CAIE IGCSE Additional Mathematics Algebra and Functions Question Bank

Practise functions, quadratics, polynomial factors, modulus and simultaneous equations, then apply logarithmic and exponential methods across mixed algebra problems.

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: 2]

Figure for Question Algebra and Functions question 1 — CAIE IGCSE Additional Math

The diagram shows the graph of $f(x)=a\cos bx+c$ , for $0\leqslant x\leqslant\frac{8\pi}{3}$ radians.

(a)

Explain why f is a function.

[ 1 ]

(b)

Write down the range of f.

[ 1 ]

$-5\leqslant f(x)\leqslant1$
B1

[Maximum number: 6]

(a) Find the coordinates of the stationary point on the curve y=(x+3)(x-4).

(a)

Find the coordinates of the stationary point on the curve y=(x+3)(x-4).

[ 3 ]

$\begin{aligned} y=x^{2}-x-12 \frac{\mathrm{~d} y}{\mathrm{~d} x}=2 x-1 \text { or }(x+3)+(x-4) \text { or } y=\left(x-\frac{1}{2}\right)^{2}-\frac{49}{4} \text { or using symmetry } x=\frac{4-3}{2} \end{aligned}$

M1

For expanding the brackets and
differentiate with at least one correct term
or for using the product rule
or for completing the square
or for using symmetry

$x=\frac{1}{2}$

A1

$y=-\frac{49}{4}$ oe

A1

(b)

On the axes, sketch the graph of y=|(x+3)(x-4)|, stating the intercepts with the axes.

Figure for Question (b) — CAIE IGCSE Additional Math

[ 2 ]

(c)

Given that k>0, write down the values of k for which the equation |(x+3)(x-4)|=k has exactly 2 distinct real roots.

[ 1 ]

$k>\frac{49}{4}$ oe

B1

FT on $\mid$ their $\left.-\frac{49}{4} \right\rvert\,$ excluding k>12

[Maximum number: 6]

The polynomial p is given by $\mathrm{p}(x)=a^{2} x^{3}+2 a x^{2}+a x+2$ , where a is a positive integer. It is given that 2 x+1 is a factor of p(x).

(a)

Find the value of a.

[ 3 ]

$a^{2}\left(-\frac{1}{2}\right)^{3}+2 a\left(-\frac{1}{2}\right)^{2}+a\left(-\frac{1}{2}\right)+2=0$ oe, soi

M1

Simplifies and solves for a

M1

FT their quadratic in a providing $\mathrm{p}\left(-\frac{1}{2}\right)=0$
oe attempted

a=4

A1

(b)

Hence factorise p(x).

[ 2 ]

$2(2 x+1)\left(4 x^{2}+1\right)$ mark final answer

M1 for $(2 x+1)\left(\frac{\text { their } a^{2}}{2} x^{2} \ldots+2\right)$
or $\left(8 x^{2}+2\right)$ found as quadratic factor

(c)

Hence show that the equation p(x)=0 has only one real root.

[ 1 ]

Discriminant of $4 x^{2}+1$ is 0-16<0 oe and no real roots oe or $4 x^{2}+1=0 \rightarrow 4 x^{2}=-1$ oe and therefore no solution oe

[statement that only solution is $x=-\frac{1}{2}$ ]

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Covered syllabus points

Algebra and Functions question 1

Question (a)

Question (b)

Algebra and Functions question 2

Question (a)

Question (b)

Question (c)

Algebra and Functions question 3

Question (a)

Question (b)

Question (c)