2017 Paper 2

Find $\displaystyle |\mathbf{v}|$, the magnitude of vector $\displaystyle \mathbf{v}=\begin{pmatrix}18\\ -14\\ 3\end{pmatrix}.$

A necklace is valued at £1200.
Its value is expected to increase by 4.5% per year over the next 3 years.
Calculate the expected value of the necklace after this time.
Give your answer to the nearest pound.

A piece of land is in the shape of a triangle as shown.
PQ = 250 m, PR = 180 m, angle QPR = $\displaystyle 147^{\circ}.$
The owner wishes to build a fence along the side QR.

Calculate the length of the fence.

Solve the equation $\displaystyle 2x^{2}+5x-4=0.$
Give your answers correct to one decimal place.

A theatre group sold 4830 tickets for their show.
This was 15% more than they sold last year.
How many tickets did they sell last year?

A spherical sweet is made by coating a caramel sphere evenly with chocolate.
The diameter of the sweet is 24 mm and the thickness of the chocolate coating is 3 mm.

Calculate the volume of the chocolate coating.
Give your answer correct to 3 significant figures.

Triangles A and B are shown below.

The triangles are placed together to form the larger triangle shown below.

Is this larger triangle right-angled?
Justify your answer.

In the diagram below, $\displaystyle \vec{RQ}$ and $\displaystyle \vec{PQ}$ represent the vectors $\displaystyle \mathbf{c}$ and $\displaystyle \mathbf{d}$ respectively.

(a)  Express $\displaystyle \vec{PR}$ in terms of $\displaystyle \mathbf{c}$ and $\displaystyle \mathbf{d}.$

The line QP is extended to T.

$\displaystyle \bullet \quad \text{TP = PQ}$
$\displaystyle \bullet \quad \text{V is the midpoint of PR}$
(b)  Express $\displaystyle \vec{TV}$ in terms of $\displaystyle \mathbf{c}$ and $\displaystyle \mathbf{d}.$ Give your answer in simplest form.

(a)  Factorise $\displaystyle 4x^{2}-25.$

(b)  Hence simplify $\displaystyle \frac{4x^{2}-25}{2x^{2}-x-10}.$

In the diagram below D, E and F represent the positions of Dunbridge, Earlsford and Fairtown respectively.
Dunbridge is 15 km west of Earlsford.
From Dunbridge, the bearing of Fairtown is $\displaystyle 126^{\circ}.$
From Earlsford the bearing of Fairtown is $\displaystyle 230^{\circ}.$

Calculate the distance between Dunbridge and Fairtown.

A straight line has equation $\displaystyle 3x-5y-10=0.$
Find the gradient of this line.

Express $\displaystyle \frac{1}{\sqrt[3]{x}}$ in the form $\displaystyle x^{n}.$

Two identical shapes are used to form a logo.
Each shape is part of a circle.
The circles have centres $\displaystyle C_1$ and $\displaystyle C_2.$
The radius of each circle is 14 cm.
The logo has half-turn symmetry about the mid-point of AB.
AB is 48 cm long.

Calculate the height of the logo.

The diagram shows part of a circle, centre O.
The radius of the circle is 6.4 cm.
Major arc AB has length 31.5 cm.

Calculate the size of the reflex angle AOB.

A wind turbine has three blades.
The height, $\displaystyle h$ metres, of the tip of blade A above the ground is given by $\displaystyle h=40+23\cos x^{\circ}$, $\displaystyle 0\le x<360.$

(a)  Calculate the height of the tip of blade A after it has turned through an angle of $\displaystyle 60^{\circ}.$

(b)  Find the minimum height of the tip of blade A above the ground.

(c)  Calculate the values of $\displaystyle x$ for which the tip of blade A is 61 metres above the ground.