Maths Paper 2

How do you calculate a percentage of an amount?

Amount × percentage ÷ 100.

How do you increase an amount by p%?

Multiply by 1 + p/100.

How do you decrease an amount by p%?

Multiply by 1 − p/100.

How do you reverse a percentage increase?

Original = final ÷ (1 + p/100).

How do you reverse a percentage decrease?

Original = final ÷ (1 − p/100).

What is the simple interest formula?

Interest = P × r × t / 100; P is principal, r annual percentage rate, t years.

What is the compound interest formula?

Final amount = P(1 + r/100)^n; for depreciation use 1 − r/100.

How do you find percentage change?

(New − original) ÷ original × 100%.

How do you share an amount in a ratio a:b?

Divide total by a+b, then multiply by a and by b.

How do you find the multiplier for a 15% discount?

1 − 0.15 = 0.85.

How do you find the multiplier for 8% growth?

1 + 0.08 = 1.08.

What is direct proportion?

y ∝ x means y = kx; find k using known values.

What is inverse proportion?

y ∝ 1/x means y = k/x; find k using known values.

What if y is directly proportional to x squared?

y = kx².

What if y is inversely proportional to x squared?

y = k/x².

How do you calculate speed?

Speed = distance ÷ time.

How do you calculate density?

Density = mass ÷ volume.

How do you calculate pressure?

Pressure = force ÷ area.

How do you convert km/h to m/s?

Divide by 3.6.

How do you convert m/s to km/h?

Multiply by 3.6.

What is the rule for multiplying powers with the same base?

a^m × a^n = a^(m+n).

What is the rule for dividing powers with the same base?

a^m ÷ a^n = a^(m−n).

What is the power-of-a-power rule?

(a^m)^n = a^(mn).

What does a negative index mean?

a^(−n) = 1/a^n.

What does a fractional index mean?

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

What is standard form?

A × 10^n, where 1 ≤ |A| < 10 and n is an integer.

How do you multiply surds?

√a × √b = √(ab), for non-negative a and b.

How do you simplify a surd?

Factor out the largest square factor: √72 = √(36×2) = 6√2.

How do you rationalise 1/√a?

Multiply numerator and denominator by √a: √a/a.

How do you estimate a calculation?

Round numbers to convenient values, often 1 significant figure, then calculate.

How do you find lower and upper bounds for a rounded number?

Subtract/add half the rounding unit.

For positive a and b, how do you find the maximum of a/b?

Upper bound of a ÷ lower bound of b.

For positive a and b, how do you find the minimum of a/b?

Lower bound of a ÷ upper bound of b.

How do you convert a recurring decimal to a fraction?

Set x equal to decimal, multiply by power(s) of 10 to align repeats, subtract, solve for x.

How do you expand two brackets?

Multiply every term in the first bracket by every term in the second; collect like terms.

What is difference of two squares?

a² − b² = (a−b)(a+b).

How do you factorise a quadratic x²+bx+c?

Find two numbers that multiply to c and add to b.

What is the quadratic formula?

x = (−b ± √(b²−4ac))/(2a) for ax²+bx+c=0.

What is the discriminant?

b² − 4ac; positive: 2 distinct real roots; zero: 1 repeated root; negative: no real roots.

How do you complete the square for x²+bx+c?

(x+b/2)² + c − (b/2)².

How do you solve simultaneous linear equations by elimination?

Multiply equations if necessary so one variable has matching/opposite coefficients; add/subtract and substitute back.

How do you solve a linear inequality?

Solve like an equation, but reverse the inequality when multiplying or dividing by a negative.

How do you solve an equation with algebraic fractions?

Factor denominators, note excluded values, multiply by the lowest common denominator, solve and check.

How do you find the nth term of a linear sequence?

Common difference gives coefficient of n; compare with first term to find constant.

How do you recognise a quadratic sequence?

Constant second differences.

How do you find coefficient of n² in a quadratic sequence?

a = second difference ÷ 2.

How do you find an inverse function?

Write y=f(x), swap x and y, rearrange for y.

How do you find fg(x)?

Apply g first, then f: f(g(x)).

How do you find a straight-line gradient?

m = (y₂−y₁)/(x₂−x₁).

How do you find the midpoint of two coordinates?

((x₁+x₂)/2, (y₁+y₂)/2).

How do you find distance between two points?

√[(x₂−x₁)² + (y₂−y₁)²].

What is the equation of a straight line?

y=mx+c; m is gradient and c is y-intercept.

How do you find the gradient of a perpendicular line?

Negative reciprocal: m₂ = −1/m₁, for nonzero finite m₁.

How do you find a turning point by completing the square?

Rewrite y=a(x−h)²+k; turning point is (h,k).

How do you find the x-coordinate of a quadratic turning point?

For y=ax²+bx+c, x=−b/(2a); substitute back to find y.

How do you tell if a quadratic turning point is a minimum or maximum?

a>0: minimum; a<0: maximum.

How do you find roots on a graph?

Read the x-coordinates where y=0.

How do you solve two equations graphically?

Draw both graphs; read x and y coordinates of intersections.

How do you estimate gradient of a curve?

Draw a tangent at the point and calculate rise/run using two points on tangent.

What does gradient of a distance–time graph give?

Speed (or signed velocity for displacement–time).

What does gradient of a speed–time graph give?

Acceleration.

What does area under a speed–time graph give?

Distance travelled.

What is the sum of angles in a triangle?

180°.

What is the sum of angles in a quadrilateral?

360°.

What are the parallel-line angle rules?

Corresponding equal; alternate equal; co-interior sum to 180°.

What is the interior-angle sum of an n-sided polygon?

(n−2) × 180°.

What is one exterior angle of a regular n-gon?

360°/n.

What is one interior angle of a regular n-gon?

180° − 360°/n.

How do you find number of sides from exterior angle?

n=360° ÷ exterior angle.

What is angle at centre theorem?

Angle at centre is twice angle at circumference on same arc.

What is the angle in a semicircle?

90°.

What is same-segment theorem?

Angles at circumference standing on same chord/arc are equal.

What is cyclic quadrilateral rule?

Opposite angles sum to 180°.

What is tangent-radius rule?

Tangent is perpendicular to radius at contact point.

What is the equal tangents rule?

Tangents drawn from same external point are equal in length.

What is alternate segment theorem?

Angle between tangent and chord equals angle in opposite segment.

What are the triangle congruence tests?

SSS, SAS, ASA/AAS and RHS.

How do you find length scale factor for similar shapes?

Corresponding new length ÷ original length.

How does area scale with length scale factor k?

Area factor = k².

How does volume scale with length scale factor k?

Volume factor = k³.

How do you get length factor from area factor?

Take square root.

How do you get length factor from volume factor?

Take cube root.

What information describes a rotation?

Centre, angle and direction.

What information describes an enlargement?

Centre and scale factor.

How do you translate a point by vector (a,b)?

(x,y) → (x+a,y+b).

What happens in enlargement with negative scale factor?

Image lies opposite side of centre; distance multiplied by absolute value of factor.

What is area of a trapezium?

A = (a+b)h/2, with a and b parallel.

What is circumference of a circle?

C = 2πr = πd.

What is area of a circle?

A = πr².

What is arc length for angle θ in degrees?

L = (θ/360) × 2πr.

What is sector area for angle θ in degrees?

A = (θ/360) × πr².

What is volume of a prism?

Cross-sectional area × length.

What is volume of a cylinder?

πr²h.

What is total surface area of a closed cylinder?

2πr² + 2πrh.

What is volume of a pyramid?

Base area × perpendicular height ÷ 3.

What is volume of a cone?

πr²h/3.

What is curved surface area of a cone?

πrl, where l is slant height.

What is volume of a sphere?

4πr³/3.

What is surface area of a sphere?

4πr².

What is volume of a hemisphere?

2πr³/3.