2016 Paper 1

A sequence is defined by the recurrence relation
$\displaystyle u_{n+1}=\frac{1}{3}u_{n}+10$
with $\displaystyle u_{3}=6.$

(a)  Find the value of $\displaystyle u_{4}.$
(b)  Explain why this sequence approaches a limit as $\displaystyle n\rightarrow\infty.$
(c)  Calculate this limit.

A and B are the points $\displaystyle (-7, 3)$ and $\displaystyle (1, 5).$
AB is a diameter of a circle.

Find the equation of this circle.

Functions $\displaystyle f$ and $\displaystyle g$ are defined on $\displaystyle \mathbb{R}$, the set of real numbers.
The inverse functions $\displaystyle f^{-1}$ and $\displaystyle g^{-1}$ both exist.

(a)  Given $\displaystyle f(x)=3x+5$ find $\displaystyle f^{-1}(x).$
(b)  If $\displaystyle g(2)=7$, write down the value of $\displaystyle g^{-1}(7).$

Three vectors can be expressed as follows:

$\displaystyle \vec{FG}=-2\mathbf{i}-6\mathbf{j}+3\mathbf{k}$
$\displaystyle \vec{GH}=3\mathbf{i}+9\mathbf{j}-7\mathbf{k}$
$\displaystyle \vec{EH}=2\mathbf{i}+3\mathbf{j}+\mathbf{k}$

(a)  Find $\displaystyle \vec{FH}$.
(b)  Hence, or otherwise, find $\displaystyle \vec{FE}$.

(a)  Find the x-coordinates of the stationary points on the graph with equation $\displaystyle y=f(x)$ where
$\displaystyle f(x)=x^{3}+3x^{2}-24x$
(b)  Hence determine the range of values of $\displaystyle x$ for which the function $\displaystyle f$ is strictly increasing.

The diagram below shows the graph of the function $\displaystyle f(x)=\log_{4}x$, where $\displaystyle x \gt 0.$

The inverse function, $\displaystyle f^{-1}$, exists.
On the diagram in your answer booklet, sketch the graph of the inverse function.

(a)  A and C are the points $\displaystyle (1, 3, -2)$ and $\displaystyle (4, -3, 4)$ respectively.
Point B divides AC in the ratio 1 : 2.
Find the coordinates of B.
(b)  $\displaystyle k\vec{AC}$ is a vector of magnitude 1, where $\displaystyle k \gt 0.$
Determine the value of $\displaystyle k.$

The functions $\displaystyle f$ and $\displaystyle g$ are defined on $\displaystyle \mathbb{R}$, the set of real numbers by
$\displaystyle f(x)=2x^{2}-4x+5$ and $\displaystyle g(x)=3-x$

(a)  Given $\displaystyle h(x)=f(g(x))$, show that $\displaystyle h(x)=2x^{2}-8x+11.$
(b)  Express $\displaystyle h(x)$ in the form $\displaystyle p(x+q)^{2}+r.$

Triangle ABD is right-angled at B with angles $\displaystyle BAC=p$ and $\displaystyle BAD=q$ and lengths as shown in the diagram below.

Show that the exact value of $\displaystyle \cos(q-p)$ is $\displaystyle \frac{19\sqrt{17}}{85}.$

(a)  Evaluate $\displaystyle \log_{5} 25.$

(b)  Hence solve
$\displaystyle \log_{4}x+\log_{4}(x-6)=\log_{5}25$
where $\displaystyle x \gt 6.$

The diagram below shows the graph with equation $\displaystyle y=f(x)$, where
$\displaystyle f(x)=k(x-a)(x-b)^{2}$

(a)  Find the values of $\displaystyle a$, $\displaystyle b$ and $\displaystyle k.$
(b)  For the function $\displaystyle g(x)=f(x)-d$, where $\displaystyle d$ is positive, determine the range of values of $\displaystyle d$ for which $\displaystyle g(x)$ has exactly one real root.