Algebra & Surds Challenge (Calculator)

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Algebra & Surds Challenge (Calculator)

Time allowed: 75 minutesTotal: 57 marks8 questions

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 should use a calculator where appropriate.
Give non-exact answers correct to 3 significant figures, or 1 decimal place for angles in degrees, 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.

1.5 x + 2.4 y = 9.3

0.5 x - 1.2 y = - 0.9

x =

y =

[4]

Question 2

A shop sells small bottles that hold 0.75 litres and large bottles that hold 2 litres. A customer buys 12 bottles, which hold 15.25 litres altogether.

(a)

Write down two equations, using

s

for the number of small bottles and

l

for the number of large bottles.

[2]

(b)

Solve your equations to find how many of each size the customer bought.

Small:

Large:

[3]

(c)

A small bottle costs $1.35 and a large bottle costs $2.80. Work out the total cost.

[1]

Question 3

(a)

Solve

2 x^{2} - 7 x + 1 = 0

. Give your answers correct to 3 significant figures.

x =

x =

[4]

(b)

Solve

0.4 x^{2} + 1.3 x - 2 = 0

. Give your answers correct to 3 significant figures.

x =

x =

[4]

Question 4

A rectangular garden is 4 m longer than it is wide. The width is

x

m and the area is 50 m

^{2}

(a)

Show that

x^{2} + 4 x - 50 = 0

[2]

(b)

Solve the equation to find the width of the garden. Give your answer correct to 3 significant figures.

[3]

(c)

Work out the perimeter of the garden. Give your answer correct to 3 significant figures.

[2]

Question 5

(a)

Write

2 x^{2} + 7 x - 3

in the form

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

a =

b =

c =

[3]

(b)

Hence solve

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

. Give your answers correct to 2 decimal places.

x =

x =

[3]

Question 6

(a)

Calculate the discriminant of

1.2 x^{2} - 3.5 x + 2.6 = 0

and hence state the number of real roots.

Discriminant =

Number of real roots =

[3]

(b)

The equation

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

has one repeated root, where

k > 0

. Find the value of

k

k =

[3]

(c)

The equation

x^{2} + 3 x + 1 = k

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

k

[3]

Question 7

Show all your working in this question. Give exact answers — do not use decimals.

(a)

Expand

(4 - 7)^{2}

, giving your answer in the form

a + b 7

a =

b =

[2]

(b)

Write

\frac{5 + 7}{3 - 7}

in the form

a + b 7

a =

b =

[3]

(c)

p = 3 + 2

and

q = 3 - 2

. Find the exact value of

p^{2} + q^{2}

[2]

(d)

Using the values of

p

and

q

from part (c), write

\frac{p}{q}

in the form

\frac{a + b 2}{c}

[3]

Question 8

A ball is thrown upwards. After

t

seconds its height,

h

metres, is

h = 20 t - 4.9 t^{2} + 1.5

(a)

Find the two times when the ball is at a height of 15 m. Give your answers correct to 3 significant figures.

t =

and

t =

[4]

(b)

Use the discriminant to show that the ball never reaches a height of 25 m.

[3]

End of paper

Mark scheme

Algebra & Surds Challenge (Calculator) · 57 marks

Question	Answer	Marks	Partial marks
1(a)	$x = 3$ , $y = 2$	4	M1 for multiplying so that the coefficients of one variable are equal, e.g. $x - 2.4 y = - 1.8$ M1 for correctly adding (or subtracting) to eliminate a variable, e.g. $2.5 x = 7.5$ A1 for $x = 3$ A1 for $y = 2$
2(a)	$s + l = 12$ and $0.75 s + 2 l = 15.25$	2	B1 for each correct equation
2(b)	7 small, 5 large	3	M1 for a correct method to eliminate one variable (FT their equations) A1 for $s = 7$ A1 for $l = 5$
2(c)	23.45 FT their (b)	1
3(a)	3.35 and 0.149	4	B1 for $41$ or 41 seen M1 for $\frac{7 \pm 41}{4}$ oe A1 for each answer
3(b)	1.14 and −4.39	4	B1 for $4.89$ or 4.89 seen M1 for $\frac{- 1.3 \pm 4.89}{0.8}$ oe A1 for each answer
4(a)	$x (x + 4) = 50$ , so $x^{2} + 4 x - 50 = 0$	2	M1 for $x (x + 4) = 50$ A1 for correct rearrangement with no errors
4(b)	5.35	3	M1 for $\frac{- 4 \pm 4 ^{2} - 4 \times 1 \times ( - 50 )}{2}$ B1 for $216$ or 14.7 seen
4(c)	29.4 FT their (b)	2	M1 for $2 \times (their width + their width + 4)$
5(a)	$2 (x + \frac{7}{4})^{2} - \frac{73}{8}$ oe	3	B1 for $a = 2$ B1 for $b = \frac{7}{4}$ B1 for $c = - \frac{73}{8}$
5(b)	0.39 and −3.89	3	M1 for $(x + \frac{7}{4})^{2} = \frac{73}{16}$ oe (FT their (a)) A1 for each answer
6(a)	$- 0.23$ ; no real roots	3	M1 for $3. 5^{2} - 4 \times 1.2 \times 2.6$ A1 for −0.23 B1 for 0 roots (FT their discriminant)
6(b)	3	3	M1 for $(- 6)^{2} - 4 \times k \times k = 0$ M1 for $k^{2} = 9$
6(c)	$k > - \frac{5}{4}$ oe	3	M1 for $x^{2} + 3 x + (1 - k) = 0$ M1 for $9 - 4 (1 - k) > 0$
7(a)	$23 - 87$	2	M1 for $16 - 47 - 47 + 7$ oe
7(b)	$11 + 47$	3	M1 for multiplying by $\frac{3 + 7}{3 + 7}$ M1 for $\frac{22 + 8 7}{2}$
7(c)	22	2	M1 for $11 + 62$ or $11 - 62$ seen
7(d)	$\frac{11 + 6 2}{7}$	3	M1 for multiplying by $\frac{3 + 2}{3 + 2}$ M1 for numerator $11 + 62$ or denominator 7
8(a)	0.853 and 3.23	4	M1 for $4.9 t^{2} - 20 t + 13.5 = 0$ oe M1 for $\frac{20 \pm 135.4}{9.8}$ oe A1 for each answer
8(b)	$4.9 t^{2} - 20 t + 23.5 = 0$ has $b^{2} - 4 a c = 400 - 460.6 = - 60.6 < 0$ , so no real solutions	3	M1 for $4.9 t^{2} - 20 t + 23.5 = 0$ oe M1 for $2 0^{2} - 4 \times 4.9 \times 23.5$ A1 for −60.6 with the conclusion

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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