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ESAT Diagnostic

Fifteen mock ESAT questions, five each from Mathematics 1, Mathematics 2 and Physics, taken from the sample papers published here. Answer all fifteen, then open each answer, count your score in each module, and read the full worked solution for every question you missed. Full timed papers, 27 questions a module and marked the moment you finish, are in ESAT practice, free to start.

How to sit it

15 questions, five from each of Mathematics 1, Mathematics 2 and Physics.
About 20 minutes. The real test allows about 89 seconds a question. Note the time when you start; running over does not change your score here.
No calculator, no formula sheet, exactly as on the day.
Commit before you look. Write down a letter for all fifteen first, then open the answers. An answer opened before you commit tells you nothing.

There is no negative marking on the ESAT. If you do not know, narrow it down and commit to a guess.

Mathematics 1

1. Number

A metering pump delivers water at a steady rate, putting out $(3 + \sqrt{5}) L$ in $(9 - \sqrt{5}) min$ . How many litres does it deliver in $19 min$ , correct to 3 significant figures?

A. 24.5
B. 673
C. 14.7
D. 99.5
E. 518
F. 1.31
G. 8.00
H. 13.8

Answer

C. 14.7 The worked solution and why each other option is wrong (Paper 1, question 1).

2. Ratio and proportion

A right-angled triangular bracket has an upright edge of $40 c m$ , and the angle between its sloping and horizontal edges has tangent $\frac{5}{2}$ . A second bracket, mathematically similar and cut from the same steel sheet, has an upright edge of $10 c m$ . Find the area of the second one in $c m^{2}$ .

A. 20
B. 80
C. 5
D. 320
E. 125

Answer

A. 20 The worked solution and why each other option is wrong (Paper 2, question 2).

3. Algebra

On a plan drawn in metres, a water main is the straight line through $(4, 3)$ and $(- 2, 15)$ . A sewer connection is set out perpendicular to it through the point $(4, 3)$ . Find the $y$ intercept of the sewer connection.

A. 11
B. 1
C. 5
D. 16
E. -2

Answer

B. 1 The worked solution and why each other option is wrong (Paper 3, question 3).

4. Algebra

In a freezing tunnel at a food factory, $t$ hours after the conveyor is started, the temperature in degrees Celsius is $T = t^{2} - 20 t + 51$ . How many hours pass between the temperature first falling to $0^{\circ} C$ and its return to $0^{\circ} C$ ?

A. 14
B. 7
C. 10
D. 17
E. 3

Answer

A. 14 The worked solution and why each other option is wrong (Paper 4, question 4).

5. Algebra

A printing firm's chart records how the time $T$ to finish a fixed print run, in minutes, varies with the printing rate $r$ , in pages per minute. The graph is a smooth curve lying entirely in the first quadrant, and both coordinate axes are asymptotes to it. At $r = 6$ the curve passes through $T = 114$ . Reading from the curve, what is $T$ , in minutes, when $r = 4$ ?

A. 171
B. 76
C. 256.5
D. 342
E. 116
F. 2736

Answer

A. 171 The worked solution and why each other option is wrong (Paper 1, question 5).

Mathematics 2

1. Algebra and functions

For some $x > 0$ it is given that $\frac{x^{2} \cdot x^{- 1 / 3}}{x^{4 / 3}} = 3$ . Correct to 3 significant figures, what does $x^{- 2}$ equal?

A. 0.0185
B. 0.111
C. 0.00137
D. 27.0
E. 729
F. 0.481
G. -0.00137

Answer

C. 0.00137 The worked solution and why each other option is wrong (Paper 2, question 1).

2. Algebra and functions

Simplify $\frac{\sqrt{21} - \sqrt{11}}{\sqrt{21} + \sqrt{11}}$ by clearing every surd out of its denominator. Which of the values below is the result, correct to 3 significant figures?

A. 6.24
B. 3.36
C. 0.0501
D. -2.04
E. 0.160
F. 0.127
G. 1.68
H. 0.559

Answer

E. 0.160 The worked solution and why each other option is wrong (Paper 3, question 2).

3. Algebra and functions

The horizontal line $y = k$ is a tangent to the curve $y = x^{2} + 8 x + 4$ . Find the value of $k$ .

A. -12
B. 20
C. 4
D. -60
E. 12
F. 48

Answer

A. -12 The worked solution and why each other option is wrong (Paper 4, question 3).

4. Algebra and functions

The curve $y = 2 x^{2} - 8 x - 2$ is cut in two places by the line $y = - x - 3$ . Give the horizontal distance between the two points of intersection, correct to 3 significant figures.

A. 4.47
B. 4.03
C. 3.74
D. 3.77
E. 3.20
F. 1.60

Answer

E. 3.20 The worked solution and why each other option is wrong (Paper 1, question 4).

5. Sequences and series

A sequence satisfies $u_{n + 1} = \frac{u_{n} + 9}{2}$ , with sixth term $12$ . Find $u_{1}$ .

A. 9
B. 105
C. 33
D. 96
E. 57

Answer

B. 105 The worked solution and why each other option is wrong (Paper 2, question 5).

Physics

1. Electricity

In a laboratory exercise on rectification, a $14 V$ supply of negligible internal resistance is connected across two vertical rails with its positive terminal at the left-hand rail, and three parallel branches bridge the rails. The top branch carries a $1 Ω$ resistor and a diode whose arrow points away from the left-hand rail. The middle branch carries a $4 Ω$ resistor and nothing else. The bottom branch carries a $7 Ω$ resistor and a diode whose arrow points back towards the left-hand rail. The diodes are ideal, and the leads and the ammeter in the main lead have no resistance. What is the ammeter reading in amperes?

A. 19.5
B. 5.5
C. 3.5
D. 17.5
E. 0.8
F. 11.2
G. 245.0
H. 1.3

Answer

D. 17.5 The worked solution and why each other option is wrong (Paper 3, question 1).

2. Magnetism and electromagnetism

One wire is folded so that its 5 straight strands all pass through a uniform magnetic field at right angles to it, each over a length of $0.22 m$ , while the bends joining them lie outside the field. The flux density is $1.25 T$ , the current has the same direction in every strand, and the total force on the wire is $3.3 N$ . Determine the current.

A. 12
B. 0.48
C. 3
D. 2.64
E. 2.4
F. 0.66
G. 24
H. 60

Answer

E. 2.4 The worked solution and why each other option is wrong (Paper 4, question 2).

3. Mechanics

A cyclist sets out along a straight canal towpath. The cyclist travels $600 m$ in the first $100 s$ , then waits $200 s$ at a lock gate without moving, and finally travels a further $1200 m$ in $300 s$ before arriving. What is the average speed for the journey as a whole, in $m s^{- 1}$ ?

A. 3
B. 2
C. 4
D. 4.5
E. 6
F. 10

Answer

A. 3 The worked solution and why each other option is wrong (Paper 1, question 3).

4. Mechanics

A bag of nails of mass $9 kg$ falls vertically through the air. At one instant the drag on the bag is $18 N$ , and the drag and the weight are the only forces acting. Taking $g = 10 {N kg}^{- 1}$ , find the acceleration of the bag at that instant.

A. $2 {m s}^{- 2}$ downwards
B. $12 {m s}^{- 2}$ downwards
C. $10 {m s}^{- 2}$ downwards
D. $8 {m s}^{- 2}$ downwards
E. $7.2 {m s}^{- 2}$ downwards
F. $0.8 {m s}^{- 2}$ downwards
G. $9.8 {m s}^{- 2}$ downwards

Answer

D. $8 {m s}^{- 2}$ downwards The worked solution and why each other option is wrong (Paper 2, question 4).

5. Waves

A steady longitudinal wave is set up in a long stretched spring by a vibration generator clamped to one end, running at a frequency three times the frequency of a reference oscillator set to $4 H z$ . The wave travels at $2.4 m s^{- 1}$ , and a detector is moved slowly along its line of travel. What is the shortest distance in metres from one compression to the next compression?

A. 0.2
B. 0.1
C. 0.6
D. 0.05
E. 1.8
F. 0.3
G. 5.4

Answer

A. 0.2 The worked solution and why each other option is wrong (Paper 3, question 5).

Score yourself

Count the answers you got right in each module, out of five. The module with the lowest count is the one to open first.

Mathematics 1: practice questions by topic
Mathematics 2: practice questions by topic
Physics: practice questions by topic

Each question's heading names its topic. For a question you missed, read its worked solution before anything else: the line on the option you chose says which mistake leads there.

What a 15-question sample can and cannot do

Being straight about this matters, because acting on a bad signal costs a candidate weeks.

Module scores are usable. Five questions per module is a coarse instrument, but the gap between five out of five and one out of five is real information, and it is the fastest way to find out which module is quietly costing you the most.
A topic is a pointer, not a verdict. Each topic here carries one or two questions. Treat a miss as a suggestion about what to open first, not as evidence that the topic is a weakness.
Speed here is not speed on the day. You are sitting fifteen questions at a desk you chose, not twenty-seven under exam conditions in a test centre.
There is no predicted grade. ESAT modules are scored 1.0 to 9.0 and the scaling is not published. Anyone converting fifteen questions into a predicted band is guessing.

How this is put together

Nothing written for the diagnostic

Every one of the fifteen is taken word for word from the sample papers published here, each with a full worked solution. Nothing was written to make the diagnostic look good, and nothing sits behind a signup.

Five, five and five

Mathematics 1 is compulsory for every candidate. Mathematics 2 and Physics are the two most commonly required options. Equal weight means the module comparison is a fair one.

Nothing is sent

This page has no form and no script. Your answers stay on your paper.

Read how every solution here is checked before it is published, or go to the ESAT guide.

For individual preparation. These questions are free for a candidate working on their own. A school or college looking for mock papers for its candidates wants our schools page.

ESAT Diagnostic

How to sit it

Mathematics 1

1. Number

2. Ratio and proportion

3. Algebra

4. Algebra

5. Algebra

Mathematics 2

1. Algebra and functions

2. Algebra and functions

3. Algebra and functions

4. Algebra and functions

5. Sequences and series

Physics

1. Electricity

2. Magnetism and electromagnetism

3. Mechanics

4. Mechanics

5. Waves

Score yourself

What a 15-question sample can and cannot do

How this is put together

Nothing written for the diagnostic

Five, five and five

Nothing is sent

Keep going