2019 Paper 1

A sequence is generated by the recurrence relation $\displaystyle u_{n+1}=mu_{n}+c$, where the first three terms of the sequence are 6, 9 and 11.

(a)  Find the values of $\displaystyle m$ and $\displaystyle c.$
(b)  Hence, calculate the fourth term of the sequence.

(a)  Show that the points $\displaystyle A(1,5,-3)$, $\displaystyle B(4,-1,0)$ and $\displaystyle C(8,-9,4)$ are collinear.
(b)  State the ratio in which B divides AC.

The graphs of $\displaystyle y=x^{2}+2x+3$ and $\displaystyle y=2x^{2}+x+1$ are shown below. The graphs intersect at the points where $\displaystyle x=-1$ and $\displaystyle x=2$.

(a)  Express the shaded area, enclosed between the curves, as an integral.
(b)  Evaluate the shaded area.

Vectors $\displaystyle \mathbf{u}$ and $\displaystyle \mathbf{v}$ have components
$\displaystyle \begin{pmatrix}p\\-2\\4\end{pmatrix}$ and $\displaystyle \begin{pmatrix}2p+16\\-3\\6\end{pmatrix}$
$\displaystyle p\in\mathbb{R}.$

(a)  (i) Find an expression for $\displaystyle \mathbf{u}\cdot\mathbf{v}.$
      (ii) Determine the values of $\displaystyle p$ for which $\displaystyle \mathbf{u}$ and $\displaystyle \mathbf{v}$ are perpendicular.
(b)  Determine the value of $\displaystyle p$ for which $\displaystyle \mathbf{u}$ and $\displaystyle \mathbf{v}$ are parallel.

The diagram shows the graphs with equations $\displaystyle y=f(x)$ and $\displaystyle y=kf(x)+a.$

(a)  State the value of $\displaystyle a.$
(b)  Find the value of $\displaystyle k.$

Functions $\displaystyle f$ and $\displaystyle g$ are defined by
$\displaystyle f(x)=\frac{1}{\sqrt{x}}$, where $\displaystyle x \gt 0$
and
$\displaystyle g(x)=5-x$, where $\displaystyle x\in\mathbb{R}.$
(a)  Determine an expression for $\displaystyle f(g(x)).$
(b)  State the range of values of $\displaystyle x$ for which $\displaystyle f(g(x))$ is undefined.

Triangles ABC and ADE are both right angled. Angles $\displaystyle p$ and $\displaystyle q$ are as shown in the diagram.

(a)  Determine the value of:
      (i) $\displaystyle \cos p$
      (ii) $\displaystyle \cos q$
(b)  Hence determine the value of $\displaystyle \sin(p+q).$

(a)  Evaluate
$\displaystyle \log_{10}4+2\log_{10}5.$

(b)  Solve
$\displaystyle \log_{2}(7x-2)-\log_{2}3=5$
$\displaystyle x \ge 1.$

(a)  Solve the equation
$\displaystyle \sin 2x^{\circ}+6\cos x^{\circ}=0$
for $\displaystyle 0\le x \lt 360.$
(b)  Hence solve
$\displaystyle \sin 4x^{\circ}+6\cos 2x^{\circ}=0$
for $\displaystyle 0\le x \lt 360.$

The point P has coordinates $\displaystyle (4,k)$. $\displaystyle C$ is the centre of the circle with equation
$\displaystyle (x-1)^{2}+(y+2)^{2}=25$
(a)  Show that the distance between the points P and C is given by $\displaystyle \sqrt{k^{2}+4k+13}.$
(b)  Hence, or otherwise, find the range of values of $\displaystyle k$ such that P lies outside the circle.

(a)  Express $\displaystyle (\sin x-\cos x)^{2}$ in the form $\displaystyle p+q\sin rx$ where $\displaystyle p,q$ and $\displaystyle r$ are integers.
(b)  Hence, find $\displaystyle \int(\sin x-\cos x)^{2}dx.$