ECAA 2020 Economics Admissions Assessment D563/31: Questions & Worked Solutions

Question 1

1 mark

The admission charge to a cinema is different for adults and children.

Admission for 2 adults and 3 children costs £20.

Admission for 4 adults and 4 children costs £34.

What does admission cost for 6 adults and 2 children?

A.£27
B.£29
C.£33
D.£39
E.£44
F.£48
G.£72

Answer: D

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Question 2

1 mark

The

n^{th}

term of a sequence is

2n – 5

.

Which row in the table is correct for this sequence?

A.subtract 5, 11th
B.subtract 5, 29th
C.subtract 2, 11th
D.subtract 2, 29th
E.add 5, 11th
F.add 5, 29th
G.add 2, 11th
H.add 2, 29th

Answer: G

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Question 3

1 mark

A fair spinner has eight equal sections. Each section has one number written on it, as shown. The spinner has three sections with '1', four sections with '3', and one section with '2'.

The spinner is spun twice, and the two numbers scored are added.

What is the probability that the sum of the two numbers is 5?

A. $\frac{1}{8}$
B. $\frac{5}{8}$
C. $\frac{1}{16}$
D. $\frac{3}{16}$
E. $\frac{25}{64}$
F. $\frac{55}{64}$

Answer: A

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Question 4

1 mark

PQRS is a square with side length

x

.

M

is the midpoint of side

PS

.

A circular arc, with centre

M

, is drawn inside the square from

S

to

P

.

Another circular arc, with centre

P

, is drawn inside the square from

S

to

Q

.

What is the area of the shaded region in terms of

x

?

A. $\frac{1}{8}\pi x^2$
B. $\frac{3}{16}\pi x^2$
C. $\frac{1}{4}\pi x^2$
D. $\frac{5}{16}\pi x^2$
E. $\frac{3}{8}\pi x^2$
F. $\frac{7}{16}\pi x^2$
G. $\frac{1}{2}\pi x^2$

Answer: A

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Question 5

1 mark

A balloon contains 5000 cm³ of gas.

The gas in the balloon gradually escapes so that the volume of the balloon decreases.

60% of the volume of the balloon is lost each week.

What is the volume of the balloon, in cm³, after 3 weeks?

A.0
B.128
C.320
D.800
E.1080

Answer: C

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Question 6

1 mark

Consider the four lines with the following equations.

1

2x + 6y = 3

2

9y = 3x - 4

3

2y = 6x + 3

4

4x+6y-9 = 0

Which two lines are perpendicular?

A.1 and 2
B.1 and 3
C.1 and 4
D.2 and 3
E.2 and 4
F.3 and 4

Answer: B

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Question 7

1 mark

The equilateral triangle PQR has sides of length 8 cm.

A circle, centre O, passes through each of the vertices of the triangle.

Find an expression for the circumference of the circle, in cm.

A. $\frac{\sin 60°}{8\pi}$
B. $\frac{8\pi}{\sin 60°}$
C. $\frac{\cos 60°}{8\pi}$
D. $\frac{8\pi}{\cos 60°}$
E. $\frac{\tan 60°}{8\pi}$
F. $\frac{8\pi}{\tan 60°}$

Answer: B

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Question 8

1 mark

Find the sum of the solutions of

2\left(\frac{x}{4} + 3\right)^2 - \left(\frac{x}{4} + 3\right) - 36 = 0

A.2
B. $\frac{3}{2}$
C. $\frac{1}{2}$
D.-4
E.-13
F.-22
G.-26
H.-34

Answer: F

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Question 9

1 mark

When the expression

(2x + 3)^2 – (x – 3)^2

is written in the form

p(x + q)^2 + r

, where

p, q

and

r

are constants, what is the value of

r

?

A.-27
B.-9
C.0
D.3
E.15

Answer: A

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Question 10

1 mark

Which one of the following expressions is equivalent to

\frac{a}{b/c} - \frac{a/b}{c}

A.0
B. $\frac{a(b^2 - 1)}{bc}$
C. $\frac{a(b^2 - c^2)}{bc}$
D. $\frac{a^2b^2 - c^2}{abc}$
E. $\frac{a(c^2 - 1)}{bc}$
F. $\frac{a^2c^2-b^2}{abc}$
G. $\frac{b^2 - a^2}{abc}$

Answer: E

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Question 11

1 mark

The table shows statistics relating to the test marks of two groups of students.

The results for the two groups of students are combined.

What can be deduced about the mean and range of the combined results?

A.mean = 40, range $\leq$ 16
B.mean = 40, 16 < range < 21
C.mean = 40, range $\geq$ 21
D.mean = 44, range $\leq$ 16
E.mean = 44, 16 < range < 21
F.mean = 44, range $\geq$ 21

Answer: F

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Question 12

1 mark

The number of pairs of winter boots sold on a day is inversely proportional to the cube of the outside temperature on that day, measured in °C.

On a day when the outside temperature is 8°C, 250 pairs of boots are sold.

The next day, when the outside temperature is

x

°C, the number of pairs of boots sold is 700% more than on the previous day.

What is the value of

x

?

A.2
B.4
C. $\frac{8}{\sqrt[3]{7}}$
D. $8\sqrt[3]{7}$
E.16

Answer: B

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Question 13

1 mark

In a sale, all prices are reduced by 25%.

A customer calculates the pre-sale price of a bicycle incorrectly by increasing the marked sale price by 25%.

The customer's calculated pre-sale price is incorrect by £15.

What is the correct pre-sale price of the bicycle?

A.£180
B.£195
C.£210
D.£225
E.£240

Answer: E

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Question 14

1 mark

A paint colour is a mixture of red paint, blue paint and yellow paint.

The ratio of red paint to blue paint in the mixture is 18:5

The ratio of blue paint to yellow paint in the mixture is

p:3

The ratio of red paint to yellow paint in the mixture is 12:5

What is the value of

p

?

A.2
B.4.5
C.5
D.7.5
E.12

Answer: A

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Question 15

1 mark

In the diagram, QS is perpendicular to PR.

PS =

x

cm
PQ =

y

cm
QR =

z

cm

angle QRS = 61°

PSR is a straight line.

Which one of the following is an expression for the length

z

, in cm?

A. $\sqrt{y^2 + x^2} \sin 61°$
B. $\sqrt{y^2 – x^2} \sin 61°$
C. $\sqrt{y^2 + x^2} \cos 61°$
D. $\sqrt{y^2 - x^2} \cos 61°$
E. $\frac{\sqrt{y^2 + x^2}}{\sin 61°}$
F. $\frac{\sqrt{y^2 - x^2}}{\sin 61°}$
G. $\frac{\sqrt{y^2 + x^2}}{\cos 61°}$
H. $\frac{\sqrt{y^2 - x^2}}{\cos 61°}$

Answer: F

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Question 16

1 mark

Two identical fair six-sided dice each have their faces numbered from 1 to 6, with one number on each face.

Both dice are thrown, and the number on each of the dice is recorded.

They are then both thrown again, and the number on each of the dice is recorded.

What is the probability that at least one of the four recorded numbers is even?

A. $\frac{1}{4}$
B. $\frac{1}{2}$
C. $\frac{9}{16}$
D. $\frac{3}{4}$
E. $\frac{15}{16}$

Answer: E

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Question 17

1 mark

The quadratic equation

2x^2 – px – 4 = 0

, where

p

is a positive constant, has two solutions that differ by 6.

What is the value of

p

?

A.2
B. $4\sqrt{7}$
C.12
D. $4\sqrt{11}$
E. $4\sqrt{34}$
F. $6\sqrt{30}$

Answer: B

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Question 18

1 mark

Two vertices of a square are at (1, 1) and (3, 5).

What is the difference between the perimeters of the largest and smallest possible squares that can be drawn with these points as two of their vertices?

A.0
B. $4\sqrt{3}(2-\sqrt{2})$
C. $4\sqrt{3}(\sqrt{2}-1)$
D. $4\sqrt{5}(2-\sqrt{2})$
E. $4\sqrt{5}(\sqrt{2}-1)$
F. $4\sqrt{13}(2-\sqrt{2})$
G. $4\sqrt{13}(\sqrt{2}-1)$
H. $4\sqrt{3}\sqrt{5}(2-\sqrt{2})$

Answer: D

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Question 19

1 mark

The point M is (2, 5) and the point N is (-3,-1).

The line segment MN is transformed to the line segment TU by two transformations:

MN is rotated 90° clockwise about the origin to give the line segment RS.

RS is then translated by the vector

\begin{pmatrix} p \ q \end{pmatrix}

to give the line segment TU.

The coordinates of the midpoint of TU are (7, -2.5).

Find the vector

\begin{pmatrix} p \ q \end{pmatrix}

.

A. $\begin{pmatrix} 2 \ 0.5 \end{pmatrix}$
B. $\begin{pmatrix} 0.5 \ 2 \end{pmatrix}$
C. $\begin{pmatrix} 5 \ -3 \end{pmatrix}$
D. $\begin{pmatrix} -3 \ 5 \end{pmatrix}$
E. $\begin{pmatrix} 9 \ -2 \end{pmatrix}$
F. $\begin{pmatrix} -2 \ 9 \end{pmatrix}$

Answer: C

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Question 20

1 mark

A solid cone has a base radius

x

cm.

The ratio of the perpendicular height of the cone to the radius of the cone is 5:2

A solid hemisphere of radius

\frac{y}{2}

cm is made from the same material as the cone.

Which one of the following is a correct expression for

\frac{\text{volume of the cone}}{\text{volume of the hemisphere}}

(Volume of a cone =

\frac{1}{3}\pi r^2 h

where

r

is the radius and

h

is the perpendicular height.)
(Volume of a sphere =

\frac{4}{3}\pi r^3

where

r

is the radius.)

A. $\frac{5x^3}{y^3}$
B. $\frac{5x^3}{4y^3}$
C. $\frac{8x^3}{5y^3}$
D. $\frac{10x^3}{y^3}$
E. $\frac{14x^3}{y^3}$

Answer: D

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Question 21

1 mark

Given that

\int_{4}^{9} f(x)dx = 3

and that

g(x) = 2f(x) + 1

find

\int_{4}^{9} g(x)dx

A.6
B.7
C.8
D.9
E.10
F.11

Answer: F

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Question 22

1 mark

(x – 1)

and

(x – 2)

are both factors of

x^4 + ax^3 + bx^2 – 12x + 4

What are the values of

a

and

b

?

A. $a = -6$ and $b = -23$
B. $a = -6$ and $b = 13$
C. $a = 6$ and $b = -11$
D. $a = 6$ and $b = 1$

Answer: B

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Question 23

1 mark

The area of the regular octagon is

32\sqrt{2}

cm².

What is the length, in cm, of the straight line PT? The octagon is labelled PQRSTUVW.

A.8
B.16
C. $4\sqrt{2}$
D. $8\sqrt{2}$
E. $4\sqrt{2\sqrt{2}}$
F. $16\sqrt{2\sqrt{2}}$

Answer: A

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Question 24

1 mark

What is the area of the region enclosed between the curve

y = \frac{1}{2}x^2

, the line

y = -x

, and the lines

x=1

and

x=3

?

A. $\frac{1}{3}$
B.2
C.4
D.6
E. $\frac{25}{3}$
F. $\frac{28}{3}$

Answer: E

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Question 25

1 mark

A wall is used to mark out the perimeter of a rectangular field WXYZ.

Walls are also used to divide WXYZ into four identical rectangular fields, P, Q, R and S, as shown.

The total length of wall used is 260 m.

What is the length of WZ that maximises the area P?

A.20 m
B.26 m
C.32.5 m
D.40 m
E.52 m
F.65 m

Answer: B

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Question 26

1 mark

A line with non-zero gradient

m

is reflected in the line

y=x

What is the gradient of the reflected line?

A. $m$
B. $-m$
C. $\frac{1}{m}$
D. $- \frac{1}{m}$

Answer: C

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Question 27

1 mark

There are two red balls and two blue balls in a bag.

Two balls are removed at random without replacement.

Given that at least one of them is red, what is the probability that one of them is blue?

A. $\frac{1}{2}$
B. $\frac{2}{3}$
C. $\frac{4}{5}$
D. $\frac{5}{6}$
E.1

Answer: C

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Question 28

1 mark

The sum of the first 20 terms of an arithmetic progression is 50.

The sum of the next 20 terms of the arithmetic progression is –50.

What is the sum of the first 100 terms of the arithmetic progression?

A.-750
B.-350
C.-50
D. $- \frac{159}{8}$
E. $\frac{159}{8}$
F.50
G.350
H.750

Answer: A

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Question 29

1 mark

A sequence is generated by

x_{n+1} = -\frac{12}{x_n}

n

is an integer, where

n \geq 1

.

The 50th term of the sequence is 6.

What is the sum of the first fifteen terms of the sequence?

A.-2
B.6
C.10
D.22
E.26
F.34
G.58

Answer: E

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Question 30

1 mark

The line L with equation

y = mx + c

, where

m > 0

and

c \geq 0

, passes through the point (2, 4).

A line is drawn through the point (2, 4) perpendicular to L.

The triangle enclosed between the two lines and the y-axis has area 5 square units.

What is the larger of the two possible values of

m

?

A.-0.5
B.0.5
C.1.25
D.2
E.5

Answer: D

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Question 31

1 mark

A list of

n

numbers has mean

m

and a unique mode

d

.

Two numbers are removed from the list.

The remaining list of numbers also has a unique mode, but this unique mode is not equal to

d

.

The mean of the remaining

n – 2

numbers is

m + 2

.

What was the unique mode,

d

, of the original list?

A. $n-m+2$
B. $n-m-2$
C. $n+m-2$
D. $m+n+2$
E. $m-n+2$
F. $m-n-2$

Answer: E

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Question 32

1 mark

P

and

Q

are two different geometric progressions.

The 3rd term of each geometric progression is 4.

The 5th term of each geometric progression is 2.

What is the modulus of the difference between the sums to infinity of

P

and

Q

?

A.0
B.8
C. $8\sqrt{2}$
D.16
E. $16\sqrt{2}$
F.32
G. $32\sqrt{2}$

Answer: E

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Question 33

1 mark

Two circles have centres

P

and

Q

.

The radius of each circle is 1 cm.

The distance

PQ

is 1 cm.

What is the area of overlap, in cm², of the two circles?

A. $\frac{\pi}{3} - \frac{1}{4}$
B. $\frac{\pi}{3} - \frac{\sqrt{3}}{4}$
C. $\frac{2\pi}{3} - \frac{1}{2}$
D. $\frac{2\pi}{3} - \frac{\sqrt{3}}{2}$
E. $\frac{4\pi}{3} - \frac{1}{4}$
F. $\frac{4\pi}{3} - \frac{\sqrt{3}}{2}$

Answer: D

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Question 34

1 mark

The curve

y = x^3 + 3\sqrt{5}px^2 + 3px + 13

has two distinct turning points.

What are all the possible values of

p

?

A. $p < 0, p > 0.2$
B. $p \leq 0, p \geq 0.2$
C. $0 < p < 0.2$
D. $0 \leq p \leq 0.2$
E. $p < 0, p > 1.2$
F. $p \leq 0, p \geq 1.2$
G. $0 < p < 1.2$
H. $0 \leq p \leq 1.2$

Answer: A

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Question 35

1 mark

Find the complete set of values of

x

for which

x^4 + 36 < 13x^2

A. $4 < x < 9$
B. $x < 4, x > 9$
C. $-9 < x < -4, 4 < x < 9$
D. $x < -9, -4 < x < 4, x > 9$
E. $2 < x < 3$
F. $x < 2, x > 3$
G. $-3 < x < -2, 2 < x < 3$
H. $x < -3, -2 < x < 2, x > 3$

Answer: G

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Question 36

1 mark

Find the number of solutions of the equation

14 \cos^3 x + 10 \sin^2 x \cos x = 13 \cos x

in the range

-2\pi \leq x \leq 2\pi

A.4
B.6
C.8
D.10
E.12
F.14

Answer: E

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Question 37

1 mark

It is given that

2^x = 3^y

and

x + y = 2

Which one of the following is an expression for

x

?

A. $2\log_{10}\frac{1}{2}$
B. $\log_{10} 3$
C. $\frac{6}{5}$
D. $\frac{3}{2}$
E. $\frac{\log_{10} 9}{\log_{10} 5}$
F. $\frac{\log_{10} 9}{\log_{10} 6}$
G. $\frac{\log_{10} 3}{\log_{10} 2}$

Answer: F

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Question 38

1 mark

Find the product of the real roots of the equation

(\log_{10} x^2)^2 + \log_{10} x^2 = 3

A. $10^{-\frac{3}{2}}$
B. $10^{-1}$
C. $10^{-\frac{1}{2}}$
D. $10^{-\frac{1}{4}}$
E. $10^{\frac{3}{5}}$
F. $10^1$

Answer: D

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Question 39

1 mark

Find the

y

-coordinate of the points on the curve

y=x^2

that are closest to the point

\left(0, \frac{9}{2}\right)

A.0
B. $\frac{1}{4}$
C. $\frac{4}{3}$
D.4
E. $\frac{9}{2}$

Answer: D

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Question 40

1 mark

Find the maximum value of the gradient of the curve with equation

y = 2 – 4x + 4x^{\frac{3}{2}} - x^2

where

x > 0

A.-4
B. $\frac{8}{9}$
C. $\frac{1}{2}$
D.2
E.4

Answer: C

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