May/June 2026 Paper 12

2026 · 10 questions · 19 parts · 75 marks

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2(a)2(b)

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4(a)(i)4(a)(ii)4(b)

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10(a)10(b)(i)10(b)(ii)

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10(a)Quadratic Equations InequalitiesMedium3 marks

A function

f

is defined by

f (x) = p x^{2} - 2 x + p + 4

for

x \in R

, where

p

is a constant. Find the exact values of

p

for which the equation

f (x) = 0

has repeated roots.

Show your working

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Smaller value of p

Larger value of p

Context for 10(b)

A function

g

is defined by

g (x) = p x^{2} - 2 x + p + 4

for

x \geq a

, where

p

and

a

are positive constants.

10(b)(i)Functions Composites InversesEasy2 marks

Show that the least value of

a

for which the function

g^{- 1}

exists is

\frac{1}{p}

Draw with the mouse or a finger.

10(b)(ii)Functions Composites InversesMedium4 marks

Given that

p = 2

and

g^{- 1}

does exist, find an expression for

g^{- 1} (x)

Show your working

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g^{- 1} (x) =