Questions - Edexcel C4 - A-Level Maths

Find an expression for $\frac { \mathrm { d } y } { \mathrm {~d} x }$ in terms of $t$.
Find the coordinates of all the points on $C$ where $\frac { \mathrm { d } y } { \mathrm {~d} x } = 0$

Edexcel C4 2012 January Q6

6. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{8c963567-d751-4898-b7a7-7095d90514f0-09_639_1179_246_386} \captionsetup{labelformat=empty} \caption{Figure 3}

\end{figure} Figure 3 shows a sketch of the curve with equation $y = \frac { 2 \sin 2 x } { ( 1 + \cos x ) } , 0 \leqslant x \leqslant \frac { \pi } { 2 }$.
The finite region $R$, shown shaded in Figure 3, is bounded by the curve and the $x$-axis. The table below shows corresponding values of $x$ and $y$ for $y = \frac { 2 \sin 2 x } { ( 1 + \cos x ) }$.

$x$	0	$\frac { \pi } { 8 }$	$\frac { \pi } { 4 }$	$\frac { 3 \pi } { 8 }$	$\frac { \pi } { 2 }$
$y$	0		1.17157	1.02280	0

Complete the table above giving the missing value of $y$ to 5 decimal places.
Use the trapezium rule, with all the values of $y$ in the completed table, to obtain an estimate for the area of $R$, giving your answer to 4 decimal places.
Using the substitution $u = 1 + \cos x$, or otherwise, show that $$\int \frac { 2 \sin 2 x } { ( 1 + \cos x ) } d x = 4 \ln ( 1 + \cos x ) - 4 \cos x + k$$ where $k$ is a constant.
Hence calculate the error of the estimate in part (b), giving your answer to 2 significant figures.

Edexcel C4 2012 January Q7

7. Relative to a fixed origin $O$, the point $A$ has position vector ( $2 \mathbf { i } - \mathbf { j } + 5 \mathbf { k }$ ), the point $B$ has position vector $( 5 \mathbf { i } + 2 \mathbf { j } + 10 \mathbf { k } )$, and the point $D$ has position vector $( - \mathbf { i } + \mathbf { j } + 4 \mathbf { k } )$. The line $l$ passes through the points $A$ and $B$.

Find the vector $\overrightarrow { A B }$.
Find a vector equation for the line $l$.
Show that the size of the angle $B A D$ is $109 ^ { \circ }$, to the nearest degree. The points $A , B$ and $D$, together with a point $C$, are the vertices of the parallelogram $A B C D$, where $\overrightarrow { A B } = \overrightarrow { D C }$.
Find the position vector of $C$.
Find the area of the parallelogram $A B C D$, giving your answer to 3 significant figures.
Find the shortest distance from the point $D$ to the line $l$, giving your answer to 3 significant figures.

Edexcel C4 2012 January Q8

(a) Express $\frac { 1 } { P ( 5 - P ) }$ in partial fractions.

A team of conservationists is studying the population of meerkats on a nature reserve. The population is modelled by the differential equation $$\frac { \mathrm { d } P } { \mathrm {~d} t } = \frac { 1 } { 15 } P ( 5 - P ) , \quad t \geqslant 0$$ where $P$, in thousands, is the population of meerkats and $t$ is the time measured in years since the study began. Given that when $t = 0 , P = 1$,
(b) solve the differential equation, giving your answer in the form, $$P = \frac { a } { b + c \mathrm { e } ^ { - \frac { 1 } { 3 } t } }$$ where $a$, $b$ and $c$ are integers.
(c) Hence show that the population cannot exceed 5000

Edexcel C4 2013 January Q1

Given

$$f ( x ) = ( 2 + 3 x ) ^ { - 3 } , \quad | x | < \frac { 2 } { 3 }$$ find the binomial expansion of $\mathrm { f } ( x )$, in ascending powers of $x$, up to and including the term in $x ^ { 3 }$. Give each coefficient as a simplified fraction.

Edexcel C4 2013 January Q2

2. (a) Use integration to find $$\int \frac { 1 } { x ^ { 3 } } \ln x \mathrm {~d} x$$ (b) Hence calculate $$\int _ { 1 } ^ { 2 } \frac { 1 } { x ^ { 3 } } \ln x \mathrm {~d} x$$

Edexcel C4 2013 January Q3

3. Express $\frac { 9 x ^ { 2 } + 20 x - 10 } { ( x + 2 ) ( 3 x - 1 ) }$ in partial fractions.

Edexcel C4 2013 January Q4

4. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{a98d4a7f-1e6d-4294-9b5c-c945e8fbe83e-05_650_1143_223_427} \captionsetup{labelformat=empty} \caption{Figure 1}

\end{figure} Figure 1 shows a sketch of part of the curve with equation $y = \frac { x } { 1 + \sqrt { } x }$. The finite region $R$, shown shaded in Figure 1, is bounded by the curve, the $x$-axis, the line with equation $x = 1$ and the line with equation $x = 4$.

Complete the table with the value of $y$ corresponding to $x = 3$, giving your answer to 4 decimal places.
(1)
$x$ 1 2 3 4
$y$ 0.5 0.8284 1.3333
Use the trapezium rule, with all the values of $y$ in the completed table, to obtain an estimate of the area of the region $R$, giving your answer to 3 decimal places.
Use the substitution $u = 1 + \sqrt { } x$, to find, by integrating, the exact area of $R$.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{a98d4a7f-1e6d-4294-9b5c-c945e8fbe83e-07_743_1568_219_182} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} Figure 2 shows a sketch of part of the curve $C$ with parametric equations $$x = 1 - \frac { 1 } { 2 } t , \quad y = 2 ^ { t } - 1$$ The curve crosses the $y$-axis at the point $A$ and crosses the $x$-axis at the point $B$.

Edexcel C4 2013 January Q6

6. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{a98d4a7f-1e6d-4294-9b5c-c945e8fbe83e-09_862_1534_219_205} \captionsetup{labelformat=empty} \caption{Figure 3}

\end{figure} Figure 3 shows a sketch of part of the curve with equation $y = 1 - 2 \cos x$, where $x$ is measured in radians. The curve crosses the $x$-axis at the point $A$ and at the point $B$.

Find, in terms of $\pi$, the $x$ coordinate of the point $A$ and the $x$ coordinate of the point $B$. The finite region $S$ enclosed by the curve and the $x$-axis is shown shaded in Figure 3. The region $S$ is rotated through $2 \pi$ radians about the $x$-axis.
Find, by integration, the exact value of the volume of the solid generated.

Edexcel C4 2013 January Q7

7. With respect to a fixed origin $O$, the lines $l _ { 1 }$ and $l _ { 2 }$ are given by the equations $$\begin{aligned} & l _ { 1 } : \mathbf { r } = ( 9 \mathbf { i } + 13 \mathbf { j } - 3 \mathbf { k } ) + \lambda ( \mathbf { i } + 4 \mathbf { j } - 2 \mathbf { k } )
& l _ { 2 } : \mathbf { r } = ( 2 \mathbf { i } - \mathbf { j } + \mathbf { k } ) + \mu ( 2 \mathbf { i } + \mathbf { j } + \mathbf { k } ) \end{aligned}$$ where $\lambda$ and $\mu$ are scalar parameters.

Given that $l _ { 1 }$ and $l _ { 2 }$ meet, find the position vector of their point of intersection.
Find the acute angle between $l _ { 1 }$ and $l _ { 2 }$, giving your answer in degrees to 1 decimal place. Given that the point $A$ has position vector $4 \mathbf { i } + 16 \mathbf { j } - 3 \mathbf { k }$ and that the point $P$ lies on $l _ { 1 }$ such that $A P$ is perpendicular to $l _ { 1 }$,
find the exact coordinates of $P$.

Edexcel C4 2013 January Q8

8. A bottle of water is put into a refrigerator. The temperature inside the refrigerator remains constant at $3 ^ { \circ } \mathrm { C }$ and $t$ minutes after the bottle is placed in the refrigerator the temperature of the water in the bottle is $\theta ^ { \circ } \mathrm { C }$. The rate of change of the temperature of the water in the bottle is modelled by the differential equation, $$\frac { \mathrm { d } \theta } { \mathrm {~d} t } = \frac { ( 3 - \theta ) } { 125 }$$

By solving the differential equation, show that, $$\theta = A \mathrm { e } ^ { - 0.008 t } + 3$$ where $A$ is a constant. Given that the temperature of the water in the bottle when it was put in the refrigerator was $16 ^ { \circ } \mathrm { C }$,
find the time taken for the temperature of the water in the bottle to fall to $10 ^ { \circ } \mathrm { C }$, giving your answer to the nearest minute.

Edexcel C4 2014 January Q1

(a) Find the binomial expansion of

$$\frac { 1 } { ( 4 + 3 x ) ^ { 3 } } , \quad | x | < \frac { 4 } { 3 }$$ in ascending powers of $x$, up to and including the term in $x ^ { 3 }$.
Give each coefficient as a simplified fraction. In the binomial expansion of $$\frac { 1 } { ( 4 - 9 x ) ^ { 3 } } , \quad | x | < \frac { 4 } { 9 }$$ the coefficient of $x ^ { 2 }$ is $A$.
(b) Using your answer to part (a), or otherwise, find the value of $A$. Give your answer as a simplified fraction.

Edexcel C4 2014 January Q2

2. (i) Find $$\int x \cos \left( \frac { x } { 2 } \right) \mathrm { d } x$$ (ii) (a) Express $\frac { 1 } { x ^ { 2 } ( 1 - 3 x ) }$ in partial fractions.
(b) Hence find, for $0 < x < \frac { 1 } { 3 }$ $$\int \frac { 1 } { x ^ { 2 } ( 1 - 3 x ) } \mathrm { d } x$$

Edexcel C4 2014 January Q3

The number of bacteria, $N$, present in a liquid culture at time $t$ hours after the start of a scientific study is modelled by the equation

$$N = 5000 ( 1.04 ) ^ { t } , \quad t \geqslant 0$$ where $N$ is a continuous function of $t$.

Find the number of bacteria present at the start of the scientific study.
Find the percentage increase in the number of bacteria present from $t = 0$ to $t = 2$ Given that $N = 15000$ when $t = T$,
find the value of $\frac { \mathrm { d } N } { \mathrm {~d} t }$ when $t = T$, giving your answer to 3 significant figures.

Edexcel C4 2014 January Q4

4. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{245bbe52-3a14-4494-af17-7711caf79b22-10_752_1182_226_395} \captionsetup{labelformat=empty} \caption{Figure 1}

\end{figure} Figure 1 shows a sketch of part of the curve with equation $y = \frac { 4 \mathrm { e } ^ { - x } } { 3 \sqrt { } \left( 1 + 3 \mathrm { e } ^ { - x } \right) }$
The finite region $R$, shown shaded in Figure 1, is bounded by the curve, the $x$-axis, the line $x = - 3 \ln 2$ and the $y$-axis. The table below shows corresponding values of $x$ and $y$ for $y = \frac { 4 \mathrm { e } ^ { - x } } { 3 \sqrt { } \left( 1 + 3 \mathrm { e } ^ { - x } \right) }$

$x$	$- 3 \ln 2$	$- 2 \ln 2$	$- \ln 2$	0
$y$	2.1333		1.0079	0.6667

Complete the table above by giving the missing value of $y$ to 4 decimal places.
Use the trapezium rule, with all the values of $y$ in the completed table, to obtain an estimate for the area of $R$, giving your answer to 2 decimal places.
1. Using the substitution $u = 1 + 3 \mathrm { e } ^ { - x }$, or otherwise, find $$\int \frac { 4 \mathrm { e } ^ { - x } } { 3 \sqrt { } \left( 1 + 3 \mathrm { e } ^ { - x } \right) } \mathrm { d } x$$
2. Hence find the value of the area of $R$.

Edexcel C4 2014 January Q5

Given that $y = 2$ at $x = \frac { \pi } { 8 }$, solve the differential equation

$$\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 3 y ^ { 2 } } { 2 \sin ^ { 2 } 2 x }$$ giving your answer in the form $y = \mathrm { f } ( x )$.
\includegraphics[max width=\textwidth, alt={}, center]{245bbe52-3a14-4494-af17-7711caf79b22-17_81_102_2649_1779}

Edexcel C4 2014 January Q6

6. Oil is leaking from a storage container onto a flat section of concrete at a rate of $0.48 \mathrm {~cm} ^ { 3 } \mathrm {~s} ^ { - 1 }$. The leaking oil spreads to form a pool with an increasing circular cross-section. The pool has a constant uniform thickness of 3 mm . Find the rate at which the radius $r$ of the pool of oil is increasing at the instant when $r = 5 \mathrm {~cm}$. Give your answer, in $\mathrm { cm } \mathrm { s } ^ { - 1 }$, to 3 significant figures. \includegraphics[max width=\textwidth, alt={}, center]{245bbe52-3a14-4494-af17-7711caf79b22-19_104_95_2617_1786}

Edexcel C4 2014 January Q7

7. The curve $C$ has parametric equations $$x = 2 \cos t , \quad y = \sqrt { 3 } \cos 2 t , \quad 0 \leqslant t \leqslant \pi$$ where $t$ is a parameter.

Find an expression for $\frac { \mathrm { d } y } { \mathrm {~d} x }$ in terms of $t$. The point $P$ lies on $C$ where $t = \frac { 2 \pi } { 3 }$
The line $l$ is a normal to $C$ at $P$.
Show that an equation for $l$ is $$2 x - 2 \sqrt { 3 } y - 1 = 0$$ The line $l$ intersects the curve $C$ again at the point $Q$.
Find the exact coordinates of $Q$. You must show clearly how you obtained your answers.
\includegraphics[max width=\textwidth, alt={}, center]{245bbe52-3a14-4494-af17-7711caf79b22-23_106_63_2595_1882}

Edexcel C4 2014 January Q8

8. With respect to a fixed origin $O$, the lines $l _ { 1 }$ and $l _ { 2 }$ are given by the equations $$l _ { 1 } : \mathbf { r } = \left( \begin{array} { r } 2
- 3
4 \end{array} \right) + \lambda \left( \begin{array} { r } - 1
2
1 \end{array} \right) , \quad l _ { 2 } : \mathbf { r } = \left( \begin{array} { r } 2
- 3
4 \end{array} \right) + \mu \left( \begin{array} { r } 5
- 2
5 \end{array} \right)$$ where $\lambda$ and $\mu$ are scalar parameters.

Find, to the nearest $0.1 ^ { \circ }$, the acute angle between $l _ { 1 }$ and $l _ { 2 }$ The point $A$ has position vector $\left( \begin{array} { l } 0
1
6 \end{array} \right)$.
Show that $A$ lies on $l _ { 1 }$ The lines $l _ { 1 }$ and $l _ { 2 }$ intersect at the point $X$.
Write down the coordinates of $X$.
Find the exact value of the distance $A X$. The distinct points $B _ { 1 }$ and $B _ { 2 }$ both lie on the line $l _ { 2 }$
Given that $A X = X B _ { 1 } = X B _ { 2 }$
find the area of the triangle $A B _ { 1 } B _ { 2 }$ giving your answer to 3 significant figures. Given that the $x$ coordinate of $B _ { 1 }$ is positive,
find the exact coordinates of $B _ { 1 }$ and the exact coordinates of $B _ { 2 }$
\includegraphics[max width=\textwidth, alt={}, center]{245bbe52-3a14-4494-af17-7711caf79b22-28_96_59_2478_1834}

Edexcel C4 2005 June Q1

Use the binomial theorem to expand

$$\sqrt { } ( 4 - 9 x ) , \quad | x | < \frac { 4 } { 9 }$$ in ascending powers of $x$, up to and including the term in $x ^ { 3 }$, simplifying each term.

Edexcel C4 2005 June Q2

2. A curve has equation $$x ^ { 2 } + 2 x y - 3 y ^ { 2 } + 16 = 0 .$$ Find the coordinates of the points on the curve where $\frac { \mathrm { d } y } { \mathrm {~d} x } = 0$.

Edexcel C4 2005 June Q3

3. (a) Express $\frac { 5 x + 3 } { ( 2 x - 3 ) ( x + 2 ) }$ in partial fractions.
(b) Hence find the exact value of $\int _ { 2 } ^ { 6 } \frac { 5 x + 3 } { ( 2 x - 3 ) ( x + 2 ) } \mathrm { d } x$, giving your answer as a single logarithm.

Edexcel C4 2005 June Q4

4. Use the substitution $x = \sin \theta$ to find the exact value of $$\int _ { 0 } ^ { \frac { 1 } { 2 } } \frac { 1 } { \left( 1 - x ^ { 2 } \right) ^ { \frac { 3 } { 2 } } } \mathrm {~d} x$$

Edexcel C4 2005 June Q5

5. \begin{figure}[h]

\captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{7fa2c564-d1e5-4fd0-a690-e3189daea332-06_586_1079_260_427}

\end{figure} Figure 1 shows the graph of the curve with equation $$y = x \mathrm { e } ^ { 2 x } , \quad x \geqslant 0$$ The finite region $R$ bounded by the lines $x = 1$, the $x$-axis and the curve is shown shaded in Figure 1.

Use integration to find the exact value for the area of $R$.
Complete the table with the values of $y$ corresponding to $x = 0.4$ and 0.8 .
$x$ 0 0.2 0.4 0.6 0.8 1
$y = x \mathrm { e } ^ { 2 x }$ 0 0.29836 1.99207 7.38906
Use the trapezium rule with all the values in the table to find an approximate value for this area, giving your answer to 4 significant figures.

Edexcel C4 2005 June Q6

A curve has parametric equations

$$x = 2 \cot t , \quad y = 2 \sin ^ { 2 } t , \quad 0 < t \leqslant \frac { \pi } { 2 }$$

Find an expression for $\frac { \mathrm { d } y } { \mathrm {~d} x }$ in terms of the parameter $t$.
Find an equation of the tangent to the curve at the point where $t = \frac { \pi } { 4 }$.
Find a cartesian equation of the curve in the form $y = \mathrm { f } ( x )$. State the domain on which the curve is defined.

\(x\)	0	\(\frac { \pi } { 8 }\)	\(\frac { \pi } { 4 }\)	\(\frac { 3 \pi } { 8 }\)	\(\frac { \pi } { 2 }\)
\(y\)	0		1.17157	1.02280	0

\(x\)	\(- 3 \ln 2\)	\(- 2 \ln 2\)	\(- \ln 2\)	0
\(y\)	2.1333		1.0079	0.6667

\(x\)	0	0.2	0.4	0.6	0.8	1
\(y = x \mathrm { e } ^ { 2 x }\)	0	0.29836		1.99207		7.38906

\(x\)	1	2	3	4
\(y\)	0.5	0.8284		1.3333

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