ecaa 2020 specimen - step maths

ECONOMICS ADMISSIONS ASSESSMENT

Specimen Paper – New Format for 2020 60 minutes SECTION 1 INSTRUCTIONS TO CANDIDATES Please read these instructions carefully, but do not open this question paper until you are told that you may do so. This paper is Section 1 of 2. A separate answer sheet is provided for this paper. Please check you have one. You also require a soft pencil and an eraser. Please complete the answer sheet with your candidate number, centre number, date of birth, and name. At the end of 60 minutes, your supervisor will collect this question paper and answer sheet before giving out Section 2. This paper contains two parts: A and B and you should attempt both parts. Part A Mathematics (20 questions) Part B Advanced Mathematics (20 questions) You are strongly advised to divide your time equally between the two parts: 30 minutes on Part A and 30 minutes on Part B. The scores for Part A and Part B are reported separately. This paper contains 40 multiple-choice questions. There are no penalties for incorrect responses, only marks for correct answers, so you should attempt all 40 questions. Each question is worth one mark. For each question, choose the one option you consider correct and record your choice on the separate answer sheet. If you make a mistake, erase thoroughly and try again. You must complete the answer sheet within the time limit. You can use the question paper for rough working, but no extra paper is allowed. Only your responses on the answer sheet will be marked. Dictionaries and calculators are NOT permitted.

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This question paper consists of 28 printed pages and 4 blank pages.

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PART A Mathematics

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PART A Mathematics

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Paper content PART A Mathematics ............................................................................................................................. 5 PART B Advanced Mathematics .......................................................................................................... 17

PART A Mathematics

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PART A Mathematics

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PART A Mathematics

PART A Mathematics

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1 A square piece of metal has a semicircular piece cut out of it as shown. The area of the remaining metal is 100 cm2.

Which one of the following is a correct expression for the length of the side of the square in centimetres?

A π−8

220

B

π−4210

C

π+8220

D

π−8110

E

π−4120

2 In a right-angled triangle PQR the hypotenuse is the side PR.

The length of side PQ is 20 cm and the ratio RQ : PQ is 1 : 2 What is the length of the perpendicular from the hypotenuse to the point Q?

A 58 cm B 210 cm C 52 cm D 25 cm E 54 cm

PART A Mathematics

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3 A cube has sides of unit length. What is the length of a line joining a vertex to the midpoint of one of the opposite faces (the dashed line in the diagram below)?

A 23

B 2

C 25

D 3

E 5

4 If you look at a clock and the time is 9:45, what is the angle between the hour and the minute

hands? A 0° B 7.5° C 15° D 22.5° E 30°

PART A Mathematics

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5 The right-angled triangle shown has horizontal and vertical sides measuring ( 24 + ) cm and ( 22 − ) cm respectively.

Calculate the area of the triangle.

A ( 235 + ) cm2 B ( 23 − ) cm2 C ( 233 + ) cm2 D ( 25 − ) cm2 6 A solid sphere of radius r fits inside a hollow cylinder. The cylinder has the same internal

diameter and length as the diameter of the sphere. The volume of a sphere is

334 rπ , where r is the radius of the sphere.

What fraction of the space inside the cylinder is taken up by the sphere?

A 41

B

31

C

21

D

32

E

43

PART A Mathematics

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7 Which of the expressions below has the largest value for 0 < x < 1?

A

x1

B 2x

C

)( x+11

D

x1

E x

8 A shape is formed by drawing a triangle ABC inside the triangle ADE.

BC is parallel to DE. AB = 4 cm BC = x cm DE = x + 3 cm DB = x – 4 cm

Calculate the length of DE.

A 5 cm B 7 cm C 9 cm D cm724 + E cm727 +

PART A Mathematics

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9 Two variables are connected by the relation: P2

1Q

∝

Q is increased by 40%. To the nearest percent, describe the change in P in percentage terms.

A 29% decrease B 44% decrease C 49% decrease D 51% decrease E 80% decrease

F 96% decrease 10 Three variables x, y and z are known to be related to each other in the following ways:

x is directly proportional to the square of z. y is inversely proportional to the cube of z.

Which of the following correctly describes the relationship between x and y?

A The square of x is directly proportional to the cube of y. B The square of x is inversely proportional to the cube of y. C The cube of x is directly proportional to the square of y. D The cube of x is inversely proportional to the square of y. E x is directly proportional to y

6.

PART A Mathematics

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11 In the triangle PQR shown below:

X lies on PR ∠QXR is 90°

61

PXQX =

32

XRQX =

M is the midpoint of PR.

What is MXQX ?

A 9

1

B

125

C 94

D 21

E

65

12 Solve the inequality xx 282 −≥ A 4≥x B 2≤x and 4−≥x

C 2−≥x and 4≤x D 2≥x or 4−≤x

PART A Mathematics

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13 The total surface area of a cylinder, measured in square centimetres, is numerically the same as its volume, measured in cubic centimetres. The radius of the cylinder is r cm, the height is h cm. Express h in terms of r.

A h =2

2−rr

B h =

22+rr

C h = 2+r

D h = 2−r

E h = )( 22 −rr

14 How many different integers, n, are there such that the difference between n2 and 7 is less

than 1? A 0 B 2 C 4 D 6 E 8

PART A Mathematics

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15 The square PQRS is positioned so that its vertices are at the points with coordinates: (1, 1), (–1, 1), (–1, –1) and (1, –1). The square is rotated clockwise through 90° about the origin and then reflected in the line y = x Which transformation will return the square to its original orientation?

A A reflection in the x-axis. B A reflection in the y-axis. C A reflection in the line y = – x D A rotation of 90° clockwise about the origin. E A rotation of 90° anticlockwise about the origin. 16 To get to work, Sylvie first catches a bus and then catches a train.

The probability that the bus is on time is 0.6 The probability that the bus is late is 0.4 If the bus is on time, then the probability that she will catch the train is 0.8 If the bus is late, then the probability that she will catch the train is 0.6 Given that Sylvie catches the train, what is the probability that the bus was on time?

A 31

B 2512

C 52

D 53

E 32

F 2518

G 76

PART A Mathematics

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17 A design is set up by joining the points which are one third of the way along the sides of a square. This forms a second square as shown.

This process is repeated. Calculate the area of the fourth square as a fraction of the original square.

A 27

1

B

8155

C

729125

D

8125

E

2755

PART A Mathematics

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18 The equation connects the variables M, x, y, z, P and Q.

QP

zyxM2

)( +=

The following changes are made:

x and y are both increased by 50%

z is decreased by 20%

P is doubled and Q remains the same. What is the resulting percentage change in M ?

A 2.5% decrease B 2.5% increase C 10% decrease D 10% increase E 20% decrease F 20% increase 19 I have two six-sided dice, each with faces numbered from 1 to 6. One of the dice is fair, but the

other is not; it will land on numbers 1 to 5 with equal probability, but lands on 6 with a different probability.

When I roll the dice the probability that I get a total of 12 is 181 .

What is the probability that I get a total of 2 when I roll the dice?

A 72

1

B 451

C 361

D 181

E 91

PART A Mathematics

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20 A cross-country running track is in the shape of a regular pentagon.

[diagram not to scale] Competitors run clockwise around the track. When on the third leg of the course they run on a bearing of 110°. What bearing do they run on for the first leg?

A 034° B 038° C 106° D 178° E 182° F 244° G 322° H 326°

PART B Advanced Mathematics

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18

21 Given that 232 =xxx cba , where a, b and c are positive real numbers, then x =

A

++ cba 322

10log

B )(log

logcba 32

2

10

10

++

C )(log cba 32

2

10 ++

D

cba 322++

E

32102cab

log

F )(log

log32

10

102

cab

G )(log 32

10

2cab

H 32

2cab

22 Which one of the following is a simplification of xx

x24

2

2

−− where x ≠ 2 and x ≠ 0?

A 24

−−

xx

B

xx 2−

C

x2

D

xx 2+

E

12

++

xx


19

23

Which one of the following numbers is largest in value? (All angles are given in radians.)

A

π

43tan

B 10010log

C

π

210sin

D 102log E ( )10

12 − 24 When x = 2 is substituted in the expression x3 + px2 + qx + p2 the result is 0

When x = 1 is substituted into the same expression, the result is –3.5 Find all possible values of p.

A p = –1 ± 36

B p = 1 or p = –3 C p = 1 D p = 1 ± 7 E there are no values for p


20

25 The sum of the roots of the equation 0152822 =+×− xx is A 3 B 8 C 2 log10 2

D log10

415

E 2

15

10

10

loglog

26 A square PQRS is drawn above the x-axis with the side PQ on the x-axis.

P is the point (–5, 0) and Q is the point (1, 0). A circle is drawn inside the square with diameter equal in length to the side of the square. Which one of the following is an equation of the circle?

A x2 + y2 – 4x + 6y + 4 = 0 B x2 + y2 – 4x + 6y + 9 = 0 C x2 + y2 + 4x – 6y + 4 = 0 D x2 + y2 + 4x – 6y + 9 = 0 E x2 + y2 – 6x – 4y + 9 = 0 F x2 + y2 – 6x + 4y + 4 = 0 G x2 + y2 + 6x – 4y + 4 = 0 H x2 + y2 + 6x + 4y + 9 = 0


21

27 For any real numbers a, b and c where a≥ b, consider these three statements:

1 ab −≥−

2 abba 222 ≥+

3 bcac ≥ Which of the above statements must be true?

A none of them B 1 only C 2 only D 3 only E 1 and 2 only F 1 and 3 only G 2 and 3 only H 1, 2 and 3


22

28 The first term of a convergent geometric series is 8 The fifth term is 2 The sixth term is real and positive. What is the sum to infinity of this series?

(The sum to infinity of a convergent geometric series is given by r

a−1

, where a is the first term

and r is the common ratio.) A ( )218 + B ( )218 − C ( )228 + D ( )228 − E 16

F 14

485

5

−

G

1448

5

5

+


23

29 The sequence na is given by the rule: 21 =a

( )nnn aa 11 −+=+ for 1≥n

What is ∑=

100

1nna

A 150 B 250 C − 4750 D 5150

E

−

100

2114

F

−

1

234

100


24

30 A box is a hollow pyramid. The base of the box is a square with sides 10 cm and all the slant edges of the box are 12 cm long.

What is the angle made by the slant edge TP with the base PQRS?

A 12

521−sin

B

125

1−sin

C

1225

1−sin

D

12521−cos

E 12

51−cos

F

12251−cos

31 How many real roots does the equation 01044 234 =−+− xxx have? A 0 B 1 C 2 D 3 E 4


25

32 Tangents are drawn from a point P to a circle of radius 10 cm. The centre of the circle is C and the distance PC is 20 cm.

Which one of the following is an expression for the shaded area in square centimetres?

A ( )π−33

3100

B ( )π−53

3100

C ( )π−363

50

D ( )π−563

50

E ( )π233

50 −

F ( )32

350 −π

33 Given that 7cos θ – 3tan θ sin θ = 1, which one of the following is true?

A cos θ = 53− or

21−

B cos θ =

53− or

21

C cos θ =

53 or

21

D cos θ =

53 or

21−


26

34 A triangle is to be drawn with sides that are integer lengths in centimetres, and a total perimeter of 12 cm. How many different (non-congruent) triangles can be drawn?

A 1 B 2 C 3 D 10 E 12 35 For what values of the non-zero real number a does the quadratic equation

( ) 222 =−+ xaax

have distinct real roots?

A all values of a B 2−=a C 2−>a D 2−≠a E no values of a


27

36 The complete set of values of a for which the equation 3x

2 = (a + 2) x – 3 has two real distinct roots is:

A no values of a B 2424 <<− a C 24,24 >−< aa D 84 <<− a E 8,4 >−< aa F 48 <<− a G 4,8 >−< aa H all values of a 37 The angle x is measured in radians and is such that π≤≤ x0 .

The total length of any intervals for which 11 ≤≤− xtan and 502 .sin ≥x is

A 12π

B

6π

C

4π

D

3π

E

125π

F

2π

G

65π


28

38 The straight line with equation y = mx + 3 , where m > 0 and m ≠ 1, is perpendicular to the line with equation y = px + 2 The lines cut the x-axis at the points L and M respectively. The length of LM is 5 units. What is the value of m + p given that m > 1 ?

A

38−

B

613−

C

65−

D

65

E 6

13

F

38


29

39 The curve y = x2 is translated by the vector

34

and then reflected in the line y = –1

Which one of the following is an equation of the resulting curve?

A y = – 3 – (x – 4)2 B y = – 3 + (x + 4)2 C y = 3 – (x + 4)2 D y = 3 + (x – 4)2 E y = – 5 – (x – 4)2 F y = – 5 + (x + 4)2 G y = 5 – (x + 4)2 H y = 5 + (x – 4)2 40 The triangle PQR has a right angle at R.

The length of PQ is 4 cm, correct to the nearest centimetre. The length of PR is 2 cm, correct to the nearest centimetre. Find the minimum possible length, in centimetres, of QR.

A 216 −

B

2132 −

C

2152 −

D 52

E 32

F 6

END OF TEST


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ECONOMICS ADMISSIONS ASSESSMENT

Specimen Paper – New Format for 2020 60 minutes SECTION 2 Candidate number K Centre number

d d m m y y y y Date of birth – – First name (s)

Surname / Family name

INSTRUCTIONS TO CANDIDATES Please read these instructions carefully, but do not open the question paper until you are told that you may do so. This paper is Section 2 of 2. This question paper requires you to read a single passage and complete a related task. You should write your answer in the space provided in this question paper. Please complete this section in black pen. You can use the blank inside front and back covers for rough working or notes, but no extra paper is allowed. Only answers in the space indicated in the paper will be marked. Dictionaries and calculators are NOT permitted.

Please wait to be told you may begin before turning this page.

This question paper consists of 10 printed pages and 2 blank pages. Copyright © UCLES 2020 *

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Read the article ‘Should America want a strong dollar? It’s complicated’ taken from The Economist (February 9th 2017). Based on this text, complete the task below.

TASK

• Explain why in some circumstances a rise in the value of the US dollar might benefit American residents, while in other circumstances it would be harmful to them.

• Suggest possible reasons why the US dollar might be persistently overvalued. Your answer will be assessed taking into account your ability to construct a reasoned, insightful and logically consistent argument with clarity and precision. Should America want a strong dollar? It’s complicated In an experience we have all had, Donald Trump, unable to sleep, reportedly rang his National Security Advisor at 3:00 in the morning to ask whether America should want a strong dollar or a weak one. Look, there are no stupid questions. This one is actually more interesting than you might think. There are a few ways to answer it. One way is to ask what it would mean if there were a random rise in the dollar (caused, say, by noise trading). Tyler Cowen takes this approach. Since most Americans have their incomes and wealth in dollars, a rise in the dollar would generally be good for America and bad for non-Americans: with caveats, including the fact that a stronger dollar might cause financial trouble for indebted emerging markets which could feed back into the American economy. That’s fine and all, but the dollar usually does what it does for a reason, or some set of reasons, and the answer to the question depends on why the dollar is strong or weak. Matt Yglesias gives his answer using this approach. If the dollar is rising because the Fed is inducing a deflationary recession, that’s probably not good for anyone. (This is not outside the realm of plausibility, as it happens; Britain suffered years of painful deflation in the 1920s in order to defend sterling’s peg to gold at an overvalued rate, only to then subsequently devalue during the Depression.) If the dollar is rising because America has discovered something amazing (like oil, or pluots) and the world cannot get enough of it, then that’s probably good for everyone – though it is worth remembering that in all of these cases there are winners and losers, in America and elsewhere. But another way to approach the question is to ask whether the dollar is overvalued or undervalued. If the dollar is overvalued, then a bout of weakening would be a healthy thing. How can you tell if a currency is overvalued or undervalued? Well, you can consult your nearest Big Mac index. Alternatively, you can check the current-account balance. If a country is persistently running current-account surpluses, its currency is probably undervalued; deficits, overvalued. America has been running current-account deficits since the early 1980s, therefore its currency is overvalued and a weaker dollar would be better. Ok, but hold on. How is it that the dollar hasn’t adjusted to its appropriate value after nearly 40 years. That, it turns out, is the critical question. The dollar has been overvalued all this time because the dollar is a reserve currency. Not just a reserve currency, but rather the world’s dominant reserve currency, by a long shot. It holds that position partly for reasons of path dependence: because the dollar was the anchor currency in the postwar Bretton Woods agreement, other economies accumulated lots of dollar reserves to help maintain their fixed exchange rate. It holds the position partly because of network externalities; if lots of people do their trade invoicing in dollars, it is convenient for you to also do your business stuff in dollars to keep everything simple. And it holds that position in large part because no other currency, or country, is as capable of providing reserve-currency services. America has a very large and rich and stable economy, a high tax-revenue generating capacity, an extremely credible central bank, a

4 stable democracy (knock wood), massive military and diplomatic power, broad and deep financial markets, and so on. The point is: America can provide large amounts of dollars, and government bonds, and other dollar-denominated assets without getting anywhere near the sorts of crises that might lead to sudden or unexpected changes in value in those dollars and bonds and such. Just as important, America has historically accepted the role of reserve-currency provider and the responsibilities that entails: like cooperating with other countries in crises to provide emergency dollar liquidity. America is the total package, or has been anyway. What that has tended to mean, then, is that foreign firms and governments buy American government bonds by the truckload. Foreign appetite for dollars and dollar assets – even the super boring low-yield ones – allows Americans to buy more than they produce. It seems like a win-win situation. The dollar-as-a-reserve currency is a public good of sorts, a lubricant for global trade and finance, provided by America to the rest of the world. In return, Americans get to buy more than they make, forever. As explained elsewhere in this issue of the Economist, there are downsides, though:

“An overvalued currency and persistent trade deficits are fine for America’s consumers, but painful for its producers. The reserve accumulation of the past two decades has gone hand-in-hand with a soaring current-account deficit in America. Imports have grown faster than exports; new jobs in exporting industries have not appeared in numbers great enough to absorb workers displaced by increased foreign competition. Tariffs cannot fix this problem. The current-account gap is a product of underlying financial flows, and taxing imports will simply cause the dollar to rise in an offsetting fashion. America’s privilege also increases inequality, since lost jobs in factories hurt workers while outsize investment performance benefits richer Americans with big portfolios. Because the rich are less inclined to spend an extra dollar than the typical worker, this shift in resources creates weakness in American demand – and sluggish economic growth – except when consumer debt rises as the rich lend their purchasing power to the rest. Chalk the headaches generated by low interest rates up to the dollar standard, too. Some economists reckon they reflect global appetite outstripping the supply of the safe assets America is uniquely equipped to provide – dollar-denominated government bonds. As the price of safe bonds rises, rates on those bonds fall close to zero, leaving central banks with ever less room to stimulate their economies when they run into trouble.”

So one could ask whether a strong dollar is better or a weak one. But one could also ask whether it is in America’s long-term interest to continue supporting the global dollar standard. In my view, it is neither sustainable or desirable for America to continue to play this role on its own. On the other hand, the status quo is preferable to the chaotic destruction of the dollar standard in the absence of a ready replacement. Put in the shoes of Mike Flynn, Mr Trump’s National Security Advisor, I’m not sure quite how I’d respond. But the answer would start with, ‘First, Mr President, give me your phone’.

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ecaa 2020 specimen - step maths

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