BCS Written Mathematics cheat sheet

Divisibility: by 3 if digit sum is divisible by 3; by 4 if the last two digits form a multiple of 4; by 8 if the last three digits do; by 9 if the digit sum is divisible by 9; by 11 if the alternating digit sum is divisible by 11.

HCF × LCM of two positive integers equals their product; for more than two numbers this identity fails, so compute HCF by prime factorisation with minimum exponents and LCM with maximum exponents.

Percentage change = (new − old)/old × 100; successive changes of a% then b% give a net change of a + b + ab/100 percent, which may be negative.

Simple interest I = PRT/100; compound interest amount A = P(1 + R/100)^T for annual compounding, and A = P(1 + R/(100n))^(nT) for n conversions per year.

Ratio scaling: if a : b = m : n, write a = mk and b = nk; for three-part ratios combine two-term ratios through the common middle term's LCM.

Average speed over equal distances at speeds x and y is the harmonic mean 2xy/(x + y), not the arithmetic mean (x + y)/2.

Work rate: if A finishes a job in a days and B in b days, together they take ab/(a + b) days; rates add, times do not.

Profit or loss percent is always computed on cost price: profit% = (SP − CP)/CP × 100; a discount is computed on marked price.

Quadratic ax² + bx + c = 0 has roots (−b ± √(b² − 4ac))/(2a); the discriminant b² − 4ac decides two real, one repeated, or two complex roots.

For roots α and β of ax² + bx + c = 0: α + β = −b/a and αβ = c/a; build the equation as x² − (sum)x + (product) = 0.

Factor and remainder theorem: (x − a) divides p(x) exactly when p(a) = 0; the remainder on division by (x − a) is p(a).

Arithmetic progression: nth term a + (n − 1)d, sum of n terms (n/2)[2a + (n − 1)d]; geometric progression: nth term ar^(n−1), sum a(r^n − 1)/(r − 1) for r ≠ 1.

Laws of logarithms: log(xy) = log x + log y, log(x/y) = log x − log y, log x^n = n log x; log_b a = log_c a / log_c b, and log_b a = 1/log_a b.

Binomial theorem: (x + y)^n = Σ C(n,r) x^(n−r) y^r for r = 0 to n, with general term C(n,r) x^(n−r) y^r; C(n,r) = n!/(r!(n−r)!).

Exponential and logarithmic equations are solved by isolating the power then taking logarithms of both sides, or by converting log_b N = k into N = b^k.

For inequalities, multiplying or dividing by a negative number reverses the inequality sign; the solution set of a quadratic inequality is read from the sign of the parabola between its roots.

Triangle angle sum is 180°; exterior angle equals the sum of the two remote interior angles; the angle in a semicircle is 90°.

Pythagoras: in a right triangle a² + b² = c² with c the hypotenuse; common triples are 3-4-5, 5-12-13, 8-15-17 and their multiples.

Circle: circumference 2πr, area πr²; arc length = (θ/360) × 2πr and sector area = (θ/360) × πr² for central angle θ in degrees.

Area formulas: triangle ½ × base × height or √(s(s−a)(s−b)(s−c)) with s the semiperimeter; trapezium ½(a + b)h; parallelogram bh; rhombus ½ d₁d₂.

Volume and surface: cylinder πr²h and 2πr(r + h); cone (1/3)πr²h with slant l = √(r² + h²); sphere (4/3)πr³ with surface 4πr².

Similar figures: corresponding lengths scale by k, areas by k², volumes by k³; the ratio of areas of similar triangles equals the square of the ratio of corresponding sides.

Angle properties: angles on a straight line sum to 180°, angles at a point to 360°; alternate angles in parallel lines are equal and co-interior angles sum to 180°.

Sum of interior angles of an n-sided polygon is (n − 2) × 180°; each exterior angle of a regular n-gon is 360°/n.

Right-triangle ratios: sin θ = opposite/hypotenuse, cos θ = adjacent/hypotenuse, tan θ = opposite/adjacent = sin θ/cos θ.

Standard 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.

Pythagorean identity sin²θ + cos²θ = 1, with 1 + tan²θ = sec²θ and 1 + cot²θ = cosec²θ.

Sine rule a/sin A = b/sin B = c/sin C = 2R, where R is the circumradius; use it when a side and its opposite angle are known.

Cosine rule a² = b² + c² − 2bc cos A; rearrange to cos A = (b² + c² − a²)/(2bc) when all three sides are known.

Area of any triangle = ½ ab sin C using two sides and the included angle.

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 angle: sin 2A = 2 sin A cos A; cos 2A = cos²A − sin²A = 1 − 2sin²A = 2cos²A − 1.

Radian measure: π radians = 180°; arc length s = rθ and sector area ½r²θ with θ in radians.

Derivative from first principles: f′(x) = lim(h→0) [f(x + h) − f(x)]/h.

Power rule d/dx (x^n) = n x^(n−1); product rule (uv)′ = u′v + uv′; quotient rule (u/v)′ = (u′v − uv′)/v²; chain rule dy/dx = dy/du × du/dx.

Standard derivatives: d/dx(sin x) = cos x, d/dx(cos x) = −sin x, d/dx(tan x) = sec²x, d/dx(e^x) = e^x, d/dx(ln x) = 1/x.

Stationary points occur where f′(x) = 0; classify with f″(x) > 0 for a minimum, f″(x) < 0 for a maximum, and test sign change when f″(x) = 0.

Integration reverses differentiation: ∫x^n dx = x^(n+1)/(n+1) + C for n ≠ −1; definite integral equals F(b) − F(a) for antiderivative F.

Area under a curve between x = a and x = b is ∫ y dx; area between two curves is ∫ (upper − lower) dx over the intersection interval.

Volume of revolution about the x-axis is V = π∫ y² dx between the given limits.

Substitution rule: ∫ f(g(x)) g′(x) dx = ∫ f(u) du with u = g(x); integration by parts ∫ u dv = uv − ∫ v du, choosing u by the LIATE order.

Mean of grouped data = Σfx/Σf; median class is located where cumulative frequency first reaches N/2; mode class has the highest frequency.

Variance = Σ(x − mean)²/n and standard deviation is its positive square root; for grouped data use Σfx²/Σf − (mean)².

Probability of an event lies between 0 and 1; P(A′) = 1 − P(A); for mutually exclusive events P(A ∪ B) = P(A) + P(B).

Addition rule: P(A ∪ B) = P(A) + P(B) − P(A ∩ B); independence means P(A ∩ B) = P(A)P(B).

Conditional probability P(A|B) = P(A ∩ B)/P(B), provided P(B) > 0.

Permutations of n distinct objects taken r at a time: nPr = n!/(n − r)!; combinations nCr = n!/(r!(n − r)!), where order does not matter.

Binomial distribution: P(X = r) = C(n,r) p^r (1 − p)^(n−r) for n independent trials each with success probability p.

Quartiles: Q1 and Q3 split the ordered data at 25% and 75%; interquartile range IQR = Q3 − Q1; outliers are commonly flagged beyond Q1 − 1.5 IQR or Q3 + 1.5 IQR.

A matrix of order m × n has m rows and n columns; addition and subtraction require identical orders, while product AB requires columns of A to equal rows of B.

Matrix multiplication is not commutative in general: AB ≠ BA, though (AB)C = A(BC) and A(B + C) = AB + AC.

Determinant of a 2 × 2 matrix [[a, b], [c, d]] is ad − bc; the matrix is singular when the determinant is zero.

Inverse of a 2 × 2 matrix is (1/det) [[d, −b], [−c, a]], defined only when det ≠ 0; for any invertible A, (AB)⁻¹ = B⁻¹A⁻¹.

Transpose rules: (Aᵀ)ᵀ = A, (A + B)ᵀ = Aᵀ + Bᵀ, (AB)ᵀ = BᵀAᵀ; a symmetric matrix satisfies Aᵀ = A.

Determinant properties: swapping two rows changes the sign, multiplying one row by k multiplies det by k, and a repeated row makes det zero.

Cramer's rule solves AX = B by x = det A_x/det A, where A_x replaces the relevant column with B; it requires det A ≠ 0.

A system AX = B has a unique solution when det A ≠ 0; if det A = 0 the system is either inconsistent or has infinitely many solutions.