₂ + nx₁)/(m + n), (my₂ + ny₁)/(m + n)).

To find intersections of a line and a curve, substitute the line equation into the curve equation and solve the resulting quadratic; the discriminant indicates 0, 1 (tangent) or 2 intersections.

Arithmetic progression: nth term a + (n − 1)d; sum of n terms Sₙ = n/2 [2a + (n − 1)d] = n/2 (a + l) where l is the last term.

Geometric progression: nth term ar^(n−1); sum of n terms Sₙ = a(1 − rⁿ)/(1 − r) for r ≠ 1; sum to infinity S∞ = a/(1 − r) valid only when |r| < 1.

For a GP with |r| ≥ 1 the series diverges and has no sum to infinity; for r = 1 the sum of n terms is na.

The sum of the first n natural numbers is n(n + 1)/2; the sum of the first n squares is n(n + 1)(2n + 1)/6.

Binomial expansion: (a + b)ⁿ = Σ from r = 0 to n of ⁿCᵣ a^(n−r) b^r; the general term is ⁿCᵣ a^(n−r) b^r.

ⁿCᵣ = n!/(r!(n − r)!); the term independent of x is found by setting the power of x in the general term to zero.

Sigma notation: Σ from r = 1 to n of (Ar + B) splits into A Σr + B Σ1; use the standard sums of r and r² for polynomial terms.

Convergence of an arithmetic series: it diverges unless d = 0; a geometric series converges only when |r| < 1.

Recurrence relations: a term like uₙ₊₁ = kuₙ + c defines a sequence; iterate to find terms, and solve a fixed point u = ku + c to find any limit.

Reciprocal identities: cosec θ = 1/sin θ, sec θ = 1/cos θ, cot θ = 1/tan θ; Pythagorean identities sin²θ + cos²θ = 1, 1 + tan²θ = sec²θ, 1 + cot²θ = cosec²θ.

Exact values: sin 30° = ½, cos 30° = √3/2, tan 30° = 1/√3; sin 45° = cos 45° = 1/√2; sin 60° = √3/2, cos 60° = ½, tan 60° = √3.

Sine rule: a/sin A = b/sin B = c/sin C; cosine rule: a² = b² + c² − 2bc cos A; area of triangle = ½ab sin C.

Identities: 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 angle: sin 2A = 2 sin A cos A; cos 2A = cos²A − sin²A = 2cos²A − 1 = 1 − 2sin²A; tan 2A = 2tan A/(1 − tan²A).

Radians: π radians = 180°; arc length s = rθ and sector area = ½r²θ for θ in radians.

General solutions: sin θ = k gives θ = α + 360°n or 180° − α + 360°n; cos θ = k gives θ = ±α + 360°n; tan θ = k gives θ = α + 180°n.

The equation a sin θ + b cos θ = c is solved by writing it as R sin(θ + α) with R = √(a² + b²) and tan α = b/a.

Graphs: y = sin x and y = cos x have period 360° and range [−1, 1]; y = tan x has period 180° and asymptotes at 90° + 180°n.

Derivative from first principles: f′(x) = limit as h → 0 of [f(x + h) − f(x)]/h; the derivative is the gradient of the tangent to the curve.

Power rule: d/dx (axⁿ) = anx^(n−1); the derivative of a constant is zero; d/dx (ln x) = 1/x and d/dx (eˣ) = eˣ.

Chain rule: dy/dx = dy/du × du/dx; product rule: d/dx (uv) = u dv/dx + v du/dx; quotient rule: d/dx (u/v) = (v du/dx − u dv/dx)/v².

Trig derivatives: d/dx (sin x) = cos x, d/dx (cos x) = −sin x, d/dx (tan x) = sec²x, with x in radians.

Stationary points occur where dy/dx = 0; use the second derivative to classify: d²y/dx² > 0 gives a minimum, < 0 gives a maximum, = 0 requires further investigation.

Tangents and normals: tangent gradient is dy/dx at the point; normal gradient is −1/(dy/dx); use y − y₁ = m(x − x₁).

Integration reverses differentiation: ∫ axⁿ dx = ax^(n+1)/(n + 1) + c for n ≠ −1; ∫ x⁻¹ dx = ln|x| + c; ∫ eˣ dx = eˣ + c.

Definite integrals: ∫ from a to b of f(x) dx = F(b) − F(a); the area between two curves is ∫ (upper − lower) dx between intersection points.

Volumes of revolution: rotating y = f(x) about the x-axis gives V = π ∫ y² dx; about the y-axis use V = π ∫ x² dy.

A vector has magnitude and direction; a scalar has magnitude only; vectors are equal if they have the same magnitude and direction.

Magnitude of a column vector (x, y): √(x² + y²); in three dimensions |(x, y, z)| = √(x² + y² + z²).

Unit vector in the direction of a is a/|a|; the position vector of a point relative to the origin is written using i, j, k components.

Vector addition is head-to-tail; subtraction a − b gives the vector from b to a; scalar multiplication k a changes magnitude by |k| and reverses direction if k < 0.

Scalar (dot) product: a · b = |a||b| cos θ = a₁b₁ + a₂b₂ + a₃b₃; perpendicular vectors have a · b = 0.

Angle between vectors: cos θ = (a · b)/(|a||b|); this is used to find angles in triangles and between lines.

A line has equation r = a + t b, where a is a point on the line and b is a direction vector; parallel lines have parallel direction vectors.

Velocity and displacement: if r = a + t v, the velocity is v and speed is |v|; the position at time t is found by substituting t into r.

A matrix of order m × n has m rows and n columns; addition and subtraction require the same order, and multiplication requires the inner dimensions to match.

Matrix multiplication is not commutative: AB ≠ BA in general; the identity matrix I satisfies AI = IA = A.

Determinant of a 2 × 2 matrix [[a, b], [c, d]] is ad − bc; the matrix is singular (no inverse) when the determinant is zero.

Inverse of a 2 × 2 matrix: (1/(ad − bc)) [[d, −b], [−c, a]]; the inverse satisfies AA⁻¹ = A⁻¹A = I.

Transformation matrices: reflection in the x-axis [[1, 0], [0, −1]], in the y-axis [[−1, 0], [0, 1]], in y = x [[0, 1], [1, 0]]; rotation by θ anticlockwise [[cos θ, −sin θ], [sin θ, cos θ]].

Enlargement with scale factor k has matrix [[k, 0], [0, k]]; a one-way stretch parallel to the x-axis has matrix [[k, 0], [0, 1]].

To find the matrix of a transformation, apply it to the unit vectors (1, 0) and (0, 1); the images form the columns of the matrix.

Combined transformations: if transformation A is followed by transformation B, the combined matrix is BA; order matters.

Solving simultaneous equations with matrices: write AX = B and solve X = A⁻¹B, provided A is non-singular.

Mean of a data set: x̄ = Σx/n; for grouped data use class midpoints, x̄ = Σfx/Σf; median is the middle value when data are ordered.

Variance and standard deviation: σ² = Σ(x − x̄)²/n = Σx²/n − x̄²; standard deviation is the positive square root of the variance.

For grouped data, the mean is Σfx/Σf and variance is Σfx²/Σf − x̄²; use midpoints of classes.

Probability: P(A ∪ B) = P(A) + P(B) − P(A ∩ B); for mutually exclusive events P(A ∩ B) = 0; for independent events P(A ∩ B) = P(A)P(B).

Conditional probability: P(A|B) = P(A ∩ B)/P(B); events are independent if P(A|B) = P(A).

Permutations: the number of arrangements of n distinct objects is n!; choosing r from n and ordering them is ⁿPᵣ = n!/(n − r)!.

Combinations: the number of ways of choosing r objects from n without regard to order is ⁿCᵣ = n!/(r!(n − r)!).

Binomial distribution: if X ~ B(n, p), then P(X = r) = ⁿCᵣ pʳ (1 − p)^(n−r); mean is np and variance is np(1 − p).

Tree diagrams: multiply probabilities along branches and add probabilities for mutually exclusive paths; the sum of all branch probabilities from one node is 1.