NOTE: Q1-7 are the same as Q17-23 on June 17 Foundation Paper 2
(DOESNT NEED CHECKING TWICE – CHANGE BOTH PAPERS IF THERE IS AN ERROR ON BOTH)
Question 1 [3 marks]
The table shows the probabilities that a biased dice will land 1, 2, 3, 4 and 5
Ron rolls the biased dice 500 times.
Work out an estimate for the total number of times the dice will land on 5 or on 6
Answer type: Simple text answer
ANSWER: 235
WORKING:
\text{P(Landing on a 6)} = 1 - 0.05 - 0.23 - 0.09 - 0.16 - 0.28 = 0.19
\text{P(Landing on a 5 or a 6)} = 0.28 + 0.19 = 0.47
The biased dice will land on a 5 or on a 6 approximately
500 \times 0.47 = 235 times
Question 2 [5 marks]
On Sunday, some adults and some children were in a stadium
The ratio of the number of adults to the number of children was 3:2
Each person had a seat in the North stand or the East stand.
\dfrac{4}{5} of the children had seats in the East stand.
120 children had seats in the North stand.
There are exactly 2000 seats in the stadium.
On this Sunday, what percentage of the stadium seats were filled?
Answer type: Simple text answer
ANSWER: 75 \%
WORKING:
1 - \dfrac{4}{5} = \dfrac{1}{5} of the children had seats in the North stand, which was 120 children.
Therefore there were 120 \times 5 = 600 children in the stadium.
The ratio of adults to children is 3:2, so the ratio scaled up is 900:600
So, there are 900 + 600 = 1500 people in the stadium.
The \% of the stadium that is full is \dfrac{1500}{2000} \times 100 = 75 \%
Question 3 [4 marks]
The diagram shows a 3D shape.
Choose the correct diagram below that shows the front and side elevation of the 3D shape above, drawn on a centimetre grid.
Answer type: Multiple choice type 1
A:
B:
C:
D:
ANSWER: A
Question 4
Oscar drove 100 km from Hull to Leeds.
He then drove 72 km from Leeds to Manchester.
Oscar’s average speed from Hull to Leeds was 80 km/h.
Oscar took 63 minutes to drive from Leeds to Manchester.
Question 4(a) [4 marks]
Work out Oscar’s average speed for his total drive from Hull to Manchester.
Give your answer to 1 decimal place.
Answer type: Simple text answer
ANSWER: 74.8 km/h
WORKING:
Time from Hull to Leeds: \dfrac{100}{80} = 1.25 hours =75 minutes
Total distance = 100 + 72 = 172 km
Total time = 75 + 63 = 138 minutes = 2.3 hours
Average speed = \dfrac{172}{2.3} = 74.8 km/h (1 dp)
Question 4(b) [1 mark]
Francis drove straight from Hull to Manchester.
The distance from Hull to Manchester is 160 km.
She took 2 hours.
Who had the fastest average speed?
Answer type: Multiple choice type 1
A: Oscar
B: Francis
ANSWER: B
WORKING:
The average speed for Francis is \dfrac{160}{2} = 80 km/h.
Therefore Francis had the fastest average speed.
Question 5
ABC and EDC are straight lines.
DB is parallel to EA.
EC = 7 m
DC = 2.8 m
DB = 2.2 m
Question 5(a) [2 marks]
What is the length of AE?
Answer type: Simple text answer
ANSWER: 5.5 m
WORKING:
Scale factor = \dfrac{EC}{DC} = \dfrac{7}{2.8} = 2.5
AE = 2.2 \times 2.5 = 5.5 m
Question 5(b) [2 marks]
AC = 6 m
What is the length of AB?
Answer type: Simple text answer
ANSWER: 3.6 m
WORKING:
BC = 6 \div 2.5 = 2.4 m
AB = AC - BC = 6 - 2.4 = 3.6 m
Question 6 [3 marks]
Junet wants to invest £20000 for 4 years in a bank.
Which bank should Junet choose, to maximise his investment?
Answer type: Multiple choice type 1
A: Bank A
B: Bank B
ANSWER: B
WORKING:
A: £20000 \times 1.015^4 = £21227.27
B: £20000 \times 1.04 \times 1.01^3 = £21430.26
Junet should choose bank B.
Question 7 [2 marks]
A number, x, is rounded to 2 decimal places.
The result is 8.64
What is the error interval for x?
Answer type: Multiple choice type 1
A: 8.635 \leq x < 8.645
B: 8.63 \leq x < 8.65
C: 8.6 \leq x < 8.7
D: 8.6355 \leq x < 8.6455
ANSWER: A
Question 8 [2 marks]
The cumulative frequency graph shows some information about the time spent playing video games one day, in minutes, of 80 teenagers.
Work out an estimate for the number of these teenagers that spent over 60 minutes playing video games on this day.
Answer type: Simple text answer
ANSWER: 20
WORKING:
80 - 60 = 20
Question 9 [3 marks]
The diagram shows triangle \textbf{A} drawn on the grid.
Peter reflects triangle \textbf{A} in the x axis to get triangle \textbf{B}.
He then reflects triangle \textbf{B} in the line y=x to get triangle \textbf{C}.
Jess reflects triangle \textbf{A} in the line y=x to get triangle \textbf{D}.
She is then going to reflect triangle \textbf{D} in the y axis to get triangle E.
Jess says that triangle \textbf{E} should be in the same position as triangle \text{C}.
Is Jess correct?
Answer type: Multiple choice type 1
A: Yes
B: No
ANSWER: A
WORKING:
See the diagram below:
As you can see, triangle C and E are in the same position, so Jess is correct.
Question 10
The distance shows some information about 5 planets.
Question 10(a) [1 mark]
Which planet has the greatest mass?
Answer type: Multiple choice type 2
A: Zarg 14T
B: Brobipra
C: Gros VZ7
D: Motovo
E: Quacarro
ANSWER: B
Question 10(b) [1 mark]
What is the difference between the mass of Gros VZ7 and Motovo?
Answer type: Multiple choice type 1
A: 1.4998 \times 10^{25}
B: 1.4998 \times 10^{23}
C: 1.4998 \times 10^{4}
D: 1.4998 \times 10^{2}
ANSWER: A
WORKING:
1.56 \times 10^{25} = 156 \times 10^{23}
156 \times 10^{23} - 6.02 \times 10^{23} = 149.98 \times 10^{23} = 1.4998 \times 10^{25}
Question 10(c) [2 marks]
Rusi says that Motovo is over 50 times further away from Zarg 14T than Brobipra is.
Is Rusi correct?
Answer type: Multiple choice type 1
A: Yes
B: No
ANSWER: A
WORKING:
\dfrac{2.45 \times 10^9}{4.14 \times 10^7} = 59.178... > 50
Rusi is correct.
Question 11 [4 marks]
Solve \dfrac{2x-1}{5} - \dfrac{x+6}{10} = \dfrac{3x+5}{15}
Answer type: Fraction
ANSWER: \dfrac{34}{3}
WORKING:
Common denominator =30
\dfrac{6(2x-1)}{30} - \dfrac{3(x+6)}{30} = \dfrac{2(3x+5)}{30}
6(2x-1) - 3(x+6) = 2(3x+5)
12x - 6 - 3x - 18 = 6x + 10
3x = 34
x = \dfrac{34}{3}
Question 12
There are 20 students in Mr Richards’ class.
12 of the students are boys.
Two students from the class are chosen at random.
Mr Richards draws this probability tree diagram for this information.
Question 12(a) [1 mark]
Chloe, a student in the class, believes that the tree diagram is wrong.
Choose the correct statement.
Answer type: Multiple choice type 1
A: The denominators are incorrect for the 2nd student
B: The numerators are incorrect for the 2nd student
C: The probabilities for the 1st student are wrong.
D: The tree diagram is in fact correct.
ANSWER: A
Question 12(b) [1 mark]
Ashraf and Jake play for the school rugby team.
The probability that Ashraf will score a try in the next match is 0.3
The probability that Jake will score a try in the next match is 0.15
Mr Beadle says,
“The probability that both boys will score a try in the next match is 0.3 + 0.15”
Is Mr Beadle right?
Answer type: Multiple choice type 1
A: Yes
B: No
ANSWER: B
WORKING:
The probabilities should be multiplied: 0.3 \times 0.15
Question 13 [3 marks]
The histogram shows some information about the ages of the 82 people at a park.
40 \% of the people who are under 15 years of age are male.
Calculate an estimate for the number of males there are in the park who are under the age of 15
Answer type: Simple text answer
ANSWER: 18
WORKING:
\text{Frequency} = \text{Frequency density} \times \text{Class width}
10+20+15 = 45 under the age of 15
45 \times 0.4 = 18 males under the age of 15
Question 14
Below are some graphs.
In each question, write your answer as a CAPITAL LETTER, A – I
Question 14(a) [1 mark]
Which graph has equation y = -2x^3?
Answer type: Simple text answer
ANSWER: G
Question 14(b) [1 mark]
Which graph has equation y = \cos(x)?
Answer type: Simple text answer
ANSWER: C
Question 14(c) [1 mark]
Which graph has equation y = - \dfrac{3}{x}?
Answer type: Simple text answer
ANSWER: B
Question 15 [3 marks]
A,B,C and D are four points on the circumference of a circle.
AEC and BED are straight lines.
\angle BEC = 110 \degree
\angle ABE = 50 \degree
Find the value of \angle CDE
Answer type: Simple text answer
ANSWER: 60 \degree
WORKING:
\angle CED = 180 - 110 = 70 \degree (angles on a straight line add up to 180 \degree)
\angle DCE = 50 \degree (angles in the same segment are equal)
\angle CDE = 180 - 70 - 50 = 60 \degree (angles in a triangle add up to 180 \degree)
Question 16 [3 marks]
Find the value of 0.3 \dot{8} \times 0.\dot{6}, giving your answer as a fraction in simplest form.
ANSWER: \dfrac{7}{27}
WORKING:
x = 0.3888...
100x = 38.88...
10x = 3.88...
100x - 10x = 90x = 35
x = \dfrac{35}{90} = \dfrac{7}{18}
y = 0.666...
10y = 6.666...
10y - y = 9y = 6
y = \dfrac{6}{9} = \dfrac{2}{3}
0.3 \dot{8} \times 0.\dot{6} = xy = \dfrac{7}{18} \times \dfrac{2}{3} = \dfrac{14}{54} = \dfrac{7}{27}
Question 17 [5 marks]
OPQ is a sector of a circle with centre O and radius 10 cm.
A is the point on OP and B is the point on OQ such that AOB is an equilateral triangle, with sides of length 4 cm.
Calculate the area of the shaded region as a percentage of the area of the sector OPQ.
Give your answer to 1 decimal place.
Answer type: Simple text answer
ANSWER: 86.8 \%
WORKING:
Area of Sector = \dfrac{60}{360} \times \pi \times 10^2 = \dfrac{50}{3} \pi = 52.359...
Area of Triangle = \dfrac{1}{2} ab \sin C = \dfrac{1}{2} \times 4 \times 4 \times \sin (60) = 4 \sqrt{3} = 6.928...
Percentage = \dfrac{52.359... - 6.928...}{52.359...} \times 100 = 86.8 \% (1 dp)
Question 18 [3 marks]
27^{\frac{1}{4}} \times 3^x = 81^{\frac{2}{5}}
Work out the exact value of x
Give your answer as a decimal.
Answer type: Simple text answer
ANSWER: 0.85
WORKING:
Common base: 27 = 3^3 and 81 = 3^4
(3^3)^{\frac{1}{4}} 3^x = (3^4)^{\frac{2}{5}}
3^{\frac{3}{4}} \times 3^x = 3^{\frac{8}{5}}
\dfrac{3}{4} + x = \dfrac{8}{5}
x = \dfrac{17}{20} = 0.85
Question 19 [4 marks]
2 - \dfrac{x+3}{x-2} - \dfrac{x-6}{x+2} can be written as a single fraction in the form \dfrac{px+q}{x^2 - 4} where p and q are integers.
Find the value of p and q.
Answer type: Multiple answers type 1
ANSWER:
p = 3
q = -26
WORKING:
\dfrac{2(x-2)(x+2)}{(x-2)(x+2)} - \dfrac{(x+3)(x+2)}{(x-2)(x+2)} - \dfrac{(x-6)(x-2)}{(x-2)(x+2)}
Each fraction has a common denominator (x-2)(x+2) = x^2 - 4 so we can combine them into one fraction
Numerator:
2(x-2)(x+2) - (x+3)(x+2) - (x-6)(x-2)
=2(x^2+2x-2x-4) - (x^2+2x+3x+6) - (x^2-2x-6x+12)
=2(x^2-4) - (x^2+5x+6) - (x^2-8x+12)
=2x^2-8-x^2-5x-6-x^2+8x-12
=3x-26
Therefore, the fraction can be written as \dfrac{3x-26}{x^2-4}
Therefore, p=3 and q = -26
*Q20(a) has been split into two parts worth 1 mark each.
Question 20
The diagram below shows part of the graph of y=x^2-2x+2
Question 20(a) [1 mark]
Which of the following graphs will help you find estimates for the solutions of x^2-x-1=0?
Answer Type: Multiple Choice
A:
B:
C:
D:
ANSWER: A
WORKING:
x^2-x-1=0
\rArr (x^2-2x+2)-(-x+3)=0
\rarr x^2-2x+2=-x+3
So we need to plot the graph of y=-x+3 against the graph y=x^2-2x+2, to see where they intersect for the solutions of x^2-x-1=0.
Question 20(b) [1 mark]
Use the correct graph from part (a) to find estimates for the solutions of x^2-x-1=0
Choose the correct answer.
Answer type: Multiple choice type 1
A: x = -0.6 and x=1.6
B: x=-0.3 and x=3.3
C: x=-1.6 and x=0.6
D: x=1 and x=2
ANSWER: A
WORKING:
Use the graph of y=x^2-2x+2 and y=-x+3, and use the points of intersection to estimate the solutions.
x = -0.6 and x=1.6 (1 dp)
Question 20(c) [3 marks]
P is the point on the graph of y=x^2-2x+2 where x=0
Find an estimate for the gradient of the graph at the point P
Answer type: Multiple choice type 1
A: -2
B: -1
C: 2
D: \dfrac{1}{2}
ANSWER: A
WORKING:
Draw a tangent to the curve at x=0
Question 21 [4 marks]
The diagram below shows 4 identical circles inside a rectangle.
Each circle touches a side of the rectangle, and the circles touch each other, as seen in the diagram.
The radius of each circle is 8mm.
What is the area of the rectangle?
Give your answer to 3 significant figures.
Answer type:
ANSWER: 1400 mm^2
WORKING:
Width =8 \times 4 = 32 mm
Height =8 \times 2 + 2x = 16 + 2x
x can be found using Pythagoras.
x= \sqrt{16^2-8^2}=13.856...Height = 16 + 2x = 16 + 2(13.856...) = 43.712... mm
Area = 32 \times 43.712... = 1400 mm^2 (3 sf)
Question 22 [3 marks]
Here are the first 5 terms of a sequence
4, 9, 16, 25, 36
What is the nth term of this sequence?
Answer type: Multiple choice type 1
A: n^2 + 2n + 1
B: n^2 + n + 1
C: 2n^2 + 2n + 1
D: 2n + 1
ANSWER: A
WORKING:
The nth term will take the form an^2 + bn + c
Firstly, find the differences between the terms, and then the difference between the differences.
The second difference is +2, a is half of the second difference, so a=1
Now, to find b and c we compare the values generated by the sequence of n^2, to the original sequence. (u_n)
u_n = 4, 9, 16, 25, 36
n^2 = 1, 4, 9, 16, 25
u_n - n^2 = 3, 5, 7, 9, 11
The difference is a linear sequence, whose nth term is precisely bn+c
The difference is 2, and the sequence starts at 3, so the nth term is 2n+1
Therefore b=2 and c=1
So, the nth term of the original sequence is n^2 + 2n + 1
Question 23 [3 marks]
C is the circle with equation x^2 + y^2 = 15
P= \bigg(\dfrac{7}{2}, \dfrac{\sqrt{11}}{2} \bigg) is a point on C
Find an equation of the tangent to L at the point P, in the form y=mx+c
Answer type: Multiple choice type 1
A: y = - \dfrac{7}{\sqrt{11}} \, x + \dfrac{30\sqrt{11}}{11}
B: y = \dfrac{7}{\sqrt{11}} \, x - \dfrac{19\sqrt{11}}{11}
C: y = - \dfrac{\sqrt{11}}{7} \, x
D: y = - \dfrac{7}{\sqrt{11}} \, x - \dfrac{19\sqrt{11}}{11}
ANSWER: A
WORKING:
Gradient of OP = \dfrac{\sqrt{11}}{2} \div \dfrac{7}{2} = \dfrac{\sqrt{11}}{7}
Gradient of tangent = - \bigg( \dfrac{\sqrt{11}}{7} \bigg)^{-1} = - \dfrac{7}{\sqrt{11}}
Equation of tangent:
y = mx+c
Substitute in the gradient and the coordinates of P to find c,
\dfrac{\sqrt{11}}{2} = - \dfrac{7}{\sqrt{11}} \bigg(\dfrac{7}{2} \bigg) + c
c = \dfrac{30 \sqrt{11}}{11}
y = - \dfrac{7}{\sqrt{11}} \, x + \dfrac{30 \sqrt{11}}{11}