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CAIE IGCSE Additional Mathematics Algebra and Functions Question Bank

CAIE IGCSE Additional Mathematics Algebra and Functions Question Bank
Cambridge IGCSE Additional Mathematics 0606 syllabus for exams in 2028, 2029 and 2030Exams 2028-2030

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

Covered syllabus points

1. Functions2. Quadratic functions3. Factors of polynomials4. Equations, inequalities and graphs5. Simultaneous equations6. Logarithmic and exponential functions

Question 1

[Maximum number: 5]

It is given that p(x)=ax3−7x2−bx+9p(x)=ax^3-7x^2-bx+9, where a and b are constants.
x-3 is a factor of p(x).
When p(x) is divided by x+2, the remainder is -35.
Find the values of a and b.

Question 1

[Maximum number: 5]

1 A curve has equation y=x2+2x−3y=x^{2}+2 x-3 .

Question 1(a)

(a)

(a)Use the method of completing the square to find the coordinates of the stationary point on the

curve.

[ 3 ]

Question 1(b)

(b)

(b)On the axes,sketch the curve,stating the intercepts with the coordinate axes.

Figure for Question 1(b) — CAIE IGCSE Additional Math
[ 2 ]

Question 1

[Maximum number: 7]

Solve the following inequalities.

Question 1(a)

(a)

x2−x−6⩾0x^2-x-6 \geqslant 0

[ 3 ]

Question 1(b)

(b)

|3x-4|<x+2

[ 4 ]

Question 2

[Maximum number: 8]

The polynomial p(x)=6x3+ax2+bx+2\mathrm{p}(x)=6 x^{3}+a x^{2}+b x+2, where a and b are integers, has a factor of x-2.

Question 2(a)

(a)

Given that p(1)=-2 p(0), find the values of a and b.

[ 4 ]

Question 2(b)

(b)

Using your values of a and b,

[ 4 ]

Question 2(b)(i)

(i)

find the remainder when p(x) is divided by 2 x-1

[ 2 ]

Question 2(b)(ii)

(ii)

factorise p(x).

[ 2 ]

Question 2

Question 2(a)(ii)

(a)

Describe the relationship between the graph of f(x) and the graph of f−1(x)f^{-1}(x).

[ 1 ]

Question 2(b)

(b)

A function g is defined by g(x)=ex−2g(x)=e^{\sqrt{x-2}} for x⩾2x\geqslant 2.

[ 4 ]

Question 2(b)(i)

(i)

Find an expression for g−1(x)g^{-1}(x).

Question 2(b)(ii)

(ii)

Write down the range of g−1g^{-1}.

[ 3 ]

Question 2(b)(iii)

(iii)

A function h is defined by h(x)=1x2+2h(x)=\frac{1}{x^2}+2 for x>0.
Find an expression for gh(x) in its simplest form.

[ 1 ]

Question 5

Question 5(a)

(a)

Solve the equation 625x3−12125x3=5\frac{625^{\frac{x^3-1}{2}}}{125^{x^3}}=5.

[ 3 ]

Question 5(b)

(b)
Figure for Question 5(b) — CAIE IGCSE Additional Math

On the axes, sketch the graph of y=4ex+3y=4e^x+3, showing the values of any intercepts with the coordinate axes.

[ 2 ]

Question 4

Question 4(a)

(a)
Figure for Question 4(a) — CAIE IGCSE Additional Math

Explain why this graph does not represent a function.

[ 1 ]

Question 4(b)

(b)

The table shows the graphs of four different functions.

Table for Question 4(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 ]

Question 4(c)

(c)

Functions f and g are defined as follows.
f: x↦sin⁡xx \mapsto \sin x \quad for 30∘⩽x⩽a∘30^{\circ} \leqslant x \leqslant a^{\circ}
g: x↦x−12x \mapsto \sqrt{x-\frac{1}{2}} \quad for x⩾12x \geqslant \frac{1}{2}
It is given that the function gf exists.

[ 5 ]

Question 4(c)(i)

(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∘=12\sin 30^{\circ}=\frac{1}{2}.

[ 2 ]

Question 4(c)(ii)

(ii)

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

[ 2 ]

Question 4(c)(iii)

(iii)

Determine whether the function g2\mathrm{g}^{2} exists.

[ 1 ]

Question 4

Question 4(a)

(a)

Solve the equation x13−x16=2x^{\frac{1}{3}}-x^{\frac{1}{6}}=2.

[ 4 ]

Question 4(b)

(b)

Solve the simultaneous equations

lg⁡(x+2y)=0x2+4xy+y=1\begin{array}{r} \lg (x+2 y)=0 \\ x^{2}+4 x y+y=1 \end{array}
[ 5 ]

Question 3

Question 3(a)(i)

(a)

Write x2−x−6x^{2}-x-6 in the form (x+a)2+b(x+a)^{2}+b where a and b are constants.

[ 2 ]

Question 3(a)(ii)

(b)

Hence write down the coordinates of the stationary point on the curve y=x2−x−6y=x^{2}-x-6.

[ 2 ]

Question 3(b)

(c)

On the axes, draw the graph of y=∣x2−x−6∣y=|x^2-x-6| for −4⩽x⩽4-4\leqslant x\leqslant 4.

Figure for Question 3(b) — CAIE IGCSE Additional Math
[ 3 ]

Question 3(c)

(d)

Use your graph to solve the inequality ∣x2−x−6∣<4\left|x^{2}-x-6\right|<4.

[ 2 ]

Question 6

[Maximum number: 5]

the equation ∣2x2+x−10∣=5\left|2 x^{2}+x-10\right|=5.