National 5 Maths 2023 Paper 1 — Questions & Video Solutions

1

Question 1

Fractions and mixed numbers

2 Marks

2023 P1 Q1
Evaluate

\displaystyle 2 \frac{1}{6} \div \frac{8}{9}.

Give your answer in its simplest form.

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\displaystyle 2\frac{7}{16}

2

Question 2

Expanding brackets

3 Marks

2023 P1 Q2

Expand and simplify $\displaystyle (x+7)^2+6(x^2-10)$

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\displaystyle 7x^2+14x-11

3

Question 3

Simultaneous equations

3 Marks

2023 P1 Q3

Solve, algebraically, the system of equations

$\begin{aligned} 2x + 3y &= 8 \\ 5x + 2y &= -2 \end{aligned}$

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\displaystyle x=-2, y=4

4

Question 4

Parabola Equation from Graph

(1, 1, 1)3 Marks

2023 P1 Q4

The graph below shows part of a parabola of the form $\displaystyle y=(x+a)^2+b$ .

Parabola with minimum turning point (3, 2)

(a) (i) State the value of $\displaystyle a.$
(ii) State the value of $\displaystyle b.$

(b) P is the point (0, $\displaystyle c$ ).
Find the value of $\displaystyle c.$

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(a)(i)

\displaystyle a=-3

(ii)

\displaystyle b=2

(b)

\displaystyle c=11

5

Question 5

Discriminant

2 Marks

2023 P1 Q5

Determine the nature of the roots of the function $\displaystyle f(x)=4x^2+6x-1.$

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two distinct real roots

6

Question 6

Cosine rule: calculate length

3 Marks

2023 P1 Q6

In triangle ABC:
• AC = 5 metres
• BC = 6 metres
$\displaystyle \bullet \quad \cos C = \frac{1}{5}$

Triangle ABC with side lengths and cosine of angle C

Calculate the length of AB.

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7 metres

7

Question 7

Straight Line EquationScatter Graph

(3, 1)4 Marks

2023 P1 Q7

A business recorded the salaries of a sample of its employees and the length of time they have worked for the business. The scattergraph shows the relationship between their salary, $\displaystyle P$ pounds, and the length of time, $\displaystyle T$ years, they have worked.

A line of the best fit has been drawn.

(a) Find the equation of the line of best fit in terms of $\displaystyle P$ and $\displaystyle T$ . Give the equation in its simplest form.

(b) Use your equation from part (a) to estimate the salary of an employee who has worked for the business for 8 years.

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(a)

\displaystyle P=1500T+12500

(b) £24,500

8

Question 8

Rationalising the denominator

2 Marks

2023 P1 Q8
Express

\displaystyle \large\frac{12}{\sqrt{15}}

with a rational denominator.
Give your answer in its simplest form.

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\displaystyle \frac{4\sqrt{15}}{5}

9

Question 9

Median/Quartiles/Interquartile RangeComparing Calculated Statistics

(3, 2)5 Marks

2023 P1 Q9

A magazine company conducted a survey of the ages of its readers. A sample of ten readers' ages, in years, are shown below.

33 55 38 47 36 41 42 41 35 31

(a) Calculate the median and interquartile range of the ages of readers for this sample.

A newspaper company also conducted a survey of the ages of its readers. The median age of a sample of its readers was 41 years and the interquartile range was 9 years.

(b) Make two valid comments comparing the ages of the readers of the magazine and the ages of the readers of the newspaper.

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(a) Median = 39.5 years, IQR = 7 years.
(b) On average, the readers of the newspaper are older, as

\displaystyle 41>39.5

. The ages of the newspaper readers are more varied, as

\displaystyle 9>7

.

10

Question 10

Pythagoras in circle diagrams

4 Marks

2023 P1 Q10

Alan buys some identical paving slabs to make a path.
Each slab is part of a circle.

The diagram below shows a single slab.

The circle, centre C, has a radius of 50 centimetres.
Length AB is 60 centimetres.
Calculate the width of the paving slab.

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10 cm

11

Question 11

sin/cos/tan of related angles

1 Mark

2023 P1 Q11

Given that $\displaystyle \sin 30^{\circ} = 0.5$ , state the value of $\displaystyle \sin 330^{\circ}.$

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-0.5

12

Question 12

Rewriting fraction or negative index in the form ax^nLaws of indices

3 Marks

2023 P1 Q12
Simplify

\displaystyle \large\frac{5c^{-2}}{c^3 \times c^4}.

Give your answer with a positive power.

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\displaystyle \frac{5}{c^9}

13

Question 13

Identify equation of trigonometric graph

(1, 1)2 Marks

2023 P1 Q13

Part of the graph of $\displaystyle y=\cos(x+a)^\circ + b$ is shown.

(a) State the value of $\displaystyle a.$
(b) State the value of $\displaystyle b.$

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(a)

\displaystyle a=-45

(b)

\displaystyle b=1

14

Question 14

Linear equations and inequations

3 Marks

2023 P1 Q14

Solve, algebraically, the inequation
$\displaystyle \large\frac{x\ +\ 1}{3}\normalsize -2\gt\large\frac{3x}{5}\normalsize$

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\displaystyle x \lt -\frac{25}{4}