CAIE IGCSE Mathematics Extended 2 Algebra and Graphs Questions

Practise algebraic manipulation, equations, inequalities, sequences, functions and coordinate or practical graphs across CAIE IGCSE Mathematics.

Covered syllabus points

E2.1 Introduction to algebra E2.2 Algebraic manipulation E2.3 Algebraic fractions E2.4 Indices II E2.5 Equations E2.6 Inequalities E2.7 Sequences E2.8 Proportion E2.9 Graphs in practical situations E2.10 Graphs of functions E2.11 Sketching curves E2.12 Differentiation E2.13 Functions

[Maximum number: 9]

Gaya spends $ 48 to buy books that cost $ x each.

(a)

Write down an expression, in terms of x, for the number of books Gaya buys.

[ 1 ]

$\frac{48}{x}$ final answer

Accept $48\div x$

(b)

Myra spends $ 60 to buy books that cost $(x+2) each. Gaya buys 4 more books than Myra.

Show that $x^{2}+5 x-24=0$ .

[ 4 ]

their (a) $-\frac{60}{x+2}=4$ oe
48(x+2)-60x=4x(x+2) oe
$48x+96-60x=4x^2+8x$ oe leading to $x^2+5x-24=0$

(c)

Solve by factorisation.

x^{2}+5 x-24=0

or x=

[ 3 ]

(d)

Find the number of books Myra buys.

[ 1 ]

[Maximum number: 16]

(a)

Simplify.

[ 6 ]

(i)

3 m-5 n-4 m+8 n

[ 2 ]

(ii)

$\left(3 a^{2} c^{3}\right)^{4}$

[ 2 ]

$81a^8c^{12}$ final answer
B1 for final answer in correct form with any two of 81, $a^8$ , $c^{12}$ correct
or for $81a^8c^{12}$ seen and then spoiled

(iii)

$\frac{4 x}{5}-\frac{3 x}{10}+\frac{2 x}{15}$

[ 2 ]

$\frac{19x}{30}$ final answer
M1 for $\frac{6\times4x-3\times3x+2\times2x}{30}$ oe

(b)

This isosceles triangle has a perimeter of 35.5 cm .

NOT TO SCALE

Find the value of a.
a=

[ 3 ]

(c)

Using the quadratic formula, solve $5 x^{2}-4 x-3=0$ .

You must show all your working.

x=\ldots \ldots \ldots \ldots \ldots \ldots . \text { or } x=\ldots \ldots \ldots \ldots \ldots \ldots . .[3]

[ 3 ]

$\frac{-(-4)\pm\sqrt{(-4)^2-4\times5\times(-3)}}{2\times5}$ oe
M1 for $(-4)^2-4\times5\times(-3)$ or better
or for $\frac{-(-4)+q}{2\times5}$ or $\frac{-(-4)-q}{2\times5}$ or better

-0.472 or -0.4718 to -0.4717
and 1.27 or 1.271 to 1.272

(d)

Solve these simultaneous equations.

\begin{aligned} & y=x^{2}-4 x+5 \\ & y=2 x-3 \end{aligned}

You must show all your working.

\begin{aligned} & x=\ldots \ldots \ldots \ldots \ldots \ldots . y=\ldots \ldots \ldots \ldots \ldots \ldots . \\ & x=\ldots \ldots \ldots \ldots \ldots \ldots . y=\ldots \ldots \ldots \ldots \ldots \ldots .[5] \end{aligned}

[ 4 ]

$x^2-6x+8[=0]$
or $y^2-6y+5[=0]$
M1 for $x^2-4x+5=2x-3$
or $y=\left(\frac{y+3}{2}\right)^2-4\left(\frac{y+3}{2}\right)+5$ (x-4)(x-2)[=0]
or (y-1)(y-5)[=0]
OR quadratic formula method
M1 FT their 3-term quadratic but not if $x^2-4x+5[=0]$ [x=]2, [y=]1[x=]4, [y=]5
B1 for one correct pair or two correct x-values or two correct y-values
If B0 scored and at least 2 method marks scored, SC1 for correct substitution of both of their values.

[Maximum number: 3]

Write as a single fraction in its simplest form.

\frac{3}{x-4}-\frac{4}{x+3}

$\frac{25-x}{(x-4)(x+3)}$
oe final answer

M1 for denominator (x-4)(x+3) or $x^2-x-12$ .
B1 for 3(x+3)-4(x-4) seen.

[Maximum number: 6]

(a)

Simplify $\left(\frac{x^{2}}{4}\right)^{\frac{3}{2}}$ .

[ 2 ]

[ $\pm$ ] $\frac{x^3}{8}$ final answer
B1 for $\frac{x^3}{k}$ or $\frac{x^k}{8}$

(b)

$16^{x} \times\left(\frac{1}{2}\right)^{x}=4^{x+3}$

Find the value of x.

[ 4 ]

6 nfww
B3 for 4x-x=2(x+3) oe
or for correctly combining to a single base on each side with no brackets in the powers
OR M1 for $(2^4)^x$ or $(2^{-1})^x$ or $2^{2(x+3)}$ or better seen
M1 for correctly forming a linear equation in x from their powers of their consistent base

All question bank results loaded

Covered syllabus points

Question 1

Question (a)

Question (b)

Question (c)

Question (d)

Question 2

Question (a)

Question (i)

Question (ii)

Question (iii)

Question (b)

Question (c)

Question (d)

Question 3

Question 4

Question (a)

Question (b)