Algebra & Surds Test

0 of 29 parts answered

Test

Time allowed: 80 minutesTotal: 68 marks12 questions29 September 2026

Candidate: Hana

Instructions

Answer all questions.
Write your answer to each part on the answer line. Use the maths keys that appear at the bottom of the screen for √, powers and other symbols.
Show all your working. You can earn marks for correct method even if your final answer is wrong.
You must not use a calculator.
Give answers in exact form — fractions or surds — unless the question says otherwise.
The number of marks for each question or part question is shown in brackets [ ].

Question 1

(a)

Solve the simultaneous equations. Show all your working.

3 x + 2 y = 18

5 x - 2 y = 14

x =

y =

[3]

Question 2

(a)

Solve the simultaneous equations. Show all your working.

2 x + 5 y = 1

3 x - 4 y = 13

x =

y =

[4]

Question 3

At a café, 3 coffees and 2 cakes cost $19. 1 coffee and 4 cakes cost $18.

(a)

Write down two equations, using

c

for the cost of a coffee and

k

for the cost of a cake, in dollars.

[2]

(b)

Solve your equations to find the cost of one coffee and the cost of one cake.

Coffee $

Cake $

[3]

Question 4

(a)

Factorise

x^{2} - x - 20

[2]

(b)

Hence solve

x^{2} - x - 20 = 0

x =

x =

[1]

(c)

Solve

2 x^{2} + 7 x + 3 = 0

by factorising.

x =

x =

[3]

Question 5

(a)

Write

x^{2} + 10 x + 7

in the form

(x + p)^{2} + q

p =

q =

[2]

(b)

Write

x^{2} - 7 x + 2

in the form

(x + p)^{2} + q

p =

q =

[2]

Question 6

(a)

Write

4 x^{2} + 24 x - 5

in the form

a (x + b)^{2} + c

a =

b =

c =

[3]

(b)

Write

2 x^{2} - 6 x + 1

in the form

a (x + b)^{2} + c

a =

b =

c =

[3]

Question 7

(a)

By completing the square, solve

x^{2} + 6 x - 3 = 0

. Give your answers in the form

a \pm b 3

x =

x =

[3]

(b)

By completing the square, solve

2 x^{2} - 8 x + 3 = 0

. Leave your answers in surd form.

x =

x =

[4]

Question 8

(a)

Find the value of the discriminant of

2 x^{2} - 5 x + 4 = 0

[2]

(b)

Hence write down the number of real roots of

2 x^{2} - 5 x + 4 = 0

[1]

(c)

Without solving the equation, show that

4 x^{2} - 12 x + 9 = 0

has one repeated root.

[2]

(d)

Use the quadratic formula to solve

2 x^{2} + 3 x - 4 = 0

. Give your answers in the form

\frac{a \pm b}{c}

x =

x =

[3]

Question 9

(a)

The equation

x^{2} + k x + 9 = 0

has one repeated root. Find the two possible values of

k

k =

k =

[3]

(b)

The equation

x^{2} + 4 x + k = 0

has two different real roots. Find the range of values of

k

[2]

Question 10

(a)

Simplify

50

[1]

(b)

Simplify

72

[1]

(c)

Write

243 - 48

in the form

k 3

[2]

(d)

Simplify

23 \times 56

, giving your answer in the form

k 2

[2]

Question 11

(a)

Expand and simplify

(4 + 3) (2 - 3)

[2]

(b)

Expand and simplify

(6 + 5) (6 - 5)

[2]

(c)

Expand

(6 - 2)^{2}

, giving your answer in the form

a + b 2

a =

b =

[2]

Question 12

(a)

Rationalise the denominator and simplify

\frac{12}{3}

[2]

(b)

Rationalise the denominator and simplify

\frac{4}{3 - 5}

[3]

(c)

Write

\frac{1 + 2}{3 - 2}

in the form

\frac{a + b 2}{c}

, where

a

b

and

c

are integers.

[3]

End of paper

Mark scheme

Algebra & Surds Test · 68 marks

Question	Answer	Marks	Partial marks
1(a)	$x = 4$ , $y = 3$	3	M1 for correctly eliminating one variable, e.g. $8 x = 32$ A1 for $x = 4$ A1 for $y = 3$
2(a)	$x = 3$ , $y = - 1$	4	M1 for multiplying one or both equations so that the coefficients of $x$ (or $y$ ) are equal, e.g. $6 x + 15 y = 3$ and $6 x - 8 y = 26$ M1 for correctly subtracting (or adding) to eliminate a variable, e.g. $23 y = - 23$ A1 for $x = 3$ A1 for $y = - 1$
3(a)	$3 c + 2 k = 19$ and $c + 4 k = 18$	2	B1 for each correct equation
3(b)	Coffee $4, cake $3.50	3	M1 for a correct method to eliminate one variable (FT their equations) A1 for $c = 4$ A1 for $k = 3.5$
4(a)	$(x - 5) (x + 4)$	2	M1 for $(x + a) (x + b)$ where $ab = - 20$ or $a + b = - 1$
4(b)	$5$ and $- 4$ FT their (a)	1
4(c)	$- \frac{1}{2}$ and $- 3$	3	M2 for $(2 x + 1) (x + 3)$ or M1 for $(2 x + a) (x + b)$ where $ab = 3$ or $a + 2 b = 7$
5(a)	$(x + 5)^{2} - 18$	2	B1 for $p = 5$ B1 for $q = - 18$
5(b)	$(x - \frac{7}{2})^{2} - \frac{41}{4}$ oe	2	B1 for $p = - \frac{7}{2}$ B1 for $q = - \frac{41}{4}$
6(a)	$4 (x + 3)^{2} - 41$	3	B1 for $a = 4$ B1 for $b = 3$ B1 for $c = - 41$
6(b)	$2 (x - \frac{3}{2})^{2} - \frac{7}{2}$ oe	3	B1 for $a = 2$ B1 for $b = - \frac{3}{2}$ B1 for $c = - \frac{7}{2}$
7(a)	$- 3 + 23$ and $- 3 - 23$	3	M1 for $(x + 3)^{2} - 9 - 3$ oe M1 for $(x + 3)^{2} = 12$ or $x + 3 = \pm 12$ A1 for both answers — accept $- 3 \pm 12$ for M2 only
7(b)	$\frac{4 + 10}{2}$ and $\frac{4 - 10}{2}$ oe	4	M1 for $x^{2} - 4 x + \frac{3}{2} = 0$ or $2 (x^{2} - 4 x) + 3$ M1 for $(x - 2)^{2} - 4$ seen M1 for $(x - 2)^{2} = \frac{5}{2}$ oe A1 for both answers, e.g. $2 \pm \frac{5}{2}$
8(a)	$- 7$	2	M1 for $(- 5)^{2} - 4 \times 2 \times 4$
8(b)	0 (no real roots) FT their (a)	1
8(c)	$b^{2} - 4 a c = (- 12)^{2} - 4 \times 4 \times 9 = 144 - 144 = 0$ , so one repeated root	2	M1 for $(- 12)^{2} - 4 \times 4 \times 9$ A1 for $= 0$ with the conclusion
8(d)	$\frac{- 3 + 41}{4}$ and $\frac{- 3 - 41}{4}$	3	M1 for $\frac{- 3 \pm 3 ^{2} - 4 \times 2 \times ( - 4 )}{2 \times 2}$ B1 for 41 seen as the discriminant
9(a)	$6$ and $- 6$	3	M1 for $k^{2} - 4 \times 1 \times 9 = 0$ M1 for $k^{2} = 36$ or B1 for one correct value only
9(b)	$k < 4$	2	M1 for $4^{2} - 4 \times 1 \times k > 0$ or B1 for $k = 4$ or $k \leq 4$
10(a)	$52$	1
10(b)	$62$	1
10(c)	$53$	2	M1 for $93$ or $43$ seen
10(d)	$302$	2	M1 for $1018$ or $18 = 32$ seen
11(a)	$5 - 23$	2	M1 for three of $8$ , $- 43$ , $+ 23$ , $- 3$ correct
11(b)	31	2	M1 for $36 - 65 + 65 - 5$ or $36 - 5$
11(c)	$38 - 122$	2	M1 for $36 - 62 - 62 + 2$ oe
12(a)	$43$	2	M1 for $\frac{12 3}{3}$
12(b)	$3 + 5$	3	M1 for multiplying by $\frac{3 + 5}{3 + 5}$ B1 for denominator $9 - 5 = 4$
12(c)	$\frac{5 + 4 2}{7}$	3	M1 for multiplying by $\frac{3 + 2}{3 + 2}$ M1 for numerator $3 + 2 + 32 + 2$ or denominator 7

M method mark · A accuracy mark (depends on the M mark) · B mark for a correct result or statement, independent of method · oe or equivalent · cao correct answer only · FT follow through after an earlier error · isw ignore subsequent working

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