National 5 Maths 2023 Paper 2 — Questions & Video Solutions

1

Question 1

Appreciation and Depreciation

3 Marks

2023 P2 Q1
A caravan was bought for £20,000. It depreciated by 11% in the first year. It then depreciated by a further 6% each year over the next two years.
Calculate the value of the caravan three years after it was bought.

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£15,728.08

2

Question 2

Scientific notation

3 Marks

2023 P2 Q2
The mass of a helium atom is 6.64

\displaystyle \times

10^–24 grams. A flask contains 300 grams of helium.
Calculate the number of helium atoms in this flask.
Give your answer in scientific notation, correct to 3 significant figures.

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\displaystyle 4.52 \times 10^{25}

atoms

3

Question 3

Arc length

3 Marks

2023 P2 Q3

The diagram shows part of a football pitch.

The penalty spot is marked at point C.
AB is an arc of a circle, centre C, radius 9.15 metres.
Calculate the length of the arc AB.

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16.9 metres

4

Question 4

Sine rule: calculate angle

3 Marks

2023 P2 Q4

The diagram shows triangle JKL.
•  Angle KJL = 25°
•  JL = 10 metres
•  KL = 7 metres

Triangle JKL with side lengths and angle

Calculate the size of acute angle JKL.

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37.1°

5

Question 5

Angles in regular polygons

2 Marks

2023 P2 Q5

A logo consists of an H shape and a regular decagon.
The diagram represents the logo.

Calculate the size of the shaded angle.

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126°

6

Question 6

Reversing a percentage change

3 Marks

2023 P2 Q6
Nadim bought a flat last year. The value of the flat has increased by 8% and it is now worth £94,500.
Calculate how much Nadim paid for the flat.

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£87,500

7

Question 7

Changing the subject of a formula

3 Marks

2023 P2 Q7

Change the subject of the formula
$\displaystyle P=\large\frac13\normalsize mn-r$ to $\displaystyle m.$

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\displaystyle m = \frac{3(P+r)}{n}

8

Question 8

Pythagoras converse

4 Marks

2023 P2 Q8

A wooden beam is used to support a wall built on horizontal ground as shown in the diagram.

The edge of the beam, AB, is 8 metres long.
C is at the foot of the wall.
A is 7 metres from C.
B is 4 metres from C.
Determine whether the wall is perpendicular to the ground.
Justify your answer.

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Not perpendicular

9

Question 9

Volume - composite shape

4 Marks

2023 P2 Q9

A concrete block is in the shape of a large pyramid with a small pyramid removed.

Concrete block in the shape of a truncated pyramid

The large pyramid has a square base of length 90 centimetres.
The small pyramid has a square base of length 40 centimetres and a height of 48 centimetres.
The block has height 60 centimetres.
Calculate the volume of the block.

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266,000 cm³

10

Question 10

Add or subtract Algebraic Fractions

3 Marks

2023 P2 Q10

Express $\displaystyle \large\frac{7}{x-3}-\frac{2}{x}\normalsize\:\:\:\:\small x\neq3,\ x\neq0\:\:$
as a single fraction in its simplest form.

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\displaystyle \frac{5x+6}{x(x-3)}

11

Question 11

Trigonometric equation

4 Marks

2023 P2 Q11

Anna has a grandfather clock in her house.

The height of the tip of the hour hand above the floor, in centimetres, is given by $\displaystyle h=20\cos x^{\circ} + 147,$ where $\displaystyle x^{\circ}$ is the angle the hour hand has rotated through since 12 o'clock.

Calculate the first two values of $\displaystyle x$ for which the tip of the hour hand is 150 centimetres above the floor.

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81.4, 278.6

12

Question 12

Simplifying algebraic fraction

3 Marks

2023 P2 Q12

Simplify $\displaystyle \large\frac{x^2-16}{x^2+x-20}.$

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\displaystyle \frac{x+4}{x+5}

13

Question 13

Trigonometric identities

2 Marks

2023 P2 Q13

Simplify $\displaystyle \sin^2 x^\circ\cos^2 x^\circ + \cos^4 x^\circ$ Show your working.

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\displaystyle \cos^2 x^\circ

14

Question 14

Create equation in geometric contextQuadratic formula

(2, 4)6 Marks

2023 P2 Q14

A storage unit, built in the shape of a cuboid, is shown.

It has length $\displaystyle (x+7)$ metres, breadth $\displaystyle x$ metres and height $\displaystyle 2$ metres.
The volume of this unit is $\displaystyle 45$ cubic metres.
(a) Show that $\displaystyle 2x^2+14x-45=0$
(b) Calculate $\displaystyle x,$ the breadth of the storage unit.
Give your answer correct to 1 decimal place.

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(a) Proof, (b)

\displaystyle x=2.4

15

Question 15

Area of a Triangle

4 Marks

2023 P2 Q15

In the diagram:
•  AC is perpendicular to BC
•  AB = 18 centimetres
•  BD = 6 centimetres
•  BC = 8 centimetres.

Triangle ADE with perpendicular AC to BC

The area of triangle ADE is 160 square centimetres.
Calculate the length of AE.

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30 cm