Questions - Edexcel C4 - A-Level Maths

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 $$8 \sqrt { } 3 y = 10 x - 25 \sqrt { } 3$$

Edexcel C4 Q5

5. \begin{figure}[h]

\captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{6e307391-198f-4ea9-99ed-6ef184fca0f7-5_846_693_246_612}

\end{figure} The curve $C$ has equation $y = \mathrm { f } ( x ) , x \in \mathbb { R }$. Figure 1 shows the part of $C$ for which $0 \leq x \leq 2$. Given that $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \mathrm { e } ^ { x } - 2 x ^ { 2 } ,$$ and that $C$ has a single maximum, at $x = k$,

show that $1.48 < k < 1.49$. Given also that the point $( 0,5 )$ lies on $C$,
find $\mathrm { f } ( x )$. The finite region $R$ is bounded by $C$, the coordinate axes and the line $x = 2$.
Use integration to find the exact area of $R$.
(4)

Edexcel C4 Q6

6. When $( 1 + a x ) ^ { n }$ is expanded as a series in ascending powers of $x$, the coefficients of $x$ and $x ^ { 2 }$ are - 6 and 27 respectively.

Find the value of $a$ and the value of $n$.
Find the coefficient of $x ^ { 3 }$.
State the set of values of $x$ for which the expansion is valid.

Edexcel C4 Q7

7. Two submarines are travelling in straight lines through the ocean. Relative to a fixed origin, the vector equations of the two lines, $l _ { 1 }$ and $l _ { 2 }$, along which they travel are $$\text { and } \quad \begin{aligned} & \mathbf { r } = 3 \mathbf { i } + 4 \mathbf { j } - 5 \mathbf { k } + \lambda ( \mathbf { i } - 2 \mathbf { j } + 2 \mathbf { k } )
& \mathbf { r } = 9 \mathbf { i } + \mathbf { j } - 2 \mathbf { k } + \mu ( 4 \mathbf { i } + \mathbf { j } - \mathbf { k } ) , \end{aligned}$$ where $\lambda$ and $\mu$ are scalars.

Show that the submarines are moving in perpendicular directions.
Given that $l _ { 1 }$ and $l _ { 2 }$ intersect at the point $A$, find the position vector of $A$. The point $b$ has position vector $10 \mathbf { j } - 11 \mathbf { k }$.
Show that only one of the submarines passes through the point $B$.
Given that 1 unit on each coordinate axis represents 100 m , find, in km , the distance $A B$.

Edexcel C4 Q8

8. In a chemical reaction two substances combine to form a third substance. At time $t , t \geq 0$, the concentration of this third substance is $x$ and the reaction is modelled by the differential equation $$\frac { \mathrm { d } x } { \mathrm {~d} t } = k ( 1 - 2 x ) ( 1 - 4 x ) \text {, where } k \text { is a positive constant. }$$

Solve this differential equation and hence show that $$\ln \left| \frac { 1 - 2 x } { 1 - 4 x } \right| = 2 k t + c \text {, where } c \text { is an arbitrary constant. }$$
Given that $x = 0$ when $t = 0$, find an expression for $x$ in terms of $k$ and $t$.
Find the limiting value of the concentration $x$ as $t$ becomes very large. END

Edexcel C4 Q1

(a) Express $1.5 \sin 2 x + 2 \cos 2 x$ in the form $R \sin ( 2 x + \alpha )$, where $R > 0$ and $0 < \alpha < \frac { 1 } { 2 } \pi$, giving your values of $R$ and $\alpha$ to 3 decimal places where appropriate.
(b) Express $3 \sin x \cos x + 4 \cos ^ { 2 } x$ in the form $a \cos 2 x + b \sin 2 x + c$, where $a , b$ and $c$ are constants to be found.
(c) Hence, using your answer to part (a), deduce the maximum value of $3 \sin x \cos x + 4 \cos ^ { 2 } x$.

\begin{figure}[h]

\captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{a1b078fe-96e3-4d62-bf0d-415294ba022f-2_668_796_863_507}

\end{figure} Figure 1 shows part of the curve with equation $y = \mathrm { f } ( x )$, where $$f ( x ) = \frac { x ^ { 2 } + 1 } { ( 1 + x ) ( 3 - x ) } , 0 \leq x < 3$$ (a) Given that $\mathrm { f } ( x ) = A + \frac { B } { 1 + x } + \frac { C } { 3 - x }$, find the values of the constants $A , B$ and $C$. The finite region $R$, shown in Fig. 1, is bounded by the curve with equation $y = \mathrm { f } ( x )$, the $x$-axis, the $y$-axis and the line $x = 2$.
(b) Find the area of $R$, giving your answer in the form $p + q \ln r$, where $p , q$ and $r$ are rational constants to be found.

Edexcel C4 Q3

3. A student tests the accuracy of the trapezium rule by evaluating $I$, where $$I = \int _ { 0.5 } ^ { 1.5 } \left( \frac { 3 } { x } + x ^ { 4 } \right) \mathrm { d } x$$

Complete the student's table, giving values to 2 decimal places where appropriate.
$x$ 0.5 0.75 1 1.25 1.5
$\frac { 3 } { x } + x ^ { 4 }$ 6.06 4.32
Use the trapezium rule, with all the values from your table, to calculate an estimate for the value of $I$.
Use integration to calculate the exact value of $I$.
Verify that the answer obtained by the trapezium rule is within $3 \%$ of the exact value.

Edexcel C4 Q4

4. Figure 1
\includegraphics[max width=\textwidth, alt={}, center]{a1b078fe-96e3-4d62-bf0d-415294ba022f-4_588_1008_242_566} Figure 1 shows a cross-section $R$ of a dam. The line $A C$ is the vertical face of the dam, $A B$ is the horizontal base and the curve $B C$ is the profile. Taking $x$ and $y$ to be the horizontal and vertical axes, then $A , B$ and $C$ have coordinates $( 0,0 ) , \left( 3 \pi ^ { 2 } , 0 \right)$ and $( 0,30 )$ respectively. The area of the cross-section is to be calculated. Initially the profile $B C$ is approximated by a straight line.

Find an estimate for the area of the cross-section $R$ using this approximation.
(1) The profile $B C$ is actually described by the parametric equations. $$x = 16 t ^ { 2 } - \pi ^ { 2 } , \quad y = 30 \sin 2 t , \quad \frac { \pi } { 4 } \leq t \leq \frac { \pi } { 2 }$$
Find the exact area of the cross-section $R$.
(7)
Calculate the percentage error in the estimate of the area of the cross-section $R$ that you found in part (a).
(2)

Edexcel C4 Q5

5. (a) Prove that, when $x = \frac { 1 } { 15 }$, the value of $( 1 + 5 x ) ^ { - \frac { 1 } { 2 } }$ is exactly equal to $\sin 60 ^ { \circ }$.
(3)
(b) Expand $( 1 + 5 x ) ^ { - \frac { 1 } { 2 } } , | x | < 0.2$, in ascending powers of $x$ up to and including the term in $x ^ { 3 }$, simplifying each term.
(c) Use your answer to part (b) to find an approximation for $\sin 60 ^ { \circ }$.
(d) Find the difference between the exact value of $\sin 60 ^ { \circ }$ and the approximation in part (c).

Edexcel C4 Q6

6. (a) Use integration by parts to show that $$\int _ { 0 } ^ { \frac { \pi } { 4 } } x \sec ^ { 2 } x \mathrm {~d} x = \frac { 1 } { 4 } \pi - \frac { 1 } { 2 } \ln 2 .$$ \begin{figure}[h]

\captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{a1b078fe-96e3-4d62-bf0d-415294ba022f-5_841_1202_459_434}

\end{figure} The finite region $R$, bounded by the equation $y = x ^ { \frac { 1 } { 2 } } \sec x$, the line $x = \frac { \pi } { 4 }$ and the $x$-axis is shown in Fig. 1. The region $R$ is rotated through $2 \pi$ radians about the $x$-axis.
(b) Find the volume of the solid of revolution generated.
(c) Find the gradient of the curve with equation $y = x ^ { \frac { 1 } { 2 } } \sec x$ at the point where $x = \frac { \pi } { 4 }$.

\(x\)	0.5	0.75	1	1.25	1.5
\(\frac { 3 } { x } + x ^ { 4 }\)	6.06	4.32

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

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