Pure maths

When decomposing partial fractions, what form is used for three distinct linear factors?

A/(ax+b) + B/(cx+d) + C/(ex+f).

What partial-fraction form is used for a repeated linear factor (ax+b)^2?

A/(ax+b) + B/(ax+b)^2.

What partial-fraction form is used when there is an irreducible quadratic factor?

(Ax+B)/(quadratic factor), together with terms for the other factors.

What must be done before partial-fraction decomposition of an improper rational function?

Use polynomial long division until the numerator degree is lower than the denominator degree.

How do you form a quadratic whose roots are p and q?

Use x^2-(p+q)x+pq=0, or multiply through by a convenient constant.

What is an arithmetic progression (AP)?

A sequence in which the same number d is added each time: a, a+d, a+2d, …

What is the nth term of an AP?

T_n=a+(n-1)d. Booklet: NOT explicitly given—memorise.

What is a geometric progression (GP)?

A sequence in which each term is multiplied by the same ratio r: a, ar, ar^2, …

What is the nth term of a GP?

T_n=ar^(n-1). Booklet: NOT explicitly given—memorise.

State the finite sum of an AP.

S_n=n/2[2a+(n-1)d]=n/2(a+l). Booklet: GIVEN.

State the finite sum of a GP.

S_n=a(1-r^n)/(1-r), r≠1. Booklet: GIVEN.

When does an infinite GP converge and what is its sum?

It converges only when |r|<1, and S_∞=a/(1-r). Booklet: not explicitly given—memorise.

What are the arithmetic and geometric means of positive numbers a and b?

Arithmetic mean=(a+b)/2; geometric mean=√(ab).

What is the convergence condition for the binomial series (1+x)^n when n is not a positive integer?

|x|<1.

What is the multiplication principle in counting?

If one stage can occur in m ways and a following stage in n ways, the combined process has mn outcomes.

What is the addition principle in counting?

If mutually exclusive alternatives can occur in m ways or n ways, there are m+n possibilities.

State nPr and nCr and explain the difference.

nPr=n!/(n-r)! for ordered selections; nCr=n!/[r!(n-r)!] for unordered selections. Booklet: GIVEN.

How many arrangements are there of n objects when some are repeated?

n! divided by the factorial of each repetition count: n!/(a!b!c!…).

How do you handle objects that must stay together in an arrangement?

Treat the required group as one block, arrange the blocks, then multiply by the internal arrangements of the group.

How do you handle two objects that must not be adjacent?

Count all arrangements, subtract arrangements where the two objects are treated as one block.

State the perpendicular distance from (x1,y1) to ax+by+c=0.

|ax1+by1+c|/√(a^2+b^2). Booklet: GIVEN.

How do you find a locus from a geometric condition?

Represent the moving point by (x,y), translate the condition into an equation using distance/slope, and simplify.

What are parametric coordinates?

Coordinates written as x=f(t), y=g(t), where t is a parameter; eliminate t only when required.

What are the centre-radius and general forms of a circle?

(x-a)^2+(y-b)^2=r^2; x^2+y^2+Dx+Ey+F=0, whose centre is (-D/2,-E/2).

How do you find the tangent to a circle at a point?

Use that the tangent is perpendicular to the radius, or use the standard tangent formula when appropriate.

What is the condition for two circles to touch externally or internally?

External: distance between centres=r1+r2. Internal: distance=|r1-r2|.

What is the condition for two circles to be orthogonal?

If centre distance is d, then d^2=r1^2+r2^2.

What is a rational function and why is long division useful?

It is a quotient of polynomials. Long division reveals polynomial and proper-fraction parts, helping find asymptotes.

What are the small-angle approximations and condition?

For x in radians and |x| small: sinx≈x, tanx≈x, cosx≈1-x²/2.

How do you rewrite a cosθ+b sinθ as R cos(θ-α)?

R=√(a²+b²), with R cosα=a and R sinα=b; then use quadrant/signs carefully.

How do moduli and arguments behave under multiplication and division?

|zw|=|z||w|; arg(zw)=argz+argw. |z/w|=|z|/|w|; arg(z/w)=argz-argw, adjusted to the principal range.

How do you write z in modulus-argument form?

z=r(cosθ+i sinθ), where r=|z| and θ=argz.

How do you find square roots of a complex number?

Let (a+bi)²=z, expand and equate real/imaginary parts, or use polar form and halve the arguments.

Why do non-real roots of a real-coefficient polynomial occur in conjugate pairs?

If f has real coefficients and f(a+bi)=0, conjugating gives f(a-bi)=0.

State de Moivre's theorem.

[cosθ+i sinθ]^n=cos(nθ)+i sin(nθ) for integer n. Booklet: GIVEN.

State Euler's formula and the exponential form of a complex number.

e^(iθ)=cosθ+i sinθ; z=re^(iθ).

How are the nth roots of z=re^(iθ) found?

z_k=r^(1/n)e^[i(θ+2πk)/n], k=0,…,n-1; arguments differ by 2π/n.

What locus is represented by |z-a|=c?

A circle with centre a and radius c on the Argand diagram.

What does |z-a|≤c represent?

The closed disc centred at a with radius c; < gives the interior only.

How do you solve a first-order separable differential equation?

Separate all y terms with dy and x terms with dx, integrate both sides, include a constant, then use any initial condition.

What must be checked after solving a differential equation?

Whether constant/equilibrium solutions were lost during division and whether the solution satisfies initial/domain conditions.

State the magnitude and unit vector of a vector a.

|a|=√(a·a); unit vector in its direction is a/|a|.

State the scalar product and angle formula.

a·b=|a||b|cosθ; hence cosθ=(a·b)/(|a||b|). Booklet: GIVEN.

What is the vector equation of a line?

r=a+λd, where a is a position vector of a point and d is a direction vector.

How do you test whether two lines intersect or are skew?

Equate their vector components and solve parameters consistently. No common solution in 3D can indicate skew or parallel lines.

What is a matrix and when can matrices be multiplied?

A rectangular array. AB exists when columns of A equal rows of B; multiplication is row-by-column.

State the inverse of a 2×2 matrix.

For A=[[a,b],[c,d]], A^-1=1/(ad-bc)[[d,-b],[-c,a]], if ad-bc≠0.

Which matrix represents a counterclockwise rotation through θ about the origin?

[[cosθ,-sinθ],[sinθ,cosθ]].

What matrix represents reflection in the line y=x?

[[0,1],[1,0]].

How are compound transformations ordered in matrix multiplication?

If transformation A is followed by B, the combined matrix is BA; the rightmost matrix acts first.

State the Maclaurin series formula.

f(x)=f(0)+xf'(0)+x²f''(0)/2!+…+x^n f^(n)(0)/n!+…

State the Maclaurin series for e^x and its convergence.

e^x=1+x+x²/2!+x³/3!+…; converges for all real x.

State the Maclaurin series for sinx and cosx.

sinx=x-x³/3!+x⁵/5!-…; cosx=1-x²/2!+x⁴/4!-…; both converge for all x.

State the series for ln(1+x) and its main convergence interval.

ln(1+x)=x-x²/2+x³/3-…; -1<x≤1 (with endpoint behaviour considered).

State the general binomial series and convergence condition.

(1+x)^n=1+nx+n(n-1)x²/2!+…; for non-integer n, |x|<1.

What is the method of differences in summation?

Rewrite the term so successive terms cancel (telescoping), write several terms, then identify the uncancelled endpoints.

How can partial fractions help sum a series?

Decompose the nth term into a difference such as A/n - A/(n+1), producing cancellation.

What are the four standard steps of mathematical induction?

State P(n); prove base case; assume P(k); prove P(k+1); conclude P(n) holds for all required n.

What is the induction hypothesis?

The temporary assumption that P(k) is true, used only to prove P(k+1).

What makes an induction proof incomplete?

Skipping the base case, failing to use the hypothesis, or not explicitly concluding the result for all n.

State the vector product magnitude and direction.

|a×b|=|a||b|sinθ; direction is perpendicular by the right-hand rule; a×b=-(b×a).

How do you find the area of a parallelogram or triangle using vectors?

Parallelogram area=|a×b|; triangle area=1/2|a×b|.

State the triple scalar product uses.

|a·(b×c)| gives parallelepiped volume; divide by 6 for a tetrahedron formed by three edge vectors.

What are vector and Cartesian forms of a plane?

Vector: r·n=d or r=a+λb+μc. Cartesian: ax+by+cz=d, where n=(a,b,c).

How is the angle between two planes found?

Find the acute angle between their normal vectors using the dot product.

How is the angle between a line direction d and a plane normal n related to the line-plane angle α?

sinα=|d·n|/(|d||n|).

State the distance from point (x0,y0,z0) to plane ax+by+cz=d.

|ax0+by0+cz0-d|/√(a²+b²+c²).

How are vertical, horizontal and oblique asymptotes usually found?

Vertical: denominator zeros not cancelled; horizontal: limiting value as x→±∞; oblique: polynomial division when numerator degree is one more.

How is y=1/f(x) related to y=f(x)?

Zeros of f become vertical asymptotes; where |f| is large the reciprocal is near 0; sign is preserved.

How is y²=f(x) interpreted?

Require f(x)≥0 and take y=±√f(x), giving symmetry about the x-axis.

State the standard volume-of-revolution formula about the x-axis.

V=π∫ y² dx between the required limits. Booklet: GIVEN.

State the polar-coordinate conversions.

x=r cosθ; y=r sinθ; r²=x²+y²; tanθ=y/x with quadrant care.

State the area enclosed by a polar curve.

A=1/2∫ r² dθ between the relevant angular limits. Booklet: GIVEN.

For x=r cosθ, y=r sinθ, how is dy/dx found?

(dy/dθ)/(dx/dθ). Horizontal tangent: dy/dθ=0; vertical tangent: dx/dθ=0, with denominator/numerator checks.

What is a singular matrix?

A square matrix with determinant 0; it has no inverse.

What are elementary row operations?

Swap rows; multiply a row by a nonzero constant; add a multiple of one row to another.

How are three simultaneous linear equations classified geometrically?

Unique solution: three planes meet at one point; infinitely many: common line/plane or coincident dependence; none: inconsistent planes.

What is an invariant point, line or plane under a transformation?

One that maps onto itself under the transformation, though individual points on an invariant line/plane need not all remain fixed.

How is the first-order linear ODE y'+P(x)y=Q(x) solved?

Use integrating factor I=e^(∫P dx), giving d(Iy)/dx=IQ; integrate and divide by I.

What is the complementary-function method for a constant-coefficient second-order ODE?

Solve the auxiliary equation ar²+br+c=0; root type determines exponential, repeated-root, or sine/cosine terms.

How is a particular integral found in the syllabus?

Use the trial form indicated by f(x), substitute, and solve coefficients; a failure-case trial is given when needed.

How do you prove a root lies in (a,b) using a sign change?

If f is continuous and f(a)f(b)<0, the intermediate value theorem guarantees at least one root in (a,b).

State the Newton-Raphson iteration.

x_(n+1)=x_n-f(x_n)/f'(x_n). The syllabus limits questions to two iterations.

What is the main condition for Simpson's rule?

The number of strips must be even; ordinates are weighted 1,4,2,4,…,2,4,1.

State conditional probability.

P(A|B)=P(A∩B)/P(B), provided P(B)>0. Booklet: GIVEN.

How do mutually exclusive and independent events differ?

Mutually exclusive: cannot both occur, P(A∩B)=0. Independent: one does not affect the other, P(A∩B)=P(A)P(B).

State the addition rule for two events.

P(A∪B)=P(A)+P(B)-P(A∩B). Booklet: GIVEN.

When should a tree diagram be used?

For sequential events, especially conditional probabilities and sampling with or without replacement.

What do ℕ, ℤ, ℚ, ℝ and ℂ represent?

ℕ natural numbers; ℤ integers; ℚ rational numbers; ℝ real numbers; ℂ complex numbers.

What is the key method for rationalising a denominator containing one surd?

Multiply numerator and denominator by the surd or by the conjugate so that the denominator becomes rational.

State the index laws for multiplying and dividing powers with the same base.

a^m a^n = a^(m+n); a^m/a^n = a^(m-n), for a ≠ 0.

What do negative and fractional indices mean?

a^(-n)=1/a^n; a^(1/n)=the nth root of a; a^(m/n)=the nth root of a^m.

State the three main logarithm laws.

log_a(xy)=log_a x+log_a y; log_a(x/y)=log_a x-log_a y; log_a(x^k)=k log_a x.

State the change-of-base formula.

log_a x = log_b x / log_b a. It is useful when the calculator has only ln or log.

What restrictions must be checked in logarithmic equations?

Every logarithm argument must be positive, and the base must be positive and not equal to 1.

State the remainder theorem.

When f(x) is divided by x-a, the remainder is f(a).

State the factor theorem.

x-a is a factor of f(x) exactly when f(a)=0.

State the sum and difference of cubes factorisations.

a^3+b^3=(a+b)(a^2-ab+b^2); a^3-b^3=(a-b)(a^2+ab+b^2). Booklet: GIVEN.

How does the discriminant classify roots of ax^2+bx+c=0?

Δ=b^2-4ac. Δ>0: two distinct real roots; Δ=0: repeated real root; Δ<0: complex conjugate roots.

What are the sum and product of the roots α,β of ax^2+bx+c=0?

α+β=-b/a and αβ=c/a.