CAIE IGCSE Mathematics Extended Algebra and Graphs Question Bank

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: 3]

(a)

Another club sells season tickets for individuals and for families. In 2018, the number of season tickets sold is in the ratio family: individual =2: 7.

[ 3 ]

(i)

The number of family season tickets sold is x.

Write an expression, in terms of x, for the number of individual season tickets sold.

[ 1 ]

$\frac{7x}{2}$ oe

(ii)

Figure for Question (ii) — CAIE IGCSE Mathematics Extended

In 2019, the number of family season tickets sold increases by 12 and the number of individual season tickets sold decreases by 26.
Complete the table by writing expressions, in terms of x, for the number of tickets sold in 2019.

[ 2 ]

x+12, $\frac{7x}{2}-26$ oe final answer
B1 for x+12
B1 for their $\frac{7x}{2}-26$

[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
M2
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
B1

(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 $y^2-6y+5[=0]$
M2
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
B2
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: 17]

(a)

Solve.
4x+15=9x=

[ 2 ]

-1.5 or $-1\frac12$ or $-\frac32$
M1 for 4x=9-15 or $x+\frac{15}{4}=\frac94$

(b)

Factorise.
$a^2-9$

[ 1 ]

(c)

Write as a single fraction in its simplest form.
$\frac{4a}{5}\div\frac{3ad}{10c}$

[ 3 ]

$\frac{8c}{3d}$ final answer
B2 for $\frac{8ac}{3ad}$ or $\frac{40c}{15d}$
or M1 for $\frac{4a}{5}\times\frac{10c}{3ad}$ oe

(d)

$5^n+5^n+5^n+5^n+5^n=5^m$
Find an expression for m in terms of n.

n+1 final answer
M1 for $5\times5^n$ or $5^{n+1}$ seen

(e)

Solve by factorisation.
$4x^2+8x-5=0$ $x=\qquad$ or $x=\qquad$

[ 3 ]

(2x-1)(2x+5)=0 oe
B2
$x=\frac12$ and $x=-\frac52$
B1

(f)

y is directly proportional to $(x+3)^3$ .
When x=2, y=13.5.
Find x when y=108.
x=

[ 3 ]

7
M1 for $y=k(x+3)^3$ or better
M1 for $108=\text{their }k(x+3)^3$

(g)

g is inversely proportional to the square of d.
When d is halved, the value of g is multiplied by a factor n.
Find n.

4
M1 for $\left(\frac12\right)^2$ oe or equivalent inverse-square working

(h)

Expand and simplify.
(2x+3)(x-1)(x+3)

[ 3 ]

$2x^3+7x^2-9$ final answer
B2 for correct expansion unsimplified or simplified four-term expression with three terms correct
or B1 for one pair of brackets expanded

(i)

Find the derivative, $\frac{dy}{dx}$ , of $y=3x^2+4x-1$ .

[ 2 ]

[Maximum number: 1]

$2p^{1/3}=6$
Find the value of p.
p=

[Maximum number: 8]

(a)

Expand and simplify.

4(2 x-1)-6(3-x)

[ 2 ]

(b)

Factorise completely.

[ 2 ]

(i)

6 x^2 y+9 x y

[ 2 ]

3xy(2x+3) final answer
M1 for answer $3(2x^2y+3xy)$ , 3x(2xy+3y), $3y(2x^2+3x)$ , or xy(6x+9)
B1 for correct answer seen and spoilt

(c)

Antonio travels 100 km at an average speed of $x \mathrm{~km} / \mathrm{h}$ .

He then travels a further 150 km at an average speed of (x+10) km / h.
The time taken for the whole journey is 4 hours 20 minutes.

[ 4 ]

(i)

Show that $13x^2-620x-3000=0$ .

[ 4 ]

$\frac{100}{x}+\frac{150}{x+10}=4\frac13$ oe or $150=\left(\frac{13}{3}-\frac{100}{x}\right)(x+10)$
M1
$\frac{100(x+10)+150x}{x(x+10)}$ or better
M1
$300x+3000+450x=13x^2+130x$ oe or better
B1
Allow correct multiples of $13x^2-620x-3000=0$

(ii)

Solve $13x^2-620x-3000=0$ to find the speed Antonio travels for the first 100 km of the journey.
You must show all your working and give your answer correct to 1 decimal place.

$\frac{620\pm\sqrt{(-620)^2-4(13)(-3000)}}{2(13)}$
or equivalent quadratic formula
M2
M1 for $(-620)^2-4\times13\times(-3000)$ oe
x=52.1 to 52.2 final answer
B1
Reject negative speed

[Maximum number: 6]

(a)

Figure for Question (a) — CAIE IGCSE Mathematics Extended

Write down the inequality shown by the number line.

[ 1 ]

$-2<x\leqslant4$ oe

(b)

$-3\leqslant2x+3<9$

[ 5 ]

(i)

Solve the inequality.

[ 3 ]

$-3\leqslant x<3$ final answer
M2 for correct intermediate inequality; SC1 for one side correct

(ii)

Write down all the integer values of x that satisfy the inequality.

[ 2 ]

(c)

Solve the equations.

(i)

$\frac{3(3-x)-2(x+2)}{5}=1$ x=

$\frac{36}{17}$ oe
B3 for -15x-2x=5+4-45 or better; lower method marks as in markscheme

(ii)

$\frac{5}{x+3}=\frac{3}{x+5}$ x=

All question bank results loaded

Covered syllabus points

2. Algebra and graphs question 1

Question (a)

Question (i)

Question (ii)

2. Algebra and graphs question 2

Question (a)

Question (i)

Question (ii)

Question (iii)

Question (b)

Question (c)

Question (d)

2. Algebra and graphs question 3

Question (a)

Question (b)

Question (c)

Question (d)

Question (e)

Question (f)

Question (g)

Question (h)

Question (i)

2. Algebra and graphs question 4

2. Algebra and graphs question 5

Question (a)

Question (b)

Question (i)

Question (c)

Question (i)

Question (ii)

2. Algebra and graphs question 6

Question (a)

Question (b)

Question (i)

Question (ii)

Question (c)

Question (i)

Question (ii)