National 5 Maths 2014 Paper 1 — Questions & Video Solutions

1

Question 1

Fractions and mixed numbers

2 Marks

2014 P1 Q1
Evaluate

\displaystyle \frac{5}{12} \times 2\frac{2}{9}.

Give the answer in simplest form.

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\displaystyle \frac{25}{27}

2

Question 2

Expanding brackets

2 Marks

2014 P1 Q2

Multiply out the brackets and collect like terms: $\displaystyle (2x-5)(3x+1)$

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\displaystyle 6x^2-13x-5

3

Question 3

Completing the square

2 Marks

2014 P1 Q3

Express $\displaystyle x^2-14x+44$ in the form $\displaystyle \left(x-a\right)^2+b$

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\displaystyle (x-7)^2-5

4

Question 4

Adding and Subtracting vector components

2 Marks

2014 P1 Q4
Find the resultant vector

\displaystyle 2\mathbf{u}-\mathbf{v}

when

\displaystyle \mathbf{u}=\begin{pmatrix}-2\\ 3\\ 5\end{pmatrix}

and

\displaystyle \mathbf{v}=\begin{pmatrix}0\\ -4\\ 7\end{pmatrix}

.
Express your answer in component form.

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\displaystyle \begin{pmatrix}-4\\ 10\\ 3\end{pmatrix}

5

Question 5

Sine rule: calculate length

3 Marks

2014 P1 Q5

In triangle KLM
•  KM = 18 centimetres
•  sin K = 0·4
•  sin L = 0·9

Triangle KLM with side KM, sin K, and sin L

Calculate the length of LM.

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8 cm

6

Question 6

Scatter GraphStraight Line Equation

(3, 1)4 Marks

2014 P1 Q6

McGregor's Burgers sells fast food. The graph shows the relationship between the amount of fat, $\displaystyle F$ grams, and the number of calories, $\displaystyle C$ , in some of their sandwiches.

Scattergraph of sandwich fat vs calories

A line of best fit has been drawn.
Point A represents a sandwich which has 5 grams of fat and 200 calories.
Point B represents a sandwich which has 25 grams of fat and 500 calories.

(a) Find the equation of the line of best fit in terms of $\displaystyle F$ and $\displaystyle C.$

(b) A Super Deluxe sandwich contains 40 grams of fat. Use your answer to part (a) to estimate the number of calories this sandwich contains.

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(a)

\displaystyle C=15F+125

, (b) 725 calories

7

Question 7

Parabola Equation from Graph

2 Marks

2014 P1 Q7

The diagram below shows part of the graph of $\displaystyle y=ax^2$ .

The point $\displaystyle (-3,45)$ lies on the graph.
Find the value of $\displaystyle a$ .

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\displaystyle a=5

8

Question 8

Simplifying surds

3 Marks

2014 P1 Q8
Express

\displaystyle \sqrt{40}+4\sqrt{10}+\sqrt{90}

as a surd in its simplest form.

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\displaystyle 9\sqrt{10}

9

Question 9

Reversing a percentage change

3 Marks

2014 P1 Q9
480 000 tickets were sold for a tennis tournament last year. This represents 80% of all the available tickets.
Calculate the total number of tickets that were available for this tournament.

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600,000 tickets

10

Question 10

Identify equation of trigonometric graph

2 Marks

2014 P1 Q10

The graph of $\displaystyle y=a\sin(x+b)^\circ$ , $\displaystyle 0\le x\le360$ , is shown below.

Write down the values of $\displaystyle a$ and $\displaystyle b$ .

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\displaystyle a=3

,

\displaystyle b=-40

11

Question 11

Coordinate Geometry with straight line equationStraight Line Equation

(2, 2)4 Marks

2014 P1 Q11

(a) A straight line has equation $\displaystyle 4x+3y=12$ .
Find the gradient of this line.

(b) Find the coordinates of the point where this line crosses the $\displaystyle x$ -axis.

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(a)

\displaystyle m=-\frac{4}{3}

, (b)

\displaystyle (3,0)

12

Question 12

Pythagoras in circle diagrams

4 Marks

2014 P1 Q12

The diagram below shows a circle, centre C.

The radius of the circle is 15 centimetres.
A is the mid-point of chord PQ.
The length of AB is 27 centimetres.
Calculate the length of PQ.

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18 cm

13

Question 13

Quadratic equation by factorising

(4, 3)7 Marks

2014 P1 Q13

The diagram below shows the path of a small rocket which is fired into the air. The height, $\displaystyle h$ metres, of the rocket after $\displaystyle t$ seconds is given by $\displaystyle h(t)=16t-t^2.$

(a) After how many seconds will the rocket first be at a height of 60 metres?

(b) Will the rocket reach a height of 70 metres? Justify your answer.

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(a) 6 seconds, (b) No, because its maximum height is 64 metres