2015 Paper 2

The vertices of triangle ABC are $\displaystyle A(-5,7)$, $\displaystyle B(-1,-5)$ and $\displaystyle C(13,3)$ as shown in the diagram.
The broken line represents the altitude from C.

(a)  Show that the equation of the altitude from C is $\displaystyle x-3y=4.$
(b)  Find the equation of the median from B.
(c)  Find the coordinates of the point of intersection of the altitude from C and the median from B.

Functions $\displaystyle f$ and $\displaystyle g$ are defined on suitable domains by
$\displaystyle f(x)=10+x$
and
$\displaystyle g(x)=(1+x)(3-x)+2$

(a)  Find an expression for $\displaystyle f(g(x))$
(b)  Express $\displaystyle f(g(x))$ in the form $\displaystyle p(x+q)^{2}+r$
(c)  Another function $\displaystyle h$ is given by
$\displaystyle h(x)=\frac{1}{f(g(x))}$
What values of $\displaystyle x$ cannot be in the domain of $\displaystyle h$?

A frog and a toad fall to the bottom of a well that is 50 feet deep.
Each day, the frog climbs 32 feet and then rests overnight. During the night, it slides down $\displaystyle \frac{2}{3}$ of its height above the floor of the well.
The toad climbs 13 feet each day before resting. Overnight, it slides down $\displaystyle \frac{1}{4}$ of its height above the floor of the well.
Their progress can be modelled by the recurrence relations
$\displaystyle f_{n+1}=\frac{1}{3}f_{n}+32$, $\displaystyle f_{1}=32$
and
$\displaystyle t_{n+1}=\frac{3}{4}t_{n}+13$, $\displaystyle t_{1}=13$
where $\displaystyle f_{n}$ and $\displaystyle t_{n}$ are the heights reached by the frog and the toad at the end of the $\displaystyle n$th day after falling in.

(a)  Calculate $\displaystyle t_{2}$, the height of the toad at the end of the second day.
(b)  Determine whether or not either of them will eventually escape from the well.

A wall plaque is to be made to commemorate the 150th anniversary of the publication of "Alice's Adventures in Wonderland".
The edges of the wall plaque can be modelled by parts of the graphs of four quadratic functions as shown in the sketch.

$\displaystyle \bullet \quad f(x)=\frac{1}{4}x^{2}-\frac{1}{2}x+3$
$\displaystyle \bullet \quad g(x)=\frac{1}{4}x^{2}-\frac{3}{2}x+5$
$\displaystyle \bullet \quad h(x)=\frac{3}{8}x^{2}-\frac{9}{4}x+3$
$\displaystyle \bullet \quad k(x)=\frac{3}{8}x^{2}-\frac{3}{4}x$

(a)  Find the x-coordinate of the point of intersection of the graphs with equations $\displaystyle y=f(x)$ and $\displaystyle y=g(x)$.
The graphs of the functions $\displaystyle f(x)$ and $\displaystyle h(x)$ intersect on the y-axis. The plaque has a vertical line of symmetry.
(b)  Calculate the area of the wall plaque.

Circle $\displaystyle C_{1}$ has equation
$\displaystyle x^{2}+y^{2}+6x+10y+9=0$
The centre of circle $\displaystyle C_{2}$ is $\displaystyle (9, 11)$
Circles $\displaystyle C_{1}$ and $\displaystyle C_{2}$ touch externally.

(a)  Determine the radius of $\displaystyle C_{2}.$
(b)  A third circle, $\displaystyle C_{3}$, is drawn such that both $\displaystyle C_{1}$ and $\displaystyle C_{2}$ touch $\displaystyle C_{3}$ internally and the centres of $\displaystyle C_{1}$, $\displaystyle C_{2}$ and $\displaystyle C_{3}$ are collinear.
Determine the equation of $\displaystyle C_{3}.$

Vectors $\displaystyle \mathbf{p}, \mathbf{q}$ and $\displaystyle \mathbf{r}$ are represented on the diagram as shown.
•  BCDE is a parallelogram
•  ABE is an equilateral triangle
$\displaystyle \bullet \quad |\mathbf{p}|=3$
•  Angle $\displaystyle ABC=90^{\circ}$

(a)  Evaluate $\displaystyle \mathbf{p}.(\mathbf{q}+\mathbf{r}).$
(b)  Express $\displaystyle \vec{EC}$ in terms of $\displaystyle \mathbf{p}, \mathbf{q}$ and $\displaystyle \mathbf{r}.$
(c)  Given that $\displaystyle \vec{AE}.\vec{EC}=9\sqrt{3}-\frac{9}{2}$, find $\displaystyle |\mathbf{r}|.$

(a)  Find $\displaystyle \int(3 \cos 2x+1)dx.$
(b)  Show that
$\displaystyle 3 \cos 2x+1=4 \cos^{2}x-2 \sin^{2}x$
(c)  Hence, or otherwise, find
$\displaystyle \int(\sin^{2}x-2 \cos^{2}x)dx.$

A crocodile is stalking prey located 20 metres further upstream on the opposite bank of a river.
Crocodiles travel at different speeds on land and in water.
The time taken for the crocodile to reach its prey can be minimised if it swims to a particular point, P, $\displaystyle x$ metres upstream on the other side of the river as shown in the diagram.

The time taken, $\displaystyle T$, measured in tenths of a second, is given by
$\displaystyle T(x)=5\sqrt{36+x^{2}}+4(20-x)$

(a)  (i) Calculate the time taken if the crocodile does not travel on land.
      (ii) Calculate the time taken if the crocodile swims the shortest distance possible.
(b)  Between these two extremes there is one value of $\displaystyle x$ which minimises the time taken. Find this value of $\displaystyle x$ and hence calculate the minimum possible time.

The blades of a wind turbine are turning at a steady rate.
The height, $\displaystyle h$ metres, of the tip of one of the blades above the ground at time, $\displaystyle t$ seconds, is given by the formula
$\displaystyle h=36 \sin(1.5t)-15 \cos(1.5t)+65$
Express
$\displaystyle 36 \sin(1.5t)-15 \cos(1.5t)$
in the form $\displaystyle k \sin(1.5t-a)$, where $\displaystyle k \gt 0$ and $\displaystyle 0 \lt a \lt \frac{\pi}{2}$, and hence find the two values of $\displaystyle t$ for which the tip of this blade is at a height of 100 metres above the ground during the first turn.