1. Further Pure Mathematics 1

For ax²+bx+c=0 with roots α and β, α+β=−b/a and αβ=c/a. These relations are often faster and safer than applying the quadratic formula twice.

For higher-degree polynomials, factorisation and the factor theorem identify roots; repeated roots occur when a factor is squared or the polynomial and its derivative share a root. Keep the equation in standard form before reading coefficients.

For 2x²−5x+3=0, the roots have sum 5/2 and product 3/2, so their average is 5/4 without finding either root.

The coefficient relationships depend on signs and the leading coefficient; they are not simply “sum=b, product=c”.

If α is a root of f(x)=0, a new equation with roots such as α+k, kα or 1/α is obtained by substituting the inverse transformation into f, then simplifying.

For a shift y=x−k, replace x by y+k; for a scale y=x/k, replace x by ky. Check whether the question asks for roots or for an equation whose roots are transformed values.

If α satisfies α²−3α+2=0, the roots α+1 satisfy (y−1)²−3(y−1)+2=0, which simplifies to y²−5y+6=0.

Adding k to every root does not mean adding k to every coefficient, and reciprocal roots require multiplying by a suitable power to remove denominators.

For a rational function, factorise numerator and denominator first. A denominator zero may create a vertical asymptote or a removable hole; the degrees and leading coefficients determine horizontal or oblique behaviour.

Find excluded x-values, intercepts and the sign of each factor on intervals. Long division gives the polynomial asymptote when the numerator degree is at least the denominator degree.

For (x+1)/(x−2), x=2 is a vertical asymptote, x=−1 is the x-intercept, and y=1 is the horizontal asymptote because the leading coefficients match.

A cancelled factor creates a hole rather than a vertical asymptote, and an asymptote is a limiting feature—not necessarily a line the graph can never cross in every case.

Graph transformations change coordinates systematically: y=f(x)+a shifts vertically, y=f(x−a) shifts right, y=−f(x) reflects in the x-axis, and y=f(−x) reflects in the y-axis.

For y=af(bx+c)+d, apply the inside transformation before the outside one: horizontal scale/reflection and shift act on x, while a and d act on y. Track a known point to check the direction.

A point (2,3) on y=f(x) becomes (5,3) on y=f(x−3), because the input required to produce the same output is three larger.

The sign inside f changes the horizontal direction opposite to the visible sign outside; y=f(x−3) shifts right, not left.

The arithmetic series 1+2+…+n equals n(n+1)/2. The geometric series a+ar+…+ar^{n−1} equals a(1−r^n)/(1−r) when r≠1, and tends to a/(1−r) when |r|<1.

Index the first and last terms carefully. For an infinite series, convergence requires the terms to tend to zero; a ratio with |r|≥1 does not give a finite sum.

The sum 3+6+12+24 is a=3, r=2, n=4, so it is 3(1−2^4)/(1−2)=45. The corresponding infinite series diverges because |r|>1.

Do not use the infinite formula for a finite or divergent series, and do not confuse the number of terms with the final index.

A telescoping or method-of-differences series is written as differences such as u_r−u_{r+1}, so most intermediate terms cancel when the terms are added.

Expand the first few and last few terms before simplifying. The answer is determined by the uncancelled boundary terms; if the cancellation pattern changes at a singular index, split the sum.

Σ_{r=1}^{n}(1/r−1/(r+1)) = 1−1/(n+1), because every middle reciprocal cancels.

Do not cancel across a plus sign or ignore the final boundary term; write at least three terms before jumping to the pattern.

For an infinite series to converge, its terms must tend to zero and the partial sums must approach a finite limit. A geometric series converges when |r|<1; otherwise its partial sums do not settle to a finite sum.

The term test is necessary but not sufficient for every series: terms tending to zero does not by itself prove convergence. Use an appropriate comparison, ratio, integral or known-series test when required.

Σ(1/2)^n converges because the ratio has magnitude below one. Σ1/n has terms tending to zero but still diverges, so checking only the terms is insufficient.

“The terms get small” is not a complete convergence argument, and a finite partial sum is not the value of an infinite series.

To multiply A by B, the number of columns of A must equal the number of rows of B. Each entry of AB is the dot product of a row of A with a column of B.

The order matters: AB may exist while BA does not, or both may exist with different results. Matrix addition requires equal dimensions, while scalar multiplication scales every entry.

For A=[[1,2],[3,4]] and B=[[0,1],[2,0]], AB=[[4,1],[8,3]], whereas BA=[[3,4],[1,2]].

Matrix multiplication is not entry-by-entry multiplication, and AB=BA is a special property rather than a default.

For a 2×2 matrix [[a,b],[c,d]], det A=ad−bc. A square matrix has an inverse only when its determinant is non-zero; determinant magnitude gives the scale factor for oriented area or volume.

A zero determinant means the transformation collapses dimension and distinct vectors become dependent. For larger matrices, expand by a row or column or use row operations while tracking determinant changes.

For [[2,1],[3,2]], det=4−3=1, so the matrix is invertible and preserves area magnitude. For [[1,2],[2,4]], det=0, so no inverse exists.

A negative determinant does not mean “no inverse”; it indicates orientation reversal, while only zero prevents invertibility.

If A is invertible, AA⁻¹=A⁻¹A=I. To solve AX=B, multiply on the left by A⁻¹: X=A⁻¹B. The side matters because matrix multiplication is not generally commutative.

For a product, (AB)⁻¹=B⁻¹A⁻¹: the operations are undone in reverse order. Check dimensions and multiply back to verify the result.

If a transformation first applies B and then A, its matrix is AB; reversing the process uses B⁻¹A⁻¹, not A⁻¹B⁻¹.

Do not divide by a matrix as if it were a scalar, and do not move A⁻¹ across B without changing the order.

A 2×2 matrix maps a vector (x,y) to a new vector by combining its components. Its columns are the images of the standard basis vectors, so the matrix describes rotation, reflection, stretch, shear or a combination.

Composition is matrix multiplication, and the order records which transformation happens first. Determinant describes area scaling; eigenvectors, when present, keep their direction under the transformation.

The matrix [[0,−1],[1,0]] sends (1,0) to (0,1) and (0,1) to (−1,0), a 90° anticlockwise rotation.

Read the coordinate convention carefully: multiplying on the left by a column-vector matrix is not interchangeable with using row vectors.

An invariant point maps to itself. An invariant line maps every point on the line to a point on the same line, even if individual points move along it.

For a matrix transformation, solve T(v)=v for invariant vectors or use the eigenvalue 1 condition. For an invariant line, check that its direction is mapped to a parallel direction and that the line’s location is preserved.

A reflection leaves every point on its mirror line invariant; a rotation about the origin leaves the origin fixed but usually has no other invariant points.

A line being drawn in the same place is not enough: points must map within it, and “invariant” does not mean every point is fixed.

A polar point is written (r,θ), where r is the directed distance from the pole and θ is the angle from the initial line. Cartesian coordinates satisfy x=r cosθ and y=r sinθ.

The same point can have infinitely many representations: (r,θ)=(r,θ+2π) and (−r,θ)=(r,θ+π). Decide whether the problem uses a signed radius or restricts r≥0.

The point (2,π/3) has Cartesian coordinates (1,√3). The representation (−2,4π/3) describes the same point.

Do not treat θ as a length or assume a unique polar pair; angle periodicity and negative r matter.

A polar curve is given by r=f(θ). To understand its shape, identify the interval, zeros of r, symmetries and key angles before plotting points.

If f(−θ)=f(θ), the curve is symmetric about the initial line; if f(π−θ)=f(θ), it is symmetric about the line θ=π/2. Negative r plots in the opposite direction, so sign changes are structural.

For r=2cosθ, r is zero at π/2 and negative for π/2<θ<3π/2; the curve is a circle of diameter 2 on the initial-line side after the negative-radius points are interpreted.

Joining a table of positive radii without checking sign and symmetry can produce the wrong curve.

The area swept by a polar curve from θ=a to θ=b is A=½∫_a^b r² dθ, provided the interval traces the intended region once.

Square r, so a negative radius contributes positive area. Find intersections and choose limits carefully; if the curve crosses the pole or overlaps itself, split the region to avoid counting it twice.

For r=2cosθ from −π/2 to π/2, A=½∫4cos²θ dθ=π, the area of the corresponding circle.

Do not integrate r rather than r², and do not assume the stated interval is automatically one non-overlapping traversal.

A plane through point a with normal vector n has equation n·(r−a)=0, or equivalently n·r=d. The normal is perpendicular to every direction lying in the plane.

To find a normal, use a cross product of two non-parallel direction vectors in the plane. Check a candidate point by substitution and keep vector notation consistent.

A plane through (1,0,2) with normal (2,−1,3) satisfies 2(x−1)−y+3(z−2)=0.

A vector lying in the plane is perpendicular to the normal, not parallel to it; one point alone does not determine a unique plane.

The vector or cross product a×b is perpendicular to both a and b, with magnitude |a×b|=|a||b|sinθ. Its magnitude is the area of the parallelogram spanned by the two vectors.

The order reverses direction: a×b=−(b×a). It is zero when vectors are parallel, so it can test parallelism and construct a plane normal.

(1,0,0)×(0,2,0)=(0,0,2), whose magnitude 2 is the area of the rectangle formed by the vectors.

The cross product is not a scalar dot product, and changing the order is not harmless.

A line is r=a+λd; a plane is n·(r−b)=0. A line is parallel to a plane when n·d=0, lies in it when a point also satisfies the plane equation, and intersects it otherwise.

To find a line–plane intersection, substitute the line into the plane equation to solve λ. For two planes, the cross product of their normals gives the direction of their line of intersection when the normals are not parallel.

For r=(1,0,0)+λ(1,2,−1) and plane x+y+z=3, substitution gives 1+λ+2λ−λ=3, so λ=2/2=1 and the intersection point is (2,2,−1).

Parallel normals do not prove two planes are the same; compare a point or the constants, and keep line direction distinct from position.

To prove P(n) for all integers n≥n₀, prove P(n₀), assume P(k) for an arbitrary k≥n₀, then use that assumption to prove P(k+1).

The assumption is used only for the arbitrary case, not asserted for every value without proof. State the domain and simplify the target P(k+1) until the expression contains the inductive hypothesis.

For 1+2+…+n=n(n+1)/2, the base case n=1 is true; adding k+1 to the assumed sum k(k+1)/2 gives (k+1)(k+2)/2.

Checking many numerical cases is not induction, and proving P(k)→P(k+1) without a valid base case proves nothing about the starting chain.

A conjecture is a statement suggested by patterns in initial cases. Induction can then prove it if the base case and inductive implication are valid for the stated domain.

Use small values to identify the likely formula, factorisation or invariant, but do not treat pattern matching as evidence of universal truth. Check parity, starting index and exceptional values before writing the proof.

The first differences of a sequence may suggest a quadratic formula. Test several terms, formulate P(n), then prove P(n₀) and P(k)⇒P(k+1) rather than extrapolating from the table.

A pattern that works for ten values can fail at the next; conjecture and proof are different learning jobs.