2018 Paper 2

The diagram shows the curve with equation
$\displaystyle y=3+2x-x^{2}$

Calculate the shaded area.

Vectors $\displaystyle \mathbf{u}$ and $\displaystyle \mathbf{v}$ are defined by
$\displaystyle \mathbf{u}=\begin{pmatrix}-1\\4\\-3\end{pmatrix}$ and $\displaystyle \mathbf{v}=\begin{pmatrix}-7\\8\\5\end{pmatrix}$
(a)  Find $\displaystyle \mathbf{u}\cdot\mathbf{v}.$
(b)  Calculate the acute angle between $\displaystyle \mathbf{u}$ and $\displaystyle \mathbf{v}.$

A function, $\displaystyle f$, is defined on the set of real numbers by
$\displaystyle f(x)=x^{3}-7x-6$
Determine whether $\displaystyle f$ is increasing or decreasing when $\displaystyle x=2$.

Express $\displaystyle -3x^{2}-6x+7$ in the form $\displaystyle a(x+b)^{2}+c$.

PQR is a triangle with $\displaystyle P(3,4)$ and $\displaystyle Q(9,-2)$.

(a)  Find the equation of $\displaystyle L_{1}$, the perpendicular bisector of PQ.
The equation of $\displaystyle L_{2}$, the perpendicular bisector of PR is $\displaystyle 3y+x=25.$

(b)  Calculate the coordinates of C, the point of intersection of $\displaystyle L_{1}$ and $\displaystyle L_{2}$.
C is the centre of the circle which passes through the vertices of triangle PQR.

(c)  Determine the equation of this circle.

Functions, $\displaystyle f$ and $\displaystyle g$, are given by
$\displaystyle f(x)=3+\cos x$ and $\displaystyle g(x)=2x$
where $\displaystyle x\in\mathbb{R}$.
(a)  Find expressions for (i) $\displaystyle f(g(x))$ and (ii) $\displaystyle g(f(x)).$
(b)  Determine the value(s) of $\displaystyle x$ for which $\displaystyle f(g(x))=g(f(x))$ where $\displaystyle 0 \le x \lt 2\pi.$

(a)  (i) Show that $\displaystyle (x-2)$ is a factor of
$\displaystyle 2x^{3}-3x^{2}-3x+2$
      (ii) Hence, factorise $\displaystyle 2x^{3}-3x^{2}-3x+2$ fully.
The fifth term, $\displaystyle u_{5}$, of a sequence is $\displaystyle u_{5}=2a-3$. The terms of the sequence satisfy the recurrence relation
$\displaystyle u_{n+1}=au_{n}-1$
(b)  Show that $\displaystyle u_{7}=2a^{3}-3a^{2}-a-1$.
For this sequence, it is known that $\displaystyle u_{7}=u_{5}$ and a limit exists.
(c)  (i) Determine the value of $\displaystyle a$.
      (ii) Calculate the limit.

(a)  Express $\displaystyle 2\cos x^{\circ}-\sin x^{\circ}$ in the form $\displaystyle k\cos(x-a)^{\circ}$, $\displaystyle k \gt 0$, $\displaystyle 0 \lt a \lt 360.$
(b)  Hence, or otherwise, find (i) the minimum value of
$\displaystyle 6\cos x^{\circ}-3\sin x^{\circ}$
and (ii) the value of $\displaystyle x$ for which it occurs where $\displaystyle 0 \le x \lt 360.$

A sector with a particular fixed area has radius $\displaystyle x$ cm. The perimeter, $\displaystyle P$ cm, of the sector is given by
$\displaystyle P=2x+\frac{128}{x}$
Find the minimum value of $\displaystyle P$.

The equation $\displaystyle x^{2}+(m-3)x+m=0$ has two real and distinct roots.
Determine the range of values for $\displaystyle m$.

A supermarket has been investigating how long customers have to wait at the checkout. During any half hour period, the percentage, $\displaystyle P\%$, of customers who wait for less than $\displaystyle t$ minutes, can be modelled by
$\displaystyle P=100(1-e^{kt})$
where $\displaystyle k$ is a constant.
(a)  If 50% of customers wait for less than 3 minutes, determine the value of $\displaystyle k$.
(b)  Calculate the percentage of customers who wait for 5 minutes or longer.

Circle $\displaystyle C_{1}$ has equation
$\displaystyle (x-13)^{2}+(y+4)^{2}=100$
Circle $\displaystyle C_{2}$ has equation
$\displaystyle x^{2}+y^{2}+14x-22y+c=0$

(a)  (i) Write down the coordinates of the centre of $\displaystyle C_{1}$.
      (ii) The centre of $\displaystyle C_{1}$ lies on the circumference of $\displaystyle C_{2}$. Show that $\displaystyle c=-455$.
The line joining the centres of the circles intersects $\displaystyle C_{1}$ at P.
(b)  (i) Determine the ratio in which P divides the line joining the centres of the circles.
      (ii) Hence, or otherwise, determine the coordinates of P.
P is the centre of a third circle, $\displaystyle C_{3}$. $\displaystyle C_{2}$ touches $\displaystyle C_{3}$ internally.
(c)  Determine the equation of $\displaystyle C_{3}$.