2016 Paper 1

Given $\displaystyle \mathbf{p}=\begin{pmatrix}4\\ -6\end{pmatrix}$ and $\displaystyle \mathbf{q}=\begin{pmatrix}-5\\ -1\end{pmatrix}$.
Find the resultant vector $\displaystyle \frac{1}{2}\mathbf{p}+\mathbf{q}$.
Express your answer in component form.

Evaluate $\displaystyle \frac{3}{4}\left(\frac{1}{3}+\frac{2}{7}\right)$.
Give your answer in its simplest form.

The diagram shows a sector of a circle, centre C.

The radius of the circle is 20 centimetres and angle ACB is $\displaystyle 45^{\circ}$.
Calculate the area of the sector.
Take $\displaystyle \pi=3\cdot14$.

Charlie is making costumes for a school show.

(a)  One day he made 2 cloaks and 3 dresses. The total amount of material he used was 9.6 square metres. Write down an equation to illustrate this information.

(b)  The following day Charlie made 3 cloaks and 4 dresses. The total amount of material he used was 13.3 square metres. Write down an equation to illustrate this information.

(c)  Calculate the amount of material required to make one cloak and the amount of material required to make one dress.

A cattle farmer records the weight of some of his calves.
The scattergraph shows the relationship between the age, $\displaystyle A$ months, and the weight, $\displaystyle W$ kilograms, of the calves.

A line of best fit is drawn.
Point D represents a 3 month old calf which weighs 100 kilograms.
Point E represents a 15 month old calf which weighs 340 kilograms.

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

(b)  Use your equation from part (a) to estimate the weight of a one year old calf.

Determine the nature of the roots of the function $\displaystyle f(x)=7x^{2}+5x-1$.

The diagram shows a rectangular based pyramid, relative to the coordinate axes.

A is the point $\displaystyle (2,0,0)$.
V is the point $\displaystyle (5,2,6)$.

(a)  Write down the coordinates of B.

(b)  Calculate the length of edge AV of the pyramid.

Solve the equation $\displaystyle \frac{2x}{3}-\frac{5}{6}=2x$.
Give your answer in its simplest form.

The function $\displaystyle f(x)$ is defined by $\displaystyle f(x)=\frac{2}{\sqrt{x}}$, $\displaystyle x>0$.
Express $\displaystyle f(5)$ as a fraction with a rational denominator.

Sketch the graph of $\displaystyle y=(x-3)^{2}+1$.
On your sketch, show clearly the coordinates of the turning point and the point of intersection with the y-axis.

Simplify $\displaystyle \tan^{2}x^{\circ}\cos^{2}x^{\circ}$.

The diagrams below show a rectangle and a triangle. All measurements are in centimetres.

(a)  Find an expression for the area of the rectangle.

(b)  Given that the area of the rectangle is equal to the area of the triangle, show that $\displaystyle x^{2}-2x-8=0$.

(c)  Hence find, algebraically, the length and breadth of the rectangle.