Questions C4 - A-Level Maths

Edexcel C4 2015 June Q8

8
5
- 2 \end{array} \right) + \mu \left( \begin{array} { r } 3
4
- 5 \end{array} \right)$$ where $\lambda$ and $\mu$ are scalar parameters and $p$ is a constant. The lines $l _ { 1 }$ and $l _ { 2 }$ intersect at the point $A$.

Find the coordinates of $A$.
Find the value of the constant $p$.
Find the acute angle between $l _ { 1 }$ and $l _ { 2 }$, giving your answer in degrees to 2 decimal places. The point $B$ lies on $l _ { 2 }$ where $\mu = 1$
Find the shortest distance from the point $B$ to the line $l _ { 1 }$, giving your answer to 3 significant figures.
1. A curve $C$ has parametric equations
$$x = 4 t + 3 , \quad y = 4 t + 8 + \frac { 5 } { 2 t } , \quad t \neq 0$$
Find the value of $\frac { \mathrm { d } y } { \mathrm {~d} x }$ at the point on $C$ where $t = 2$, giving your answer as a fraction in its simplest form.
Show that the cartesian equation of the curve $C$ can be written in the form $$y = \frac { x ^ { 2 } + a x + b } { x - 3 } , \quad x \neq 3$$ where $a$ and $b$ are integers to be determined.
6. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{89d4a7a5-3f4f-4d16-b14e-a27243cedd78-11_666_993_244_392} \captionsetup{labelformat=empty} \caption{Diagram not to scale}
\end{figure} Figure 2 Figure 2 shows a sketch of the curve with equation $y = \sqrt { ( 3 - x ) ( x + 1 ) } , 0 \leqslant x \leqslant 3$
The finite region $R$, shown shaded in Figure 2, is bounded by the curve, the $x$-axis, and the $y$-axis.
Use the substitution $x = 1 + 2 \sin \theta$ to show that $$\int _ { 0 } ^ { 3 } \sqrt { ( 3 - x ) ( x + 1 ) } d x = k \int _ { - \frac { \pi } { 6 } } ^ { \frac { \pi } { 2 } } \cos ^ { 2 } \theta d \theta$$ where $k$ is a constant to be determined.
Hence find, by integration, the exact area of $R$. 7. (a) Express $\frac { 2 } { P ( P - 2 ) }$ in partial fractions. A team of biologists is studying a population of a particular species of animal. The population is modelled by the differential equation $$\frac { \mathrm { d } P } { \mathrm {~d} t } = \frac { 1 } { 2 } P ( P - 2 ) \cos 2 t , t \geqslant 0$$ where $P$ is the population in thousands, and $t$ is the time measured in years since the start of the study. Given that $P = 3$ when $t = 0$,
solve this differential equation to show that $$P = \frac { 6 } { 3 - \mathrm { e } ^ { \frac { 1 } { 2 } \sin 2 t } }$$
find the time taken for the population to reach 4000 for the first time. Give your answer in years to 3 significant figures.
8. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{89d4a7a5-3f4f-4d16-b14e-a27243cedd78-15_696_1418_287_262} \captionsetup{labelformat=empty} \caption{Figure 3}
\end{figure} Figure 3 shows a sketch of part of the curve $C$ with equation $$y = 3 ^ { x }$$ The point $P$ lies on $C$ and has coordinates $( 2,9 )$.
The line $l$ is a tangent to $C$ at $P$. The line $l$ cuts the $x$-axis at the point $Q$.
Find the exact value of the $x$ coordinate of $Q$. The finite region $R$, shown shaded in Figure 3, is bounded by the curve $C$, the $x$-axis, the $y$-axis and the line $l$. This region $R$ is rotated through $360 ^ { \circ }$ about the $x$-axis.
Use integration to find the exact value of the volume of the solid generated. Give your answer in the form $\frac { p } { q }$ where $p$ and $q$ are exact constants.
[0pt] [You may assume the formula $V = \frac { 1 } { 3 } \pi r ^ { 2 } h$ for the volume of a cone.]

Edexcel C4 2016 June Q1

Use the binomial series to find the expansion of

$$\frac { 1 } { ( 2 + 5 x ) ^ { 3 } } , \quad | x | < \frac { 2 } { 5 }$$ in ascending powers of $x$, up to and including the term in $x ^ { 3 }$.
Give each coefficient as a fraction in its simplest form.
(6)

Edexcel C4 2016 June Q2

2. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{cbfbb690-bc85-46e5-a97f-35df4b6f1c84-03_712_1091_248_470} \captionsetup{labelformat=empty} \caption{Figure 1}

\end{figure} Figure 1 shows a sketch of part of the curve with equation $y = x ^ { 2 } \ln x , x \geqslant 1$
The finite region $R$, shown shaded in Figure 1, is bounded by the curve, the $x$-axis and the line $x = 2$ The table below shows corresponding values of $x$ and $y$ for $y = x ^ { 2 } \ln x$

$x$	1	1.2	1.4	1.6	1.8	2
$y$	0	0.2625		1.2032	1.9044	2.7726

Complete the table above, 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 3 decimal places.
Use integration to find the exact value for the area of $R$.

Edexcel C4 2016 June Q3

The curve $C$ has equation

$$2 x ^ { 2 } y + 2 x + 4 y - \cos ( \pi y ) = 17$$

Use implicit differentiation to find $\frac { \mathrm { d } y } { \mathrm {~d} x }$ in terms of $x$ and $y$. The point $P$ with coordinates $\left( 3 , \frac { 1 } { 2 } \right)$ lies on $C$.
The normal to $C$ at $P$ meets the $x$-axis at the point $A$.
Find the $x$ coordinate of $A$, giving your answer in the form $\frac { a \pi + b } { c \pi + d }$, where $a , b , c$ and $d$ are integers to be determined.

Edexcel C4 2016 June Q4

4. The rate of decay of the mass of a particular substance is modelled by the differential equation $$\frac { \mathrm { d } x } { \mathrm {~d} t } = - \frac { 5 } { 2 } x , \quad t \geqslant 0$$ where $x$ is the mass of the substance measured in grams and $t$ is the time measured in days.
Given that $x = 60$ when $t = 0$,

solve the differential equation, giving $x$ in terms of $t$. You should show all steps in your working and give your answer in its simplest form.
Find the time taken for the mass of the substance to decay from 60 grams to 20 grams. Give your answer to the nearest minute.

Edexcel C4 2016 June Q5

5. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{cbfbb690-bc85-46e5-a97f-35df4b6f1c84-09_605_1131_248_466} \captionsetup{labelformat=empty} \caption{Figure 2}

\end{figure} Figure 2 shows a sketch of the curve $C$ with parametric equations $$x = 4 \tan t , \quad y = 5 \sqrt { 3 } \sin 2 t , \quad 0 \leqslant t < \frac { \pi } { 2 }$$ The point $P$ lies on $C$ and has coordinates $\left( 4 \sqrt { 3 } , \frac { 15 } { 2 } \right)$.

Find the exact value of $\frac { \mathrm { d } y } { \mathrm {~d} x }$ at the point $P$. Give your answer as a simplified surd. The point $Q$ lies on the curve $C$, where $\frac { \mathrm { d } y } { \mathrm {~d} x } = 0$
Find the exact coordinates of the point $Q$.

Edexcel C4 2016 June Q6

6. (i) Given that $y > 0$, find $$\int \frac { 3 y - 4 } { y ( 3 y + 2 ) } d y$$ (ii) (a) Use the substitution $x = 4 \sin ^ { 2 } \theta$ to show that $$\int _ { 0 } ^ { 3 } \sqrt { \left( \frac { x } { 4 - x } \right) } \mathrm { d } x = \lambda \int _ { 0 } ^ { \frac { \pi } { 3 } } \sin ^ { 2 } \theta \mathrm {~d} \theta$$ where $\lambda$ is a constant to be determined.
(b) Hence use integration to find $$\int _ { 0 } ^ { 3 } \sqrt { \left( \frac { x } { 4 - x } \right) } d x$$ giving your answer in the form $a \pi + b$, where $a$ and $b$ are exact constants.

Edexcel C4 2016 June Q7

7. (a) Find $$\int ( 2 x - 1 ) ^ { \frac { 3 } { 2 } } d x$$ giving your answer in its simplest form. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{cbfbb690-bc85-46e5-a97f-35df4b6f1c84-13_727_1177_596_370} \captionsetup{labelformat=empty} \caption{Figure 3}

\end{figure} Figure 3 shows a sketch of part of the curve $C$ with equation $$y = ( 2 x - 1 ) ^ { \frac { 3 } { 4 } } , \quad x \geqslant \frac { 1 } { 2 }$$ The curve $C$ cuts the line $y = 8$ at the point $P$ with coordinates $( k , 8 )$, where $k$ is a constant.
(b) Find the value of $k$. The finite region $S$, shown shaded in Figure 3, is bounded by the curve $C$, the $x$-axis, the $y$-axis and the line $y = 8$. This region is rotated through $2 \pi$ radians about the $x$-axis to form a solid of revolution.
(c) Find the exact value of the volume of the solid generated.

Edexcel C4 2016 June Q8

8. With respect to a fixed origin $O$, the line $l _ { 1 }$ is given by the equation $$\mathbf { r } = \left( \begin{array} { r } 8
1
- 3 \end{array} \right) + \mu \left( \begin{array} { r } - 5
4
3 \end{array} \right)$$ where $\mu$ is a scalar parameter.
The point $A$ lies on $l _ { 1 }$ where $\mu = 1$

Find the coordinates of $A$. The point $P$ has position vector $\left( \begin{array} { l } 1
5
2 \end{array} \right)$.
The line $l _ { 2 }$ passes through the point $P$ and is parallel to the line $l _ { 1 }$
Write down a vector equation for the line $l _ { 2 }$
Find the exact value of the distance $A P$. Give your answer in the form $k \sqrt { 2 }$, where $k$ is a constant to be determined. The acute angle between $A P$ and $l _ { 2 }$ is $\theta$.
Find the value of $\cos \theta$ A point $E$ lies on the line $l _ { 2 }$ Given that $A P = P E$,
find the area of triangle $A P E$,
find the coordinates of the two possible positions of $E$.

Edexcel C4 2017 June Q1

The curve $C$ has parametric equations

$$x = 3 t - 4 , y = 5 - \frac { 6 } { t } , \quad t > 0$$

Find $\frac { \mathrm { d } y } { \mathrm {~d} x }$ in terms of $t$ The point $P$ lies on $C$ where $t = \frac { 1 } { 2 }$
Find the equation of the tangent to $C$ at the point $P$. Give your answer in the form $y = p x + q$, where $p$ and $q$ are integers to be determined.
Show that the cartesian equation for $C$ can be written in the form $$y = \frac { a x + b } { x + 4 } , \quad x > - 4$$ where $a$ and $b$ are integers to be determined.

Edexcel C4 2017 June Q2

2. $\quad \mathrm { f } ( x ) = ( 2 + k x ) ^ { - 3 } , \quad | k x | < 2$, where $k$ is a positive constant The binomial expansion of $\mathrm { f } ( x )$, in ascending powers of $x$, up to and including the term in $x ^ { 2 }$ is $$A + B x + \frac { 243 } { 16 } x ^ { 2 }$$ where $A$ and $B$ are constants.

Write down the value of $A$.
Find the value of $k$.
Find the value of $B$.

Edexcel C4 2017 June Q3

3. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{cd958ff3-ed4e-4bd7-aa4b-339da6d618a6-08_560_1082_242_438} \captionsetup{labelformat=empty} \caption{Figure 1}

\end{figure} Figure 1 shows a sketch of part of the curve with equation $y = \frac { 6 } { \left( \mathrm { e } ^ { x } + 2 \right) } , x \in \mathbb { R }$
The finite region $R$, shown shaded in Figure 1, is bounded by the curve, the $y$-axis, the $x$-axis and the line with equation $x = 1$ The table below shows corresponding values of $x$ and $y$ for $y = \frac { 6 } { \left( \mathrm { e } ^ { x } + 2 \right) }$

$x$	0	0.2	0.4	0.6	0.8	1
$y$	2		1.71830	1.56981	1.41994	1.27165

Complete the table above by 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 find an estimate for the area of $R$, giving your answer to 4 decimal places.
Use the substitution $u = \mathrm { e } ^ { x }$ to show that the area of $R$ can be given by $$\int _ { a } ^ { b } \frac { 6 } { u ( u + 2 ) } \mathrm { d } u$$ where $a$ and $b$ are constants to be determined.
Hence use calculus to find the exact area of $R$. [Solutions based entirely on graphical or numerical methods are not acceptable.]

Edexcel C4 2017 June Q4

4. The curve $C$ has equation $$4 x ^ { 2 } - y ^ { 3 } - 4 x y + 2 ^ { y } = 0$$ The point $P$ with coordinates $( - 2,4 )$ lies on $C$.

Find the exact value of $\frac { \mathrm { d } y } { \mathrm {~d} x }$ at the point $P$. The normal to $C$ at $P$ meets the $y$-axis at the point $A$.
Find the $y$ coordinate of $A$, giving your answer in the form $p + q \ln 2$, where $p$ and $q$ are constants to be determined.
(3)

Edexcel C4 2017 June Q5

5. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{cd958ff3-ed4e-4bd7-aa4b-339da6d618a6-16_589_540_248_705} \captionsetup{labelformat=empty} \caption{Figure 2}

\end{figure} Diagram not drawn to scale The finite region $S$, shown shaded in Figure 2, is bounded by the $y$-axis, the $x$-axis, the line with equation $x = \ln 4$ and the curve with equation $$y = \mathrm { e } ^ { x } + 2 \mathrm { e } ^ { - x } , \quad x \geqslant 0$$ The region $S$ is rotated through $2 \pi$ radians about the $x$-axis.
Use integration to find the exact value of the volume of the solid generated. Give your answer in its simplest form.
[0pt] [Solutions based entirely on graphical or numerical methods are not acceptable.]

Edexcel C4 2017 June Q6

6. With respect to a fixed origin $O$, the lines $l _ { 1 }$ and $l _ { 2 }$ are given by the equations $$l _ { 1 } : \mathbf { r } = \left( \begin{array} { r } 4
28
4 \end{array} \right) + \lambda \left( \begin{array} { r } - 1
- 5
1 \end{array} \right) , \quad l _ { 2 } : \mathbf { r } = \left( \begin{array} { l } 5
3
1 \end{array} \right) + \mu \left( \begin{array} { r } 3
0
- 4 \end{array} \right)$$ where $\lambda$ and $\mu$ are scalar parameters. The lines $l _ { 1 }$ and $l _ { 2 }$ intersect at the point $X$.

Find the coordinates of the point $X$.
Find the size of the acute angle between $l _ { 1 }$ and $l _ { 2 }$, giving your answer in degrees to 2 decimal places. The point $A$ lies on $l _ { 1 }$ and has position vector $\left( \begin{array} { r } 2
18
6 \end{array} \right)$
Find the distance $A X$, giving your answer as a surd in its simplest form. The point $Y$ lies on $l _ { 2 }$. Given that the vector $\overrightarrow { Y A }$ is perpendicular to the line $l _ { 1 }$
find the distance $Y A$, giving your answer to one decimal place. The point $B$ lies on $l _ { 1 }$ where $| \overrightarrow { A X } | = 2 | \overrightarrow { A B } |$.
Find the two possible position vectors of $B$.

Edexcel C4 2017 June Q7

7. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{cd958ff3-ed4e-4bd7-aa4b-339da6d618a6-24_835_1160_255_529} \captionsetup{labelformat=empty} \caption{Figure 3}

\end{figure} Figure 3 shows a vertical cylindrical tank of height 200 cm containing water. Water is leaking from a hole $P$ on the side of the tank. At time $t$ minutes after the leaking starts, the height of water in the tank is $h \mathrm {~cm}$. The height $h \mathrm {~cm}$ of the water in the tank satisfies the differential equation $$\frac { \mathrm { d } h } { \mathrm {~d} t } = k ( h - 9 ) ^ { \frac { 1 } { 2 } } , \quad 9 < h \leqslant 200$$ where $k$ is a constant. Given that, when $h = 130$, the height of the water is falling at a rate of 1.1 cm per minute,

find the value of $k$. Given that the tank was full of water when the leaking started,
solve the differential equation with your value of $k$, to find the value of $t$ when $h = 50$

Edexcel C4 2017 June Q8

8. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{cd958ff3-ed4e-4bd7-aa4b-339da6d618a6-28_721_714_255_616} \captionsetup{labelformat=empty} \caption{Figure 4}

\end{figure} Diagram not drawn to scale Figure 4 shows a sketch of part of the curve $C$ with parametric equations $$x = 3 \theta \sin \theta , \quad y = \sec ^ { 3 } \theta , \quad 0 \leqslant \theta < \frac { \pi } { 2 }$$ The point $P ( k , 8 )$ lies on $C$, where $k$ is a constant.

Find the exact value of $k$. The finite region $R$, shown shaded in Figure 4, is bounded by the curve $C$, the $y$-axis, the $x$-axis and the line with equation $x = k$.
Show that the area of $R$ can be expressed in the form $$\lambda \int _ { \alpha } ^ { \beta } \left( \theta \sec ^ { 2 } \theta + \tan \theta \sec ^ { 2 } \theta \right) \mathrm { d } \theta$$ where $\lambda , \alpha$ and $\beta$ are constants to be determined.
Hence use integration to find the exact value of the area of $R$.

Edexcel C4 2018 June Q1

(a) Find the binomial series expansion of

$$\sqrt { 4 - 9 x } , | x | < \frac { 4 } { 9 }$$ in ascending powers of $x$, up to and including the term in $x ^ { 2 }$ Give each coefficient in its simplest form.
(b) Use the expansion from part (a), with a suitable value of $x$, to find an approximate value for $\sqrt { 310 }$
Show all your working and give your answer to 3 decimal places.

Edexcel C4 2018 June Q2

The curve $C$ has equation

$$x ^ { 2 } + x y + y ^ { 2 } - 4 x - 5 y + 1 = 0$$

Use implicit differentiation to find $\frac { \mathrm { d } y } { \mathrm {~d} x }$ in terms of $x$ and $y$.
Find the $x$ coordinates of the two points on $C$ where $\frac { \mathrm { d } y } { \mathrm {~d} x } = 0$ Give exact answers in their simplest form.
(Solutions based entirely on graphical or numerical methods are not acceptable.)

Edexcel C4 2018 June Q3

3. (i) Given that $$\frac { 13 - 4 x } { ( 2 x + 1 ) ^ { 2 } ( x + 3 ) } \equiv \frac { A } { ( 2 x + 1 ) } + \frac { B } { ( 2 x + 1 ) ^ { 2 } } + \frac { C } { ( x + 3 ) }$$

find the values of the constants $A , B$ and $C$.
Hence find $$\int \frac { 13 - 4 x } { ( 2 x + 1 ) ^ { 2 } ( x + 3 ) } \mathrm { d } x , \quad x > - \frac { 1 } { 2 }$$ (ii) Find $$\int \left( \mathrm { e } ^ { x } + 1 \right) ^ { 3 } \mathrm {~d} x$$ (iii) Using the substitution $u ^ { 3 } = x$, or otherwise, find $$\int \frac { 1 } { 4 x + 5 x ^ { \frac { 1 } { 3 } } } \mathrm {~d} x , \quad x > 0$$

Edexcel C4 2018 June Q4

4. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{0c4a3759-ecaa-47c3-a071-ce25fd11159f-12_978_1264_121_411} \captionsetup{labelformat=empty} \caption{Figure 1}

\end{figure} A water container is made in the shape of a hollow inverted right circular cone with semi-vertical angle of $30 ^ { \circ }$, as shown in Figure 1. The height of the container is 50 cm . When the depth of the water in the container is $h \mathrm {~cm}$, the surface of the water has radius $r \mathrm {~cm}$ and the volume of water is $V \mathrm {~cm} ^ { 3 }$.

Show that $V = \frac { 1 } { 9 } \pi h ^ { 3 }$
[0pt] [You may assume the formula $V = \frac { 1 } { 3 } \pi r ^ { 2 } h$ for the volume of a cone.] Given that the volume of water in the container increases at a constant rate of $200 \mathrm {~cm} ^ { 3 } \mathrm {~s} ^ { - 1 }$,
find the rate of change of the depth of the water, in $\mathrm { cm } \mathrm { s } ^ { - 1 }$, when $h = 15$ Give your answer in its simplest form in terms of $\pi$.

Edexcel C4 2018 June Q5

5. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{0c4a3759-ecaa-47c3-a071-ce25fd11159f-16_938_1257_125_486} \captionsetup{labelformat=empty} \caption{Figure 2}

\end{figure} Figure 2 shows a sketch of the curve $C$ with parametric equations $$x = 1 + t - 5 \sin t , \quad y = 2 - 4 \cos t , \quad - \pi \leqslant t \leqslant \pi$$ The point $A$ lies on the curve $C$. Given that the coordinates of $A$ are ( $k , 2$ ), where $k > 0$

find the exact value of $k$, giving your answer in a fully simplified form.
Find the equation of the tangent to $C$ at the point $A$. Give your answer in the form $y = p x + q$, where $p$ and $q$ are exact real values.

Edexcel C4 2018 June Q6

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

$$\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { y ^ { 2 } } { 3 \cos ^ { 2 } 2 x } \quad - \frac { 1 } { 2 } < x < \frac { 1 } { 2 }$$ giving your answer in the form $y = \mathrm { f } ( x )$.

Edexcel C4 2018 June Q7

7. The point $A$ with coordinates ( $- 3,7,2$ ) lies on a line $l _ { 1 }$ The point $B$ also lies on the line $l _ { 1 }$ Given that $\quad \overrightarrow { A B } = \left( \begin{array} { r } 4
- 6
2 \end{array} \right)$,

find the coordinates of point $B$. The point $P$ has coordinates ( $9,1,8$ )
Find the cosine of the angle $P A B$, giving your answer as a simplified surd.
Find the exact area of triangle $P A B$, giving your answer in its simplest form. The line $l _ { 2 }$ passes through the point $P$ and is parallel to the line $l _ { 1 }$
Find a vector equation for the line $l _ { 2 }$ The point $Q$ lies on the line $l _ { 2 }$ Given that the line segment $A P$ is perpendicular to the line segment $B Q$,
find the coordinates of the point $Q$.

Edexcel C4 2018 June Q8

8. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{0c4a3759-ecaa-47c3-a071-ce25fd11159f-28_680_1266_118_482} \captionsetup{labelformat=empty} \caption{Figure 3}

\end{figure}

Find $\int x \cos 4 x d x$ Figure 3 shows part of the curve with equation $y = \sqrt { x } \sin 2 x , \quad x \geqslant 0$
The finite region $R$, shown shaded in Figure 3, is bounded by the curve, the $x$-axis and the line with equation $x = \frac { \pi } { 4 }$ The region $R$ is rotated through $2 \pi$ radians about the $x$-axis to form a solid of revolution.
Find the exact value of the volume of this solid of revolution, giving your answer in its simplest form.
(Solutions based entirely on graphical or numerical methods are not acceptable.) \includegraphics[max width=\textwidth, alt={}, center]{0c4a3759-ecaa-47c3-a071-ce25fd11159f-32_2630_1828_121_121}

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