ENGAA 2022 (Paper 2)

• A.
  $$\frac{3xz^2}{y^4}$$
• B.
  $$\frac{3xz^2}{y^5}$$
• C.
  $$\frac{9x^\frac{1}{2}z^2}{y^5}$$
• D.
  $$\frac{9xz^2}{y^4}$$
• E.
  $$\frac{9xz^2}{y^5}$$
• F.
  $$\frac{9x^\frac{5}{2}z^2}{y^5}$$

Answer: E

• A.0.0060 N
• B.0.36 N
• C.0.60 N
• D.0.81 N
• E.36 N
• F.49 N
• G.81 N
• H.4900 N

Answer: A

• A.
  $$x < \frac{1}{3}$$
• B.
  $$x > \frac{1}{3}$$
• C.
  $$x < \frac{3}{5}$$
• D.
  $$x > \frac{3}{5}$$
• E.
  $$x < \frac{3}{4}$$
• F.
  $$x > \frac{3}{4}$$
• G.
  $$x < \frac{3}{2}$$
• H.
  $$x > \frac{3}{2}$$

Answer: D

• A.0.025 C
• B.0.050 C
• C.0.15 C
• D.0.30 C
• E.0.75 C
• F.1.5 C
• G.9.0 C
• H.18 C

Answer: G

• A.£15.12
• B.£35.28
• C.£117.60
• D.£200
• E.£210
• F.£300
• G.£1200

Answer: D

Ultrasound is used to find a crack inside a cuboid block of metal. An ultrasound probe is held in contact with the top surface of the metal block and perpendicular to the surface. A short pulse of ultrasound is sent into the metal block at time $t = 0$ ms and reflects from both the crack and the bottom surface of the metal block.

The times between the emission of the ultrasound pulse and the return of the reflections to the probe, and the strengths of the reflected pulses, are measured. The results are shown on the graph.

The speed of ultrasound in the metal is $5000 \text{ms}^{-1}$.

What is the distance between the $\textbf{bottom\;surface}$ of the metal block and the crack?

• A.0.2 m
• B.0.3 m
• C.0.4 m
• D.0.5 m
• E.0.6 m
• F.1.0 m

Answer: A

• A.
  $$\frac{(x-4)(x+2)}{(x-3)(x+4)}$$
• B.
  $$\frac{(x-3)(x+2)}{(5x-3)(x+3)}$$
• C.
  $$\frac{(x-4)(x+2)}{(5x-3)(x+4)}$$
• D.
  $$\frac{(x-4)(x-3)}{(5x-3)(x-6)}$$
• E.
  $$\frac{(x+2)}{(5x+3)}$$
• F.
  $$\frac{(x+4)(x-6)}{(5x+3)(x+2)}$$
• G.
  $$\frac{(x-3)(x+2)}{(5x+3)(x+3)}$$

Answer: C

• A.0.160 J
• B.0.240 J
• C.160 J
• D.240 J
• E.576 J
• F.864 J
• G.576 000 J
• H.864 000 J

Answer: G

A rectangle PQRS has length $(2x – 1)$ cm and width $(x + 1)$ cm as shown on the diagram.

A larger rectangle is made by adding 3 cm to both the length and the width of PQRS, as shown.

The larger rectangle has an area of $360 \text{cm}^2$.

[diagram not to scale]

What is the ratio of PQ to PS?

• A.1:2
• B.4:7
• C.5:8
• D.7:11
• E.10:17
• F.17:31

Answer: B

The graph shows potential difference plotted against current for a filament lamp and a resistor.

The lamp and the resistor are connected in parallel with each other to a 6.0 V power supply and the current in the lamp, $I$, is recorded.

In a second circuit, the lamp and the resistor are now connected in series with each other to the same power supply, and the current in the resistor is 0.18 A. The potential difference across the lamp, $V$, is recorded.

What are the values of $I$ in the first circuit and $V$ in the second circuit?

| | $I$ / A | $V$ / V |
|---|---|---|
| A | 0.25 | 1.6 |
| B | 0.25 | 3.0 |
| C | 0.25 | 4.4 |
| D | 0.35 | 1.6 |
| E | 0.35 | 3.0 |
| F | 0.35 | 4.4 |

• A.$I = 0.25 \text{ A}$, $V = 1.6 \text{ V}$
• B.$I = 0.25 \text{ A}$, $V = 3.0 \text{ V}$
• C.$I = 0.25 \text{ A}$, $V = 4.4 \text{ V}$
• D.$I = 0.35 \text{ A}$, $V = 1.6 \text{ V}$
• E.$I = 0.35 \text{ A}$, $V = 3.0 \text{ V}$
• F.$I = 0.35 \text{ A}$, $V = 4.4 \text{ V}$

Answer: D

PQRS is a trapezium as shown.

[diagram not to scale]

$\tan RSQ = \frac{5}{8}$

What is the length of PS, in metres?

• A.45
• B.65
• C.80
• D.120
• E.$25 + \frac{40\sqrt{3}}{3}$
• F.$40 + \frac{64\sqrt{3}}{3}$
• G.$25 + 40\sqrt{3}$
• H.$64 + 40\sqrt{3}$

Answer: G

A transverse wave on a string has a speed of $500 \text{ms}^{-1}$.

The horizontal distance between two points P and Q on the wave is 4.0 m, as shown in the diagram.

At time $t = 0$ ms, point X on the string is at its maximum displacement of 6.0 mm above equilibrium.

What is the displacement of point X at time $t = 7.0$ ms?

• A.6.0 mm above equilibrium
• B.between 0 mm and 6.0 mm above equilibrium
• C.0 mm
• D.between 0 mm and 6.0 mm below equilibrium
• E.6.0 mm below equilibrium

Answer: C

• A.$(\frac{18}{\pi} - 2)r$
• B.$(\frac{18}{\pi} - 1)r$
• C.
  $$\frac{27r}{\pi}$$
• D.$(\frac{27}{\pi} - 1)r$
• E.$(\frac{27}{4\pi} - 1)r$
• F.
  $$\frac{108r}{\pi}$$
• G.$(\frac{108}{\pi} - 1)r$
• H.$(\frac{108}{\pi} - \frac{1}{2})r$

Answer: G

• A.38 °C
• B.51 °C
• C.150 °C
• D.170 °C
• E.480 °C

Answer: D

• A.
  $$y = \sqrt[3]{\frac{x-4}{8x-48}}$$
• B.
  $$y = \sqrt[3]{\frac{x-6}{8x-32}}$$
• C.
  $$y = \sqrt[3]{\frac{x-2}{x-6}}$$
• D.
  $$y = \sqrt[3]{\frac{x-3}{x-4}}$$
• E.
  $$y = \sqrt[3]{\frac{x-4}{2x-12}}$$
• F.
  $$y = \sqrt[3]{\frac{x-6}{2x-8}}$$

Answer: E

A bar magnet is placed at position X close to one end of a coil and on the axis of the coil as shown.

The graph shows how the velocity of the magnet varies as it is then moved rapidly to position Y and back to position X.

The magnetic field of the bar magnet still affects the coil when the magnet is at position Y.

Which graph represents how the induced voltage in the coil changes as the magnet moves?

• A.Graph A
• B.Graph B
• C.Graph C
• D.Graph D
• E.Graph E
• F.Graph F

Answer: D

• A.
  $$\frac{1}{15}$$
• B.
  $$\frac{1}{10}$$
• C.
  $$\frac{3}{10}$$
• D.
  $$\frac{1}{3}$$
• E.
  $$\frac{2}{5}$$
• F.
  $$\frac{2}{3}$$
• G.
  $$\frac{4}{5}$$

Answer: E

A small slider of mass 30 g is at rest near the bottom of a frictionless slope and in contact with a light uncompressed spring as shown.

[diagram not to scale]

The spring is compressed by 5.0 cm and the slider remains in contact with it.

The spring is released and causes the slider to rise up the slope to a maximum vertical height of 20 cm.

The slider is replaced with one of mass 20 g.

The spring is now compressed by 15 cm, and the new slider remains in contact with it.

To what maximum vertical height does this new slider rise after it is released?

(the spring obeys Hooke's law; assume that air resistance is negligible)

• A.40 cm
• B.60 cm
• C.90 cm
• D.120 cm
• E.180 cm
• F.270 cm

Answer: F

• A.-14
• B.-7
• C.-5
• D.-1
• E.1
• F.5
• G.7
• H.14

Answer: D

• A.$0.960 \text{kg m}^{-3}$
• B.$1.20 \text{kg m}^{-3}$
• C.$1.25 \text{kg m}^{-3}$
• D.$1.28 \text{kg m}^{-3}$
• E.$1.50 \text{kg m}^{-3}$
• F.$4.80 \text{kg m}^{-3}$

Answer: E

• A.
  $$\frac{19}{3}$$
• B.
  $$\frac{37}{3}$$
• C.
  $$\frac{53}{3}$$
• D.
  $$\frac{73}{4}$$
• E.
  $$\frac{81}{4}$$
• F.
  $$\frac{87}{4}$$

Answer: A

The diagram represents a stationary wave in a medium.

The transverse waves that are creating the stationary wave travel at a speed of $300 \text{ms}^{-1}$ through the medium.

What is the frequency of the transverse waves?

• A.75 Hz
• B.150 Hz
• C.200 Hz
• D.450 Hz
• E.600 Hz
• F.1200 Hz

Answer: E

The diagram shows a semicircle of radius 5 units and a triangle.

The triangle has vertices at (0,0), (10,0) and $(x,y)$.

$(x,y)$ is a point on the arc of the semicircle.

Which of the following is an expression in terms of $x$ for the area of this triangle?

• A.
  $$5\sqrt{10x - x^2}$$
• B.
  $$5x\sqrt{10-x}$$
• C.
  $$5x\sqrt{10x - x^2-20}$$
• D.$15x$
• E.$25x$
• F.$5x(10-x)$

Answer: A

• A.
  $$\left( \frac{10}{1.6} \right) h$$
• B.
  $$\left( \frac{10}{1.6} \right)^2 h$$
• C.
  $$\left( \frac{10}{3.2} \right) h$$
• D.
  $$\left( \frac{10}{3.2} \right)^2 h$$
• E.
  $$\left( \frac{10}{8.4} \right) h$$
• F.
  $$\left( \frac{10}{8.4} \right)^2 h$$
• G.
  $$\left( \frac{20}{1.6} \right) h$$
• H.
  $$\left( \frac{20}{1.6} \right)^2 h$$

Answer: A

• A.W Y X Z
• B.W Y Z X
• C.W Z Y X
• D.Y W X Z
• E.Y W Z X
• F.Y Z W X

Answer: B

A ray of light is incident on a boundary between a vacuum and medium X at an angle $\theta$ as shown:

The incident ray is partially reflected and partially refracted. The angle between the reflected and refracted rays is 90°.

What is the refractive index of medium X?

• A.
  $$\sin\theta$$
• B.
  $$\frac{1}{\sin\theta}$$
• C.
  $$\cos\theta$$
• D.
  $$\frac{1}{\cos\theta}$$
• E.
  $$\tan\theta$$
• F.
  $$\frac{1}{\tan\theta}$$

Answer: E

• A.
  $$\sqrt{5}-1$$
• B.
  $$\frac{\sqrt{5}-1}{2}$$
• C.
  $$\frac{6(5\sqrt{5}-1)}{31}$$
• D.
  $$\frac{3(5\sqrt{5}-1)}{31}$$
• E.
  $$\frac{-22+10\sqrt{2}+11\sqrt{5}-4\sqrt{10}}{2}$$
• F.
  $$\frac{-22+10\sqrt{2}+11\sqrt{5}-4\sqrt{10}}{4}$$

• 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

Answer: F

• A.
  $$\frac{21}{40}$$
• B.
  $$\frac{3}{4}$$
• C.
  $$\frac{13}{12}$$
• D.
  $$\frac{157}{120}$$
• E.
  $$\frac{11}{6}$$
• F.
  $$\frac{23}{12}$$

Answer: B

• A.
  $$\frac{2}{3}R$$
• B.
  $$\frac{8}{9}R$$
• C.$2R$
• D.$3R$
• E.$8R$
• F.$9R$

Answer: B

• A.
  $$-\sqrt{3} < x < \sqrt{3}$$
• B.
  $$-\sqrt{7} < x < \sqrt{7}$$
• C.
  $$x < -\sqrt{3} \text{ \;or\; } x > \sqrt{3}$$
• D.
  $$x < -\sqrt{7} \text{ \;or\; } x > \sqrt{7}$$
• E.
  $$-\sqrt{7} < x < -\sqrt{3} \text{ \;or\; } \sqrt{3} < x < \sqrt{7}$$
• F.
  $$x < -\sqrt{7} \text{ \;or\; } -\sqrt{3} < x < \sqrt{3} \text{\; or\; } x > \sqrt{7}$$

Answer: A

An empty bucket has a mass of 1.20 kg and an internal volume of $0.0150 \text{m}^3$. The bucket is used to lift water from a well.

The bucket is attached to a light, inextensible rope which winds onto a rotating cylinder of radius 0.200 m when a handle is turned.

To lift a bucket completely full of water at constant speed, it is necessary to apply a force of 250 N to the handle that acts along the tangent to the circle of radius 0.600 m, in which the handle moves.

The energy required to lift the bucket is wasted energy.

What is the efficiency of the system in lifting water from the well?

(gravitational field strength = $10 \text{Nkg}^{-1}$; density of water = $1000 \text{kg m}^{-3}$)

• A.2.00%
• B.2.16%
• C.20.0%
• D.21.6%
• E.60.0%

Answer: C

• A.
  $$3\sqrt{2}$$
• B.
  $$2\sqrt{6}$$
• C.
  $$3\sqrt{6}$$
• D.6
• E.12
• F.18
• G.36

Answer: D

• A.15 cm
• B.20 cm
• C.25 cm
• D.35 cm
• E.65 cm
• F.75 cm
• G.80 cm
• H.85 cm

Answer: E

• A.6
• B.6.25
• C.30
• D.31.25
• E.35
• F.36.75
• G.42
• H.43.75

Answer: D

A circuit comprising two resistors and two batteries, with negligible internal resistance, is set up as shown in the diagram. The two junctions in the circuit are labelled X and Y.

What are the magnitude and direction of the current in the 3.0 V battery?

• A.magnitude of current / A: 1.0, direction of current: from X to Y
• B.magnitude of current / A: 2.0, direction of current: from X to Y
• C.magnitude of current / A: 2.25, direction of current: from X to Y
• D.magnitude of current / A: 6.0, direction of current: from X to Y
• E.magnitude of current / A: 1.0, direction of current: from Y to X
• F.magnitude of current / A: 2.0, direction of current: from Y to X
• G.magnitude of current / A: 2.25, direction of current: from Y to X
• H.magnitude of current / A: 6.0, direction of current: from Y to X

Answer: F

• A.
  $$\frac{\sqrt{2}}{5}a$$
• B.
  $$\frac{\sqrt{5}}{2}a$$
• C.
  $$\frac{2\sqrt{5}}{5}a$$
• D.
  $$\frac{4\sqrt{5}}{5}a$$
• E.
  $$\frac{3\sqrt{10}}{5}a$$

Answer: C

Two isolated spheres have masses $m$ and $2m$. They are moving towards each other along the same straight line with speeds $4v$ and $v$ respectively as shown:

The spheres collide with each other and coalesce.

What is the loss of kinetic energy during the collision?

• A.
  $$\frac{1}{3}mv^2$$
• B.
  $$\frac{2}{3}mv^2$$
• C.
  $$\frac{25}{3}mv^2$$
• D.
  $$\frac{26}{3}mv^2$$
• E.
  $$\frac{2}{9}mv^2$$
• F.
  $$\frac{79}{9}mv^2$$
• G.
  $$9mv^2$$

Answer: C

• A.8
• B.16
• C.
  $$2^{\frac{4+\sqrt{2}}{2}}$$
• D.
  $$2^{\frac{6+\sqrt{2}}{2}}$$
• E.
  $$1 + \frac{1}{2}\log_2 3$$
• F.
  $$4 + 2\log_2 3$$

Answer: F

• A.0 m
• B.
  $$8.0 \times 10^{-7} \text{m}$$
• C.
  $$8.0 \times 10^{-6} \text{m}$$
• D.
  $$1.6 \times 10^{-5} \text{m}$$
• E.
  $$3.2 \times 10^{-5} \text{m}$$
• F.
  $$6.4 \times 10^{-5} \text{m}$$
• G.
  $$3.2 \times 10^{-4} \text{m}$$
• H.
  $$6.4 \times 10^{-4} \text{m}$$

Answer: G