a) Differentiate each of the following functions with respect to \(x\). i) \(x ^ { 5 } \ln x\) ii) \(\frac { \mathrm { e } ^ { 3 x } } { x ^ { 3 } - 1 }\) iii) \(( \tan x + 7 x ) ^ { \frac { 1 } { 2 } }\) b) A function is defined implicitly by $$3 y + 4 x y ^ { 2 } - 5 x ^ { 3 } = 8$$ Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(x\) and \(y\).
□
1
The function \(f ( x )\) is defined by $$f ( x ) = \frac { \sqrt { x ^ { 2 } - 1 } } { x }$$ with domain \(x \geqslant 1\).
a) Find an expression for \(f ^ { - 1 } ( x )\). State the domain for \(f ^ { - 1 }\) and sketch both \(f ( x )\) and \(f ^ { - 1 } ( x )\) on the same diagram.
b) Explain why the function \(f f ( x )\) cannot be formed.
□
1
A chord \(A B\) subtends an angle \(\theta\) radians at the centre of a circle. The chord divides the circle into two segments whose areas are in the ratio \(1 : 2\). \includegraphics[max width=\textwidth, alt={}, center]{966abb82-ade0-4ca8-87a4-26e806d5add7-5_572_576_1197_749}
a) Show that \(\sin \theta = \theta - \frac { 2 \pi } { 3 }\).
b) i) Show that \(\theta\) lies between \(2 \cdot 6\) and \(2 \cdot 7\).
ii) Starting with \(\theta _ { 0 } = 2 \cdot 6\), use the Newton-Raphson Method to find the value of \(\theta\) correct to three decimal places. \section*{TURN OVER} Wildflowers grow on the grass verge by the side of a motorway. The area populated by wildflowers at time \(t\) years is \(A \mathrm {~m} ^ { 2 }\). The rate of increase of \(A\) is directly proportional to \(A\).
a) Write down a differential equation that is satisfied by \(A\).
b) At time \(t = 0\), the area populated by wildflowers is \(0.2 \mathrm {~m} ^ { 2 }\). One year later, the area has increased to \(1.48 \mathrm {~m} ^ { 2 }\). Find an expression for \(A\) in terms of \(t\) in the form \(p q ^ { t }\), where \(p\) and \(q\) are rational numbers to be determined.

a) Differentiate each of the following functions with respect to $x$.

i) $x ^ { 5 } \ln x$\\
ii) $\frac { \mathrm { e } ^ { 3 x } } { x ^ { 3 } - 1 }$\\
iii) $( \tan x + 7 x ) ^ { \frac { 1 } { 2 } }$\\
b) A function is defined implicitly by

$$3 y + 4 x y ^ { 2 } - 5 x ^ { 3 } = 8$$

Find $\frac { \mathrm { d } y } { \mathrm {~d} x }$ in terms of $x$ and $y$.\\
□\\
1\\
The function $f ( x )$ is defined by

$$f ( x ) = \frac { \sqrt { x ^ { 2 } - 1 } } { x }$$

with domain $x \geqslant 1$.\\
a) Find an expression for $f ^ { - 1 } ( x )$. State the domain for $f ^ { - 1 }$ and sketch both $f ( x )$ and $f ^ { - 1 } ( x )$ on the same diagram.\\
b) Explain why the function $f f ( x )$ cannot be formed.\\
□\\
1\\
A chord $A B$ subtends an angle $\theta$ radians at the centre of a circle. The chord divides the circle into two segments whose areas are in the ratio $1 : 2$.\\
\includegraphics[max width=\textwidth, alt={}, center]{966abb82-ade0-4ca8-87a4-26e806d5add7-5_572_576_1197_749}\\
a) Show that $\sin \theta = \theta - \frac { 2 \pi } { 3 }$.\\
b) i) Show that $\theta$ lies between $2 \cdot 6$ and $2 \cdot 7$.\\
ii) Starting with $\theta _ { 0 } = 2 \cdot 6$, use the Newton-Raphson Method to find the value of $\theta$ correct to three decimal places.

\section*{TURN OVER}
Wildflowers grow on the grass verge by the side of a motorway. The area populated by wildflowers at time $t$ years is $A \mathrm {~m} ^ { 2 }$. The rate of increase of $A$ is directly proportional to $A$.\\
a) Write down a differential equation that is satisfied by $A$.\\
b) At time $t = 0$, the area populated by wildflowers is $0.2 \mathrm {~m} ^ { 2 }$. One year later, the area has increased to $1.48 \mathrm {~m} ^ { 2 }$. Find an expression for $A$ in terms of $t$ in the form $p q ^ { t }$, where $p$ and $q$ are rational numbers to be determined.

\hfill \mbox{\textit{WJEC Unit 3 2019 Q10}}