Questions C4 - A-Level Maths

Find a cartesian equation of the curve $C$.
Draw a sketch of the curve $C$.

Edexcel C4 Q3

3. Use the substitution $x = \sin \theta$ to show that, for $| x | \leq 1$, $$\int \frac { 1 } { \left( 1 - x ^ { 2 } \right) ^ { \frac { 3 } { 2 } } } \mathrm {~d} x = \frac { x } { \left( 1 - x ^ { 2 } \right) ^ { \frac { 1 } { 2 } } } + c \text {, where } c \text { is an arbitrary constant. }$$

Edexcel C4 Q4

A measure of the effective voltage, $M$ volts, in an electrical circuit is given by

$$M ^ { 2 } = \int _ { 0 } ^ { 1 } V ^ { 2 } \mathrm {~d} t$$ where $V$ volts is the voltage at time $t$ seconds. Pairs of values of $V$ and $t$ are given in the following table.

$t$	0	0.25	0.5	0.75	1
$V$	- 48	207	37	- 161	- 29
$V ^ { 2 }$

Use the trapezium rule with five values of $V ^ { 2 }$ to estimate the value of $M$.
(6)

Edexcel C4 Q5

5. \begin{figure}[h]

\captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{964070ca-a2c0-4935-8a5b-f1f656495f2e-3_771_1049_251_477}

\end{figure} Figure 1 shows part of the curve with equation $y = 1 + \frac { 1 } { 2 \sqrt { x } }$. The shaded region $R$, bounded by the curve, that $x$-axis and the lines $x = 1$ and $x = 4$, is rotated through $360 ^ { \circ }$ about the $x$-axis. Using integration, show that the volume of the solid generated is $\pi \left( 5 + \frac { 1 } { 2 } \ln 2 \right)$.
(8)

Edexcel C4 Q6

6. Liquid is poured into a container at a constant rate of $30 \mathrm {~cm} ^ { 3 } \mathrm {~s} ^ { - 1 }$. At time $t$ seconds liquid is leaking from the container at a rate of $\frac { 2 } { 15 } V \mathrm {~cm} ^ { 3 } \mathrm {~s} ^ { - 1 }$, where $V \mathrm {~cm} ^ { 3 }$ is the volume of liquid in the container at that time.

Show that $$- 15 \frac { \mathrm {~d} V } { \mathrm {~d} t } = 2 V - 450$$ Given that $V = 1000$ when $t = 0$,
find the solution of the differential equation, in the form $V = \mathrm { f } ( t )$.
Find the limiting value of $V$ as $t \rightarrow \infty$.

Edexcel C4 Q7

7. The curve $C$ has equation $y = \frac { x } { 4 + x ^ { 2 } }$.

Use calculus to find the coordinates of the turning points of $C$. Using the result $\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } = \frac { 2 x \left( x ^ { 2 } - 12 \right) } { \left( 4 + x ^ { 2 } \right) ^ { 3 } }$, or otherwise,
determine the nature of each of the turning points.
Sketch the curve $C$.

Edexcel C4 Q8

8. (i) Given that $\cos ( x + 30 ) ^ { \circ } = 3 \cos ( x - 30 ) ^ { \circ }$, prove that tan $x ^ { \circ } = - \frac { \sqrt { 3 } } { 2 }$.
(ii) (a) Prove that $\frac { 1 - \cos 2 \theta } { \sin 2 \theta } \equiv \tan \theta$.
(b) Verify that $\theta = 180 ^ { \circ }$ is a solution of the equation $\sin 2 \theta = 2 - 2 \cos 2 \theta$.
(c) Using the result in part (a), or otherwise, find the other two solutions, $0 < \theta < 360 ^ { \circ }$, of the equation using $\sin 2 \theta = 2 - 2 \cos 2 \theta$.

Edexcel C4 Q9

9. The equations of the lines $l _ { 1 }$ and $l _ { 2 }$ are given by $$\begin{array} { l l } l _ { 1 } : & \mathbf { r } = \mathbf { i } + 3 \mathbf { j } + 5 \mathbf { k } + \lambda ( \mathbf { i } + 2 \mathbf { j } - \mathbf { k } ) ,
l _ { 2 } : & \mathbf { r } = - 2 \mathbf { i } + 3 \mathbf { j } - 4 \mathbf { k } + \mu ( 2 \mathbf { i } + \mathbf { j } + 4 \mathbf { k } ) , \end{array}$$ where $\lambda$ and $\mu$ are parameters.

Show that $l _ { 1 }$ and $l _ { 2 }$ intersect and find the coordinates of $Q$, their point of intersection.
Show that $l _ { 1 }$ is perpendicular to $l _ { 2 }$. The point $P$ with $x$-coordinate 3 lies on the line $l _ { 1 }$ and the point $R$ with $x$-coordinate 4 lies on the line $l _ { 2 }$.
Find, in its simplest form, the exact area of the triangle $P Q R$. END

Edexcel C4 Q1

Use integration by parts to find the exact value of $\int _ { 1 } ^ { 3 } x ^ { 2 } \ln x \mathrm {~d} x$.
(6)
Fluid flows out of a cylindrical tank with constant cross section. At time $t$ minutes, $t \geq 0$, the volume of fluid remaining in the tank is $V \mathrm {~m} ^ { 3 }$. The rate at which the fluid flows, in $\mathrm { m } ^ { 3 } \mathrm {~min} ^ { - 1 }$, is proportional to the square root of $V$.
1. Show that the depth $h$ metres of fluid in the tank satisfies the differential equation
$$\frac { \mathrm { d } h } { \mathrm {~d} t } = - k \sqrt { } h , \quad \text { where } k \text { is a positive constant. }$$
Show that the general solution of the differential equation may be written as $$h = ( A - B t ) ^ { 2 } , \quad \text { where } A \text { and } B \text { are constants. }$$ Given that at time $t = 0$ the depth of fluid in the tank is 1 m , and that 5 minutes later the depth of fluid has reduced to 0.5 m ,
find the time, $T$ minutes, which it takes for the tank to empty.
Find the depth of water in the tank at time $0.5 T$ minutes.

Edexcel C4 Q3

3. (a) Use the identity for $\cos ( A + B )$ to prove that $\cos 2 A = 2 \cos ^ { 2 } A - 1$.
(b) Use the substitution $x = 2 \sqrt { } 2 \sin \theta$ to prove that $$\int _ { 2 } ^ { \sqrt { 6 } } \sqrt { \left( 8 - x ^ { 2 } \right) } \mathrm { d } x = \frac { 1 } { 3 } ( \pi + 3 \sqrt { } 3 - 6 ) .$$ A curve is given by the parametric equations $$x = \sec \theta , \quad y = \ln ( 1 + \cos 2 \theta ) , \quad 0 \leq \theta < \frac { \pi } { 2 } .$$ (c) Find an equation of the tangent to the curve at the point where $\theta = \frac { \pi } { 3 }$.

Edexcel C4 Q4

4. \begin{figure}[h]

\captionsetup{labelformat=empty} \caption{Figure 2} \includegraphics[alt={},max width=\textwidth]{07bc7f2d-c2b9-4502-91cd-a76afb1ca6c0-3_717_863_248_737}

\end{figure} Figure 2 shows a sketch of the curve $C$ with equation $y = \frac { 4 } { x - 3 } , x \neq 3$. The points $A$ and $B$ on the curve have $x$-coordinates 3.25 and 5 respectively.

Write down the $y$-coordinates of $A$ and $B$.
Show that an equation of $C$ is $\frac { 3 y + 4 } { y } , y \neq 0$. The shaded region $R$ is bounded by $C$, the $y$-axis and the lines through $A$ and $B$ parallel to the $x$ axis. The region $R$ is rotated through $360 ^ { \circ }$ about the $y$-axis to form a solid shape $S$.
Find the volume of $S$, giving your answer in the form $\pi ( a + b \ln c )$, where $a , b$ and $c$ are integers. The solid shape $S$ is used to model a cooling tower. Given that 1 unit on each axis represents 3 metres,
show that the volume of the tower is approximately $15500 \mathrm {~m} ^ { 3 }$.

Edexcel C4 Q5

5. Relative to a fixed origin $O$, the point $A$ has position vector $3 \mathbf { i } + 2 \mathbf { j } - \mathbf { k }$, the point $B$ has position vector $5 \mathbf { i } + \mathbf { j } + \mathbf { k }$, and the point $C$ has position vector $7 \mathbf { i } - \mathbf { j }$.

Find the cosine of angle $A B C$.
Find the exact value of the area of triangle $A B C$. The point $D$ has position vector $7 \mathbf { i } + 3 \mathbf { k }$.
Show that $A C$ is perpendicular to $C D$.
Find the ratio $A D : D B$.

Edexcel C4 Q6

6. \begin{figure}[h]

\captionsetup{labelformat=empty} \caption{Figure 2} \includegraphics[alt={},max width=\textwidth]{07bc7f2d-c2b9-4502-91cd-a76afb1ca6c0-5_809_1226_201_303}

\end{figure} Figure 2 shows the cross-section of a road tunnel and its concrete surround. The curved section of the tunnel is modelled by the curve with equation $y = 8 \sqrt { \left( \sin \frac { \pi x } { 10 } \right) }$, in the interval $0 \leq x \leq$ 10. The concrete surround is represented by the shaded area bounded by the curve, the $x$-axis and the lines $x = - 2 , x = 12$ and $y = 10$. The units on both axes are metres.

Using this model, copy and complete the table below, giving the values of $y$ to 2 decimal places.
$x$ 0 2 4 6 8 10
$y$ 0 6.13 0
The area of the cross-section of the tunnel is given by $\int _ { 0 } ^ { 10 } y \mathrm {~d} x$.
Estimate this area, using the trapezium rule with all the values from your table.
Deduce an estimate of the cross-sectional area of the concrete surround.
State, with a reason, whether your answer in part (c) over-estimates or under-estimates the true value.
(2)

Edexcel C4 Q7

7. $$f ( x ) = \frac { 25 } { ( 3 + 2 x ) ^ { 2 } ( 1 - x ) } , \quad | x | < 1$$

Express $\mathrm { f } ( x )$ as a sum of partial fractions.
Hence find $\int f ( x ) d x$.
Find the series expansion of $\mathrm { f } ( x )$ in ascending powers of $x$ up to and including the term in $x ^ { 2 }$. Give each coefficient as a simplified fraction. END

Edexcel C4 Q1

The following is a table of values for $y = \sqrt { } ( 1 + \sin x )$, where $x$ is in radians.

$x$	0	0.5	1	1.5	2
$y$	1	1.216	$p$	1.413	$q$

Find the value of $p$ and the value of $q$.
(2)
Use the trapezium rule and all the values of $y$ in the completed table to obtain an estimate of $I$, where $$I = \int _ { 0 } ^ { 2 } \sqrt { } ( 1 + \sin x ) \mathrm { d } x$$ (4)

Edexcel C4 Q2

2. (a) Use integration by parts to find $$\int x \cos 2 x d x$$ (b) Prove that the answer to part (a) may be expressed as $$\frac { 1 } { 2 } \sin x ( 2 x \cos x - \sin x ) + C ,$$ where $C$ is an arbitrary constant.

Edexcel C4 Q3

3. (a) Expand $( 1 + 3 x ) ^ { - 2 } , | x | < \frac { 1 } { 3 }$, in ascending powers of $x$ up to and including the term in $x ^ { 3 }$, simplifying each term.
(b) Hence, or otherwise, find the first three terms in the expansion of $\frac { x + 4 } { ( 1 + 3 x ) ^ { 2 } }$ as a series in ascending powers of $x$.

Edexcel C4 Q4

4. Relative to a fixed origin $O$, the point $A$ has position vector $4 \mathbf { i } + 8 \mathbf { j } - \mathbf { k }$, and the point $B$ has position vector $7 \mathbf { i } + 14 \mathbf { j } + 5 \mathbf { k }$.

Find the vector $\overrightarrow { A B }$.
Calculate the cosine of $\angle O A B$.
Show that, for all values of $\lambda$, the point P with position vector $\lambda \mathbf { i } + 2 \lambda \mathbf { j } + ( 2 \lambda - 9 ) \mathbf { k }$ lies on the line through $A$ and $B$.
Find the value of $\lambda$ for which $O P$ is perpendicular to $A B$.
Hence find the coordinates of the foot of the perpendicular from $O$ to $A B$.

Edexcel C4 Q5

5. \begin{figure}[h]

\captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{cb12f63c-f4d0-4eb8-b4a5-0ad12f926b1a-3_668_1172_1231_354}

\end{figure} Figure 1 shows a graph of $y = x \sqrt { } \sin x , 0 < x < \pi$. The maximum point on the curve is $A$.

Show that the $x$-coordinate of the point $A$ satisfies the equation $2 \tan x + x = 0$. The finite region enclosed by the curve and the $x$-axis is shaded as shown in Fig. 1.
A solid body $S$ is generated by rotating this region through $2 \pi$ radians about the $x$-axis.
Find the exact value of the volume of $S$.
(7)

Edexcel C4 Q6

6. A radioactive isotope decays in such a way that the rate of change of the number $N$ of radioactive atoms present after $t$ days, is proportional to $N$.

Write down a differential equation relating $N$ and $t$.
Show that the general solution may be written as $N = A \mathrm { e } ^ { - k t }$, where $A$ and $k$ are positive constants. Initially the number of radioactive atoms present is $7 \times 10 ^ { 18 }$ and 8 days later the number present is $3 \times 10 ^ { 17 }$.
Find the value of $k$.
Find the number of radioactive atoms present after a further 8 days.

Edexcel C4 Q7

7. Given that $$\frac { 10 ( 2 - 3 x ) } { ( 1 - 2 x ) ( 2 + x ) } \equiv \frac { A } { 1 - 2 x } + \frac { B } { 2 + x }$$

find the values of the constants $A$ and $B$.
Hence, or otherwise, find the series expansion in ascending powers of $x$, up to and including the term in $x ^ { 3 }$, of $\frac { 10 ( 2 - 3 x ) } { ( 1 - 2 x ) ( 2 + x ) }$, for $| x | < \frac { 1 } { 2 }$.

Edexcel C4 Q8

8. \begin{figure}[h]

\captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{cb12f63c-f4d0-4eb8-b4a5-0ad12f926b1a-5_609_1210_248_374}

\end{figure} A table top, in the shape of a parallelogram, is made from two types of wood. The design is shown in Fig. 1. The area inside the ellipse is made from one type of wood, and the surrounding area is made from a second type of wood. The ellipse has parametric equations, $$x = 5 \cos \theta , \quad y = 4 \sin \theta , \quad 0 \leq \theta < 2 \pi$$ The parallelogram consists of four line segments, which are tangents to the ellipse at the points where $\theta = \alpha , \theta = - \alpha , \theta = \pi - \alpha , \theta = - \pi + \alpha$.

Find an equation of the tangent to the ellipse at ( $5 \cos \alpha , 4 \sin \alpha$ ), and show that it can be written in the form $$5 y \sin \alpha + 4 x \cos \alpha = 20 .$$
Find by integration the area enclosed by the ellipse.
Hence show that the area enclosed between the ellipse and the parallelogram is $$\frac { 80 } { \sin 2 \alpha } - 20 \pi$$
Given that $0 < \alpha < \frac { \pi } { 4 }$, find the value of $\alpha$ for which the areas of two types of wood are equal.

Edexcel C4 Q1

The curve $C$ has equation $5 x ^ { 2 } + 2 x y - 3 y ^ { 2 } + 3 = 0$. The point $P$ on the curve $C$ has coordinates $( 1,2 )$.
1. Find the gradient of the curve at $P$.
2. Find the equation of the normal to the curve $C$ at $P$, in the form $y = a x + b$, where $a$ and $b$ are constants.
\begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{6e307391-198f-4ea9-99ed-6ef184fca0f7-2_674_895_900_392}
\end{figure} In Fig. 1, the curve $C$ has equation $y = \mathrm { f } ( x )$, where $$\mathrm { f } ( x ) = x + \frac { 2 } { x ^ { 2 } } , \quad x > 0$$ The shaded region is bounded by $C$, the $x$-axis and the lines with equations $x = 1$ and $x = 2$. The shaded region is rotated through $2 \pi$ radians about the $x$-axis. Using calculus, calculate the volume of the solid generated. Give your answer in the form $\pi ( a + \ln b )$, where $a$ and $b$ are constants.
(8)

Edexcel C4 Q3

3.
\includegraphics[max width=\textwidth, alt={}, center]{6e307391-198f-4ea9-99ed-6ef184fca0f7-3_826_873_246_539} Figure 2 shows part of the curve with equation $$y = \mathrm { e } ^ { x } \cos x , 0 \leq x \leq \frac { \pi } { 2 }$$ The finite region $R$ is bounded by the curve and the coordinate axes.

Calculate, to 2 decimal places, the $y$-coordinates of the points on the curve where $x = 0 , \frac { \pi } { 6 } , \frac { \pi } { 3 }$ and $\frac { \pi } { 2 }$.
(3)
Using the trapezium rule and all the values calculated in part (a), find an approximation for the area of $R$.
(4)
State, with a reason, whether your approximation underestimates or overestimates the area of $R$.
(2)

Edexcel C4 Q4

4. A curve is given parametrically by the equations $$x = 5 \cos t , \quad y = - 2 + 4 \sin t , \quad 0 \leq t < 2 \pi$$

Find the coordinates of all the points at which $C$ intersects the coordinate axes, giving your answers in surd form where appropriate.
Sketch the graph at $C$.
$P$ is the point on $C$ where $t = \frac { 1 } { 6 } \pi$.
Show that the normal to $C$ at $P$ has equation $$8 \sqrt { } 3 y = 10 x - 25 \sqrt { } 3$$

\(x\)	0	2	4	6	8	10
\(y\)	0	6.13				0

\(x\)	0	0.5	1	1.5	2
\(y\)	1	1.216	\(p\)	1.413	\(q\)

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