CAIE IGCSE Additional Math Algebra and Functions Questions

Use this algebra-and-functions unit hub to connect function structure, quadratics and polynomials with equations, inequalities, simultaneous systems and logarithms.

Exam points

Analyse functions, quadratics and polynomial factors using domains, ranges, extrema, discriminants and theorems.
Solve modulus, quadratic, polynomial and simultaneous equations or inequalities with valid roots and intervals.
Use logarithmic and exponential graphs, laws and equations with domain checks.

Covered syllabus points

1. Functions 2. Quadratic functions 3. Factors of polynomials 4. Equations, inequalities and graphs 5. Simultaneous equations 6. Logarithmic and exponential functions

[Maximum number: 10]

(a)

Figure for Question (a) — CAIE IGCSE Additional Math

Explain why this graph does not represent a function.

[ 1 ]

(b)

The table shows the graphs of four different functions.

Table for Question (b) — CAIE IGCSE Additional Math

Tick $(\checkmark)$ each correct box in the table.
There may be more than one tick in a row or a column.

[ 4 ]

(c)

Functions f and g are defined as follows.
f: $x \mapsto \sin x \quad$ for $30^{\circ} \leqslant x \leqslant a^{\circ}$
g: $x \mapsto \sqrt{x-\frac{1}{2}} \quad$ for $x \geqslant \frac{1}{2}$
It is given that the function gf exists.

[ 5 ]

(i)

Find the value of a so that the domain of gf is as large as possible.

You may use the information that $\sin 30^{\circ}=\frac{1}{2}$ .

[ 2 ]

$\sin x \geqslant \frac{1}{2}$
May be implied. Allow >

a=150 seen

(ii)

For the domain found in part (i), find the range of the function gf .

[ 2 ]

$0 \leqslant \operatorname{gf}(x) \leqslant \frac{1}{\sqrt{2}}$

B1 for each correct value with correct sign associated with it

(iii)

Determine whether the function $\mathrm{g}^{2}$ exists.

[ 1 ]

No. Range of $\mathrm{g} \not \subset$ domain of g oe

[Maximum number: 10]

(a)

Show that $2 x^{2}+5 x+3$ can be written in the form $2(x+a)^{2}+b$ , where a and b are constants to be found.

[ 2 ]

$2\left(x+\frac{5}{4}\right)^{2}-\frac{1}{8}$ B1 for $2\left(x+\frac{5}{4}\right)^{2}$

B1 for $-\frac{1}{8}$

(b)

Hence write down the coordinates of the stationary point on the curve $y=2 x^{2}+5 x+3$ .

[ 2 ]

$\left(-\frac{5}{4},-\frac{1}{8}\right)$ 2 B1FT on their - a

B1FT on their b

B0 if calculus used as question says ‘Hence’

(c)

function f is such that $\mathrm{f}(x)=2 x^{2}+5 x+3$ , for $x \geqslant p$ , where p is a constant.
It is given that $\mathrm{f}^{-1}$ exists.

[ 6 ]

(i)

Write down the least possible value of p.

[ 1 ]

$-\frac{5}{4}$ or $p=-\frac{5}{4}$ or $x \geqslant-\frac{5}{4}$ B1 FT on their -a from part (a).
Allow if from calculus

(ii)

Using your value of p, sketch the graphs of $y=\mathrm{f}(x)$ and $y=\mathrm{f}^{-1}(x)$ .
Label each graph.
State the intercepts of each of the graphs with the axes.

Figure for Question (ii) — CAIE IGCSE Additional Math

[ 5 ]

Figure for Answer to Question (ii) — CAIE IGCSE Additional Math

B1 for $y=\mathrm{f}(x)$ drawn as a one-one function in the first, second and third quadrants or first and second quadrants.

Must have correct curvature

Dep B1 for $y=\mathrm{f}^{-1}(x)$ as a reflection of their $y=\mathrm{f}(x)$ in the line y=x. Maybe implied by intercepts, but must have the correct curvature

B1 for $y=\mathrm{f}(x)$ clearly passing through x=-1 correctly soi. Allow if domain is unrestricted but must have correct quadratic shape or curvature

B1 for $y=\mathrm{f}(x)$ passing through y=3 only on the y-axis soi. Allow if domain is unrestricted but must have correct quadratic shape or curvature

B1 for - 1 and 3 marked correctly for $y=\mathrm{f}^{-1}(x)$ allow for unrestricted range.

Must have correct curvature.

[Maximum number: 5]

The polynomial p is such that $\mathrm{p}(x)=x^{3}+a x^{2}+b x-2, \quad$ where a and b are constants.
It is given that:
- x+2 is a factor of p(x)
- when p(x) is divided by x-3 the remainder is 40.

Find the values of a and b.

[Maximum number: 5]

the equation $\left|2 x^{2}+x-10\right|=5$ .

$2 x^{2}+x-10=5$

leading to $2 x^{2}+x-15=0$ with a valid attempt to solve to obtain 2 values for x M1
$x=\frac{5}{2}, \quad-3$ A1 For both
Mark final answer for this equation,
A0 if - 3 is rejected
$x^{2}+x-10=-5$

leading to $2 x^{2}+x-5=0$ M1 Must be a correct 3-term quadratic
equated to zero
$x=\frac{-1 \pm \sqrt{41}}{4}$ oe 2 Dep M1 for a valid method of
solution to obtain 2 values for x.
Mark final answer for this equation,
A0 if negative root is rejected
Denominator of final answer must be positive

All question bank results loaded

Covered syllabus points

Question 1

Question (a)

Question (b)

Question (c)

Question (i)

Question (ii)

Question (iii)

Question 2

Question (a)

Question (b)

Question (c)

Question (i)

Question (ii)

Question 3

Question 4