A curve $C$ has equation $y = x ^ { 3 } - 3 x ^ { 2 }$.

a) Find the stationary points of $C$ and determine their nature.\\
b) Draw a sketch of $C$, clearly indicating the stationary points and the points where the curve crosses the coordinate axes.\\
c) Without performing the integration, state whether $\int _ { 0 } ^ { 3 } \left( x ^ { 3 } - 3 x ^ { 2 } \right) \mathrm { d } x$ is positive or\\
negative, giving a reason for your answer.

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1 & 4 \\
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In each of the two statements below, $c$ and $d$ are real numbers. One of the statements is true, while the other is false.

$$\begin{aligned}
& \text { A : } \quad ( 2 c - d ) ^ { 2 } = 4 c ^ { 2 } - d ^ { 2 } , \text { for all values of } c \text { and } d . \\
& \text { B : } \quad 8 c ^ { 3 } - d ^ { 3 } = ( 2 c - d ) \left( 4 c ^ { 2 } + 2 c d + d ^ { 2 } \right) , \text { for all values of } c \text { and } d .
\end{aligned}$$

a) Identify the statement which is false. Show, by counter example, that this statement is in fact false.\\
b) Identify the statement which is true. Give a proof to show that this statement is in fact true.

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1 & 5 \\
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The value of a car, $\pounds V$, may be modelled as a continuous variable. At time $t$ years, the value of the car is given by $V = A \mathrm { e } ^ { k t }$, where $A$ and $k$ are constants. When the car is new, it is worth $\pounds 30000$. When the car is two years old, it is worth $\pounds 20000$. Determine the value of the car when it is six years old, giving your answer correct to the nearest $\pounds 100$.

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1 & 6 \\
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The curve $C$ has equation $y = 7 + 13 x - 2 x ^ { 2 }$. The point $P$ lies on $C$ and is such that the tangent to $C$ at $P$ has equation $y = x + c$, where $c$ is a constant. Find the coordinates of $P$ and the value of $c$.

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1 & 7 \\
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a) Solve $2 \log _ { 10 } x = 1 + \log _ { 10 } 5 - \log _ { 10 } 2$.\\
b) Solve $3 = 2 \mathrm { e } ^ { 0 \cdot 5 x }$.\\
c) Express $4 ^ { x } - 10 \times 2 ^ { x }$ in terms of $y$, where $y = 2 ^ { x }$.

Hence solve the equation $4 ^ { x } - 10 \times 2 ^ { x } = - 16$.

\hfill \mbox{\textit{WJEC Unit 1 2018 Q13}}