NSSCAS Mathematics (8227) cheat sheet

Index laws: a^m × a^n = a^(m+n); (a^m)^n = a^(mn); a^(m/n) = n√(a^m); a^0 = 1 for a ≠ 0; a^(−n) = 1/a^n.

Surds: √(ab) = √a × √b and √(a/b) = √a/√b. Rationalise 1/(a+√b) by multiplying numerator and denominator by the conjugate a−√b.

Logarithms: log_a(xy) = log_a x + log_a y; log_a(x/y) = log_a x − log_a y; log_a(x^n) = n log_a x; log_a a = 1; log_a 1 = 0.

Change of base: log_a x = log_b x / log_b a. The equation a^x = b is solved as x = log b / log a.

Quadratic formula: x = (−b ± √(b²−4ac))/(2a). The discriminant b²−4ac gives two distinct real roots if positive, one repeated root if zero, no real roots if negative.

Completing the square writes x²+bx+c as (x+p)²+q with p = b/2 and q = c − b²/4; the vertex is (−p, q) and the axis of symmetry is x = −b/2.

Remainder and factor theorems: dividing f(x) by (x−a) leaves remainder f(a); (x−a) is a factor exactly when f(a) = 0.

Inequalities: multiplying or dividing by a negative number reverses the inequality sign. Solve quadratic inequalities by finding the critical roots and testing the sign in each interval.

Binomial: (a+b)^n = Σ C(n,r) a^(n−r) b^r for r = 0 to n, with C(n,r) = n!/(r!(n−r)!); the term in b^r is C(n,r) a^(n−r) b^r.

Domain: exclude any x making a denominator zero or making an expression under an even root negative; for log_a(g(x)) require g(x) > 0.

Range: determine from the graph or by completing the square; for a quadratic with a > 0 the minimum value is q in a(x+p)²+q.

Composite: fg(x) means f(g(x)); it is defined only where x lies in the domain of g and g(x) lies in the domain of f.

Inverse: set y = f(x), solve for x in terms of y, then swap x and y. Domain of f⁻¹ = range of f; range of f⁻¹ = domain of f.

f⁻¹ exists only if f is one-to-one; the graphs of f and f⁻¹ are reflections of each other in the line y = x.

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 parts below the x-axis above it.

Asymptotes: vertical asymptotes occur where the denominator is zero after cancelling; horizontal asymptotes come from the limit of f(x) as x → ±∞.

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

Gradient m = (y₂−y₁)/(x₂−x₁). Parallel lines have equal gradients; perpendicular lines satisfy m₁m₂ = −1.

Straight line forms: y = mx + c; y − y₁ = m(x − x₁); ax + by + c = 0 has gradient −a/b.

Circle with centre (a,b) and radius r: (x−a)²+(y−b)² = r². In expanded form x²+y²+2gx+2fy+c = 0, centre (−g,−f) and radius √(g²+f²−c).

A tangent to a circle is perpendicular to the radius at the point of contact, so the tangent gradient is −1/(radius gradient).

Perpendicular distance from (x₀,y₀) to ax+by+c = 0 is |ax₀+by₀+c|/√(a²+b²).

Curve sketching: find axis intercepts, stationary points via dy/dx = 0, asymptotes, and the behaviour as x → ±∞ before plotting.

Intersections of a line and a curve are found by substitution, giving a quadratic whose discriminant decides 0, 1 or 2 intersection points.

Radian measure: π radians = 180°; degrees × π/180 = radians. Arc length s = rθ and sector area A = ½r²θ for θ in radians.

Identities: sin²θ + cos²θ = 1; tanθ = sinθ/cosθ; 1 + tan²θ = sec²θ; 1 + cot²θ = cosec²θ.

Double angles: sin2θ = 2sinθcosθ; cos2θ = cos²θ − sin²θ = 2cos²θ − 1 = 1 − 2sin²θ.

Exact values: sin30° = ½, cos30° = √3/2, tan30° = 1/√3; sin45° = cos45° = 1/√2; sin60° = √3/2, cos60° = ½, tan60° = √3.

Solving sinθ = k: principal value α = sin⁻¹k, then θ = α + 360°n or θ = 180° − α + 360°n. For cosθ = k use θ = ±α + 360°n.

For tanθ = k, θ = α + 180°n because tan has period 180°. Always check which solutions lie in the required interval.

Graphs: y = sin x and y = cos x have period 360° and amplitude 1; y = tan x has period 180° with asymptotes at x = 90° + 180°n.

Small angles in radians: sinθ ≈ θ, tanθ ≈ θ and cosθ ≈ 1 − θ²/2, valid for small θ.

Arithmetic progression: nth term u_n = a + (n−1)d; sum S_n = n/2(2a+(n−1)d) = n/2(a+l) where l is the last term.

Geometric progression: nth term u_n = ar^(n−1); sum S_n = a(1−r^n)/(1−r) for r ≠ 1.

The infinite geometric series converges to a/(1−r) only when |r| < 1; if |r| ≥ 1 the series diverges.

Sigma notation: Σ1 = n; Σr = n(n+1)/2; Σr² = n(n+1)(2n+1)/6; Σr³ = [n(n+1)/2]², each sum taken from r = 1 to n.

Combine standard results with constants: Σ(ar+b) = aΣr + bn, summing from r = 1 to n; split sums before applying formulae.

To show a sequence is arithmetic, show u_(n+1) − u_n is constant; for geometric, show u_(n+1)/u_n is constant.

A sequence converges if it approaches a finite limit as n → ∞; a geometric sequence converges to 0 when |r| < 1.

Derivative of x^n is nx^(n−1); the derivative of a constant is 0; differentiate polynomials term by term.

Chain rule: dy/dx = dy/du × du/dx for y = f(u) with u = g(x), used for composite functions.

Product rule: if y = uv then dy/dx = u(dv/dx) + v(du/dx).

Quotient rule: if y = u/v then dy/dx = (v(du/dx) − u(dv/dx))/v².

Standard derivatives: sin x → cos x; cos x → −sin x; tan x → sec²x; e^x → e^x; ln x → 1/x for x > 0.

Tangent at (x₁,y₁) has gradient m = dy/dx evaluated there; its equation is y − y₁ = m(x − x₁). The normal has gradient −1/m.

Stationary points: solve dy/dx = 0, then d²y/dx² > 0 gives a minimum and d²y/dx² < 0 gives a maximum; d²y/dx² = 0 needs a sign test.

A function is increasing where f'(x) > 0 and decreasing where f'(x) < 0. Connected rates: dy/dt = dy/dx × dx/dt.

∫x^n dx = x^(n+1)/(n+1) + C for n ≠ −1; ∫x^(−1) dx = ln|x| + C.

∫(ax+b)^n dx = (ax+b)^(n+1)/(a(n+1)) + C for n ≠ −1; always divide by the inner derivative a.

∫sin x dx = −cos x + C; ∫cos x dx = sin x + C; ∫e^x dx = e^x + C; ∫(1/x) dx = ln|x| + C.

Definite integral: ∫ from a to b of f(x) dx = F(b) − F(a), where F is any antiderivative of f.

Area under a curve from a to b is ∫ y dx; parts below the x-axis give negative values, so split at the roots and take moduli.

Area between curves: ∫(upper − lower) dx between the intersection x-values a and b.

Volume of revolution about the x-axis: V = π∫y² dx from a to b; about the y-axis: V = π∫x² dy from c to d.

Trapezium rule with n strips of width h: ∫ ≈ h/2[y₀ + 2(y₁ + … + y_(n−1)) + y_n]; it overestimates for a curve concave up.

Mean: x̄ = Σx/n for raw data or Σfx/Σf for grouped data; variance = Σ(x−x̄)²/n or Σfx²/Σf − x̄².

Standard deviation is the square root of the variance, in the same units as the data. IQR = upper quartile − lower quartile.

Permutations: the number of ordered arrangements of r objects chosen from n is nPr = n!/(n−r)!.

Combinations: the number of unordered selections of r from n is nCr = n!/(r!(n−r)!); note nCr = nC(n−r).

Probability: P(A∪B) = P(A) + P(B) − P(A∩B); P(A') = 1 − P(A); if A and B are independent, P(A∩B) = P(A)P(B).

Conditional probability: P(A|B) = P(A∩B)/P(B) for P(B) > 0; rearranging gives P(A∩B) = P(B)P(A|B).

Discrete random variable: ΣP(X = x) = 1; E(X) = ΣxP(X = x); Var(X) = E(X²) − [E(X)]².

Binomial distribution B(n,p): P(X = r) = C(n,r)p^r(1−p)^(n−r), with mean np and variance np(1−p).

Normal: standardise z = (x−μ)/σ; the curve is symmetric about μ, with about 68% of values within μ±σ, 95% within μ±2σ and 99.7% within μ±3σ.

Magnitude of a = xi + yj + zk is |a| = √(x²+y²+z²); the unit vector in the direction of a is a/|a|.

Vector algebra: add or subtract componentwise; multiplying by a scalar multiplies every component.

Position vectors: if A and B have position vectors a and b, then AB = b − a, and |AB| is the distance AB.

Scalar product: a·b = |a||b|cosθ = x₁x₂ + y₁y₂ + z₁z₂, where θ is the angle between the vectors.

a·b = 0 for non-zero a and b exactly when the vectors are perpendicular; a·b > 0 means θ is acute and a·b < 0 means θ is obtuse.

Angle between vectors: cosθ = (a·b)/(|a||b|).

Points A, B, C are collinear when AB = k·BC for some scalar k; two vectors are parallel when one is a scalar multiple of the other.