1.
The formula to calculate the volume of a sphere is V 
4 r 3
.
3
(3)
Change the subject of the formula to r.
2.
The tank of a car contains 7 litres of petrol.
The car is being refuelled.
The graph shows how the volume of petrol changes as a further 32 litres are added to the
tank at a steady rate for 40 seconds.
V
Volume
(litres)
7
0
Time (seconds)
Find the equation of the straight line in terms of V and t.
40
t
(4)
3.
Solve
2 x 2  5x  4  0 .
(4)
Give your answers correct to one decimal place.
4.
A right-angled triangle has sides as shown.
(x + 7) cm
(x – 2) cm
15 cm
(3)
Calculate the value of x.
5.
A school is buying in calculators for its pupils.
There are two types of calculator available.
Calculator A is a scientific calculator.
Calculator B is similar to calculator A but also has a solar cell for power.
(a) Mr Paterson’s class order 14 of calculator A and 5 of calculator B.
The total cost is £116.
Write an algebraic equation to illustrate this.
(1)
(b) Mrs Brown’s class order 7 of calculator A and 11 of calculator B.
The total cost is £124.30.
Write an algebraic equation to illustrate this.
(1)
(c) Find algebraically the cost of each type of calculator.
(3)
(d) Mr Sharpe has £101.80 but has lost all the order slips that the pupils handed in.
He knows that 16 pupils gave him orders. How many of each type of calculator does
he need?
(1)
6.
An electronics company makes various different sized phones and tablet computers.
Their best selling tablet measures 31·5 centimetres from corner to corner.
Their best selling phone has a screen that is mathematically similar and measures 9 cm.
9 cm
31·5 cm
The tablet screen has an area of 490 square centimetres.
Find the area of the screen on the phone.
7.
(3)
A regular pentagon has lines drawn inside as shown.
B
A
C
E
Calculate the size of angle EAC.
D
(3)
8.
The diagram shows a graph of a quadratic function of the form y  k ( x  a)( x  b) .
y
10
4
10
x
(a) Find the values of k, a and b.
(2)
(b) Write down the coordinates of the turning point.
(1)
9.
The graph below shows a function in the form y  a sin x  c .
y
6
0°
180°
x
360°
-2
(a) Write down the values of a and c.
(2)
(b) Find the x-coordinates where the graph crosses the x-axis.
(3)
10.
Triangular numbers are numbers where that many objects can be arranged in a triangular
shape, as shown in the diagram.
T=
1
3
10
6
15
The nth triangular number is the number of dots required to make a triangle that has n dots
on each side.
The formula to find the nth triangular number is T 
1
nn  1 .
2
(a) Use the formula to find how many dots are in the 20th triangular number?
(1)
(b) For a triangular number that has 55 dots show that n 2  n  110  0 .
(2)
(c) Hence find algebraically the value of n.
(3)
END OF TEST PAPER
Mathematics National 5: Unit 2
Qu.
1
Assess.
Main points of expected responses
Standard
1.1
 Dealing with denominator
 Dealing with coefficient
 Dealing with root
Extended Unit Test 2013/14
 3V  4 r 3
3V

 r3
4
3V
 r3
4
Notes:
For correct answer with or without working award 3/3
For r 
2
3
3V
3V
or r 
with working award 2/3
4
4
1.1
4
or equivalent
5
 Calculate gradient
 m
 State intercept
 c=7
4
 y  x7
5
4
 V  t7
5
 Equation of straight line
 Variables in context
Notes:
3
1.3
 (5)  (5) 2  4  2  (4)
2 2
 Method (subs. into quadratic formula)
 x
 Process discriminant

 Solution
 3·14…, -0·64…
 Rounding
57
 3·1, -0·6
Notes:
4
2.1
1.1
 Apply Pythagoras’ Theorem
 Expand brackets correctly
 Simplify and solve
 ( x  7) 2  ( x  2) 2  15 2
 x 2  14 x  49  x 2  4 x  4  225
 x  10
Notes:
Where students expand brackets to x 2  49  x 2  4  225 with or without previous working award 1/3.
Qu.
Assess.
Main points of expected responses
5a
Standard
2.1
 Equation
5b
2.1
 Equation
 7 A  11B  124  30
5c
1.1
 Scale equation(s)
 Subtract equation
 Solve for both variables
 14 A  22B  148  60
 17B  132  60
 A  £5  50 , B  £7  80
 14 A  5B  116
Alternative working
154 A  55B  1276

35 A  55B  621  50
 119 A  654  50
 A  £5  50 , B  £7  80
5d
2.2
Notes:
6
1.4
 Answer
 10 of calculator A and 6 of calculator B
 Length scale factor

 Area scale factor
4
2
   or
49
7
 40 (cm2)
 Area of phone
Notes:
7
2.1
1.4
 Internal/External angle
 Another angle inside shape
 Find required angle
2
7
2
 108°
 eg. BAˆ C  36
 EAˆ C  72
Notes:
To gain full credit sufficient working must be shown.
Watch out for students who guess that ACˆE  36 .
8a
1.2
 Interpret x-axis intercepts
 Find constant
8b
1.2
Notes:
 Turning point
In 8a for y 
 a  4 , b  10
1
 k
4
9

  7,  
4

1
( x  4)( x  10) with or without working award 2/2
4
9a
1.5
 Amplitude
 Constant
 a4
 c2
9b
1.5
 Set equal to zero
 4 sin x  2  0
1
 sin x   or equivalent
2
 x  210, 330
 Rearrange for sin x
 Solve for both x-coordinates
Notes:
Assess.
Main points of expected responses
Standard
10a 1.3
 Apply formula
10b 1.3
 Substitute into formula
Qu.
 Complete process
1.3
2.2
 Factorise
 Solve
 Discard negative solution
 210
1
n(n  1)
2
 110  n 2  n and complete
 55 
 (n  10)( n  11)  0
 n  10 and n  11
 Indication that n  10 is chosen
Notes:
In 10b there must be sufficient working to show that candidate has reached the required formula
themselves to gain the second mark.
In 10c to gain marks 2 and 3 both solutions must be stated and then n  10 chosen (by underlining,
scoring out, saying “not valid”, etc.).
Total: 40