Questions C4 - A-Level Maths

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 2006 January Q1

A curve $C$ is described by the equation

$$3 x ^ { 2 } + 4 y ^ { 2 } - 2 x + 6 x y - 5 = 0$$ Find an equation of the tangent to $C$ at the point $( 1 , - 2 )$, giving your answer in the form $a x + b y + c = 0$, where $a , b$ and $c$ are integers.

Edexcel C4 2006 January Q2

2. (a) Given that $y = \sec x$, complete the table with the values of $y$ corresponding to $x = \frac { \pi } { 16 } , \frac { \pi } { 8 }$ and $\frac { \pi } { 4 }$.

$x$	0	$\frac { \pi } { 16 }$	$\frac { \pi } { 8 }$	$\frac { 3 \pi } { 16 }$	$\frac { \pi } { 4 }$
$y$	1			1.20269

(b) Use the trapezium rule, with all the values for $y$ in the completed table, to obtain an estimate for $\int _ { 0 } ^ { \frac { \pi } { 4 } } \sec x \mathrm {~d} x$. Show all the steps of your working, and give your answer to 4 decimal places. The exact value of $\int _ { 0 } ^ { \frac { \pi } { 4 } } \sec x \mathrm {~d} x$ is $\ln ( 1 + \sqrt { } 2 )$.
(c) Calculate the \% error in using the estimate you obtained in part (b).

Edexcel C4 2006 January Q3

3. Using the substitution $u ^ { 2 } = 2 x - 1$, or otherwise, find the exact value of $$\int _ { 1 } ^ { 5 } \frac { 3 x } { \sqrt { ( 2 x - 1 ) } } \mathrm { d } x$$ (8)
(8)

Edexcel C4 2006 January Q4

4. \begin{figure}[h]

\captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{9bf05d7e-7bb9-40f6-b626-69a8a6eda5a5-05_556_723_299_632}

\end{figure} Figure 1 shows the finite shaded region, $R$, which is bounded by the curve $y = x \mathrm { e } ^ { x }$, the line $x = 1$, the line $x = 3$ and the $x$-axis. The region $R$ is rotated through 360 degrees about the $x$-axis.
Use integration by parts to find an exact value for the volume of the solid generated.
(8)

Edexcel C4 2006 January Q5

5. $$f ( x ) = \frac { 3 x ^ { 2 } + 16 } { ( 1 - 3 x ) ( 2 + x ) ^ { 2 } } = \frac { A } { ( 1 - 3 x ) } + \frac { B } { ( 2 + x ) } + \frac { C } { ( 2 + x ) ^ { 2 } } , \quad | x | < \frac { 1 } { 3 } .$$

Find the values of $A$ and $C$ and show that $B = 0$.
Hence, or otherwise, find the series expansion of $\mathrm { f } ( x )$, in ascending powers of $x$, up to and including the term in $x ^ { 3 }$. Simplify each term.

Edexcel C4 2006 January Q6

6. The line $l _ { 1 }$ has vector equation $$\mathbf { r } = 8 \mathbf { i } + 12 \mathbf { j } + 14 \mathbf { k } + \lambda ( \mathbf { i } + \mathbf { j } - \mathbf { k } ) ,$$ where $\lambda$ is a parameter. The point $A$ has coordinates (4, 8, a), where $a$ is a constant. The point $B$ has coordinates ( $b , 13,13$ ), where $b$ is a constant. Points $A$ and $B$ lie on the line $l _ { 1 }$.

Find the values of $a$ and $b$. Given that the point $O$ is the origin, and that the point $P$ lies on $l _ { 1 }$ such that $O P$ is perpendicular to $l _ { 1 }$,
find the coordinates of $P$.
Hence find the distance $O P$, giving your answer as a simplified surd.

Edexcel C4 2006 January Q7

7. The volume of a spherical balloon of radius $r \mathrm {~cm}$ is $V \mathrm {~cm} ^ { 3 }$, where $V = \frac { 4 } { 3 } \pi r ^ { 3 }$.

Find $\frac { \mathrm { d } V } { \mathrm {~d} r }$. The volume of the balloon increases with time $t$ seconds according to the formula $$\frac { \mathrm { d } V } { \mathrm {~d} t } = \frac { 1000 } { ( 2 t + 1 ) ^ { 2 } } , \quad t \geqslant 0$$
Using the chain rule, or otherwise, find an expression in terms of $r$ and $t$ for $\frac { \mathrm { d } r } { \mathrm {~d} t }$.
Given that $V = 0$ when $t = 0$, solve the differential equation $\frac { \mathrm { d } V } { \mathrm {~d} t } = \frac { 1000 } { ( 2 t + 1 ) ^ { 2 } }$, to obtain $V$ in terms of $t$.
Hence, at time $t = 5$,
1. find the radius of the balloon, giving your answer to 3 significant figures,
2. show that the rate of increase of the radius of the balloon is approximately $2.90 \times 10 ^ { - 2 } \mathrm {~cm} \mathrm {~s} ^ { - 1 }$.

Edexcel C4 2006 January Q8

8. \begin{figure}[h]

\captionsetup{labelformat=empty} \caption{Figure 2} \includegraphics[alt={},max width=\textwidth]{9bf05d7e-7bb9-40f6-b626-69a8a6eda5a5-10_545_979_285_552}

\end{figure} The curve shown in Figure 2 has parametric equations $$x = t - 2 \sin t , \quad y = 1 - 2 \cos t , \quad 0 \leqslant t \leqslant 2 \pi$$

Show that the curve crosses the $x$-axis where $t = \frac { \pi } { 3 }$ and $t = \frac { 5 \pi } { 3 }$. The finite region $R$ is enclosed by the curve and the $x$-axis, as shown shaded in Figure 2.
Show that the area of $R$ is given by the integral $$\int _ { \frac { \pi } { 3 } } ^ { \frac { 5 \pi } { 3 } } ( 1 - 2 \cos t ) ^ { 2 } \mathrm {~d} t$$
Use this integral to find the exact value of the shaded area.

Edexcel C4 2007 January Q1

1. $$f ( x ) = ( 2 - 5 x ) ^ { - 2 } , \quad | x | < \frac { 2 } { 5 }$$ Find the binomial expansion of $\mathrm { f } ( x )$, in ascending powers of $x$, as far as the term in $x ^ { 3 }$, giving each coefficient as a simplified fraction.
(5)

Edexcel C4 2007 January Q2

2. \begin{figure}[h]

\captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{d366e541-15f6-4fb5-9afb-faf6120f1a1c-03_502_917_296_548}

\end{figure} The curve with equation $y = \frac { 1 } { 3 ( 1 + 2 x ) } , x > - \frac { 1 } { 2 }$, is shown in Figure 1.
The region bounded by the lines $x = - \frac { 1 } { 4 } , x = \frac { 1 } { 2 }$, the $x$-axis and the curve is shown shaded in Figure 1. This region is rotated through 360 degrees about the $x$-axis.

Use calculus to find the exact value of the volume of the solid generated. \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 2} \includegraphics[alt={},max width=\textwidth]{d366e541-15f6-4fb5-9afb-faf6120f1a1c-03_383_447_1411_753}
\end{figure} Figure 2 shows a paperweight with axis of symmetry $A B$ where $A B = 3 \mathrm {~cm}$. $A$ is a point on the top surface of the paperweight, and $B$ is a point on the base of the paperweight. The paperweight is geometrically similar to the solid in part (a).
Find the volume of this paperweight.

Edexcel C4 2007 January Q3

A curve has parametric equations

$$x = 7 \cos t - \cos 7 t , y = 7 \sin t - \sin 7 t , \quad \frac { \pi } { 8 } < t < \frac { \pi } { 3 }$$

Find an expression for $\frac { \mathrm { d } y } { \mathrm {~d} x }$ in terms of $t$. You need not simplify your answer.
Find an equation of the normal to the curve at the point where $t = \frac { \pi } { 6 }$. Give your answer in its simplest exact form.

Edexcel C4 2007 January Q4

4. (a) Express $\frac { 2 x - 1 } { ( x - 1 ) ( 2 x - 3 ) }$ in partial fractions.
(b) Given that $x \geqslant 2$, find the general solution of the differential equation $$( 2 x - 3 ) ( x - 1 ) \frac { \mathrm { d } y } { \mathrm {~d} x } = ( 2 x - 1 ) y$$ (c) Hence find the particular solution of this differential equation that satisfies $y = 10$ at $x = 2$, giving your answer in the form $y = \mathrm { f } ( x )$.

Edexcel C4 2007 January Q5

5. A set of curves is given by the equation $\sin x + \cos y = 0.5$.

Use implicit differentiation to find an expression for $\frac { \mathrm { d } y } { \mathrm {~d} x }$. For $- \pi < x < \pi$ and $- \pi < y < \pi$,
find the coordinates of the points where $\frac { \mathrm { d } y } { \mathrm {~d} x } = 0$.

Edexcel C4 2007 January Q6

6. (a) Given that $y = 2 ^ { x }$, and using the result $2 ^ { x } = \mathrm { e } ^ { x \ln 2 }$, or otherwise, show that $\frac { \mathrm { d } y } { \mathrm {~d} x } = 2 ^ { x } \ln 2$.
(b) Find the gradient of the curve with equation $y = 2 ^ { \left( x ^ { 2 } \right) }$ at the point with coordinates $( 2,16 )$.

Edexcel C4 2007 January Q7

7. The point $A$ has position vector $\mathbf { a } = 2 \mathbf { i } + 2 \mathbf { j } + \mathbf { k }$ and the point $B$ has position vector $\mathbf { b } = \mathbf { i } + \mathbf { j } - 4 \mathbf { k }$, relative to an origin $O$.

Find the position vector of the point $C$, with position vector $\mathbf { c }$, given by $$\mathbf { c } = \mathbf { a } + \mathbf { b } .$$
Show that $O A C B$ is a rectangle, and find its exact area. The diagonals of the rectangle, $A B$ and $O C$, meet at the point $D$.
Write down the position vector of the point $D$.
Find the size of the angle $A D C$.

Edexcel C4 2007 January Q8

8. $$I = \int _ { 0 } ^ { 5 } \mathrm { e } ^ { \sqrt { } ( 3 x + 1 ) } \mathrm { d } x$$

Given that $y = \mathrm { e } ^ { \sqrt { } ( 3 x + 1 ) }$, complete the table with the values of $y$ corresponding to $x = 2$, 3 and 4.
$x$ 0 1 2 3 4 5
$y$ $\mathrm { e } ^ { 1 }$ $\mathrm { e } ^ { 2 }$ $\mathrm { e } ^ { 4 }$
Use the trapezium rule, with all the values of $y$ in the completed table, to obtain an estimate for the original integral $I$, giving your answer to 4 significant figures.
Use the substitution $t = \sqrt { } ( 3 x + 1 )$ to show that $I$ may be expressed as $\int _ { a } ^ { b } k t e ^ { t } \mathrm {~d} t$, giving the values of $a , b$ and $k$.
Use integration by parts to evaluate this integral, and hence find the value of $I$ correct to 4 significant figures, showing all the steps in your working.

Edexcel C4 2008 January Q1

1. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{ac7d862f-d10d-45ed-9077-ae4c7413cbf6-02_390_675_246_630} \captionsetup{labelformat=empty} \caption{Figure 1}

\end{figure} The curve shown in Figure 1 has equation $y = \mathrm { e } ^ { x } \sqrt { } ( \sin x ) , 0 \leqslant x \leqslant \pi$. The finite region $R$ bounded by the curve and the $x$-axis is shown shaded in Figure 1.

Complete the table below with the values of $y$ corresponding to $x = \frac { \pi } { 4 }$ and $\frac { \pi } { 2 }$, giving your answers to 5 decimal places.
$x$ 0 $\frac { \pi } { 4 }$ $\frac { \pi } { 2 }$ $\frac { 3 \pi } { 4 }$ $\pi$
$y$ 0 8.87207 0
Use the trapezium rule, with all the values in the completed table, to obtain an estimate for the area of the region $R$. Give your answer to 4 decimal places.

Edexcel C4 2008 January Q2

2. (a) Use the binomial theorem to expand $$( 8 - 3 x ) ^ { \frac { 1 } { 3 } } , \quad | x | < \frac { 8 } { 3 }$$ in ascending powers of $x$, up to and including the term in $x ^ { 3 }$, giving each term as a simplified fraction.
(b) Use your expansion, with a suitable value of $x$, to obtain an approximation to $\sqrt [ 3 ] { } ( 7.7 )$. Give your answer to 7 decimal places.

Edexcel C4 2008 January Q3

3. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{ac7d862f-d10d-45ed-9077-ae4c7413cbf6-04_493_490_278_712} \captionsetup{labelformat=empty} \caption{Figure 2}

\end{figure} The curve shown in Figure 2 has equation $y = \frac { 1 } { ( 2 x + 1 ) }$. The finite region bounded by the curve, the $x$-axis and the lines $x = a$ and $x = b$ is shown shaded in Figure 2. This region is rotated through $360 ^ { \circ }$ about the $x$-axis to generate a solid of revolution. Find the volume of the solid generated. Express your answer as a single simplified fraction, in terms of $a$ and $b$.

Edexcel C4 2008 January Q4

4. (i) Find $\int \ln \left( \frac { x } { 2 } \right) \mathrm { d } x$.
(ii) Find the exact value of $\int _ { \frac { \pi } { 4 } } ^ { \frac { \pi } { 2 } } \sin ^ { 2 } x \mathrm {~d} x$.

Edexcel C4 2008 January Q5

5. A curve is described by the equation $$x ^ { 3 } - 4 y ^ { 2 } = 12 x y$$

Find the coordinates of the two points on the curve where $x = - 8$.
Find the gradient of the curve at each of these points.

Edexcel C4 2008 January Q6

6. The points $A$ and $B$ have position vectors $2 \mathbf { i } + 6 \mathbf { j } - \mathbf { k }$ and $3 \mathbf { i } + 4 \mathbf { j } + \mathbf { k }$ respectively. The line $l _ { 1 }$ passes through the points $A$ and $B$.

Find the vector $\overrightarrow { A B }$.
Find a vector equation for the line $l _ { 1 }$. A second line $l _ { 2 }$ passes through the origin and is parallel to the vector $\mathbf { i } + \mathbf { k }$. The line $l _ { 1 }$ meets the line $l _ { 2 }$ at the point $C$.
Find the acute angle between $l _ { 1 }$ and $l _ { 2 }$.
Find the position vector of the point $C$.

Edexcel C4 2008 January Q7

7. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{ac7d862f-d10d-45ed-9077-ae4c7413cbf6-09_559_864_255_530} \captionsetup{labelformat=empty} \caption{Figure 3}

\end{figure} The curve $C$ has parametric equations $$x = \ln ( t + 2 ) , \quad y = \frac { 1 } { ( t + 1 ) } , \quad t > - 1$$ The finite region $R$ between the curve $C$ and the $x$-axis, bounded by the lines with equations $x = \ln 2$ and $x = \ln 4$, is shown shaded in Figure 3.

Show that the area of $R$ is given by the integral $$\int _ { 0 } ^ { 2 } \frac { 1 } { ( t + 1 ) ( t + 2 ) } \mathrm { d } t$$
Hence find an exact value for this area.
Find a cartesian equation of the curve $C$, in the form $y = \mathrm { f } ( x )$.
State the domain of values for $x$ for this curve.
$\_\_\_\_$}

Edexcel C4 2008 January Q8

8. Liquid is pouring into a large vertical circular cylinder at a constant rate of $1600 \mathrm {~cm} ^ { 3 } \mathrm {~s} ^ { - 1 }$ and is leaking out of a hole in the base, at a rate proportional to the square root of the height of the liquid already in the cylinder. The area of the circular cross section of the cylinder is $4000 \mathrm {~cm} ^ { 2 }$.

Show that at time $t$ seconds, the height $h \mathrm {~cm}$ of liquid in the cylinder satisfies the differential equation $$\frac { \mathrm { d } h } { \mathrm {~d} t } = 0.4 - k \sqrt { } h \text {, where } k \text { is a positive constant. }$$ When $h = 25$, water is leaking out of the hole at $400 \mathrm {~cm} ^ { 3 } \mathrm {~s} ^ { - 1 }$.
Show that $k = 0.02$
Separate the variables of the differential equation $$\frac { \mathrm { d } h } { \mathrm {~d} t } = 0.4 - 0.02 \sqrt { } h$$ to show that the time taken to fill the cylinder from empty to a height of 100 cm is given by $$\int _ { 0 } ^ { 100 } \frac { 50 } { 20 - \sqrt { } h } \mathrm {~d} h$$ Using the substitution $h = ( 20 - x ) ^ { 2 }$, or otherwise,
find the exact value of $\int _ { 0 } ^ { 100 } \frac { 50 } { 20 - \sqrt { h } } \mathrm {~d} h$.
Hence find the time taken to fill the cylinder from empty to a height of 100 cm , giving your answer in minutes and seconds to the nearest second.

\(x\)	0	\(\frac { \pi } { 16 }\)	\(\frac { \pi } { 8 }\)	\(\frac { 3 \pi } { 16 }\)	\(\frac { \pi } { 4 }\)
\(y\)	1			1.20269

\(x\)	0	1	2	3	4	5
\(y\)	\(\mathrm { e } ^ { 1 }\)	\(\mathrm { e } ^ { 2 }\)				\(\mathrm { e } ^ { 4 }\)

\(x\)	0	\(\frac { \pi } { 4 }\)	\(\frac { \pi } { 2 }\)	\(\frac { 3 \pi } { 4 }\)	\(\pi\)
\(y\)	0			8.87207	0

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

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