Questions - Edexcel C4 - A-Level Maths

Edexcel C4 2013 June Q5

(a) Use the substitution $x = u ^ { 2 } , u > 0$, to show that

$$\int \frac { 1 } { x ( 2 \sqrt { x } - 1 ) } \mathrm { d } x = \int \frac { 2 } { u ( 2 u - 1 ) } \mathrm { d } u$$ (b) Hence show that $$\int _ { 1 } ^ { 9 } \frac { 1 } { x ( 2 \sqrt { x } - 1 ) } \mathrm { d } x = 2 \ln \left( \frac { a } { b } \right)$$ where $a$ and $b$ are integers to be determined.

Edexcel C4 2013 June Q6

6. Water is being heated in a kettle. At time $t$ seconds, the temperature of the water is $\theta ^ { \circ } \mathrm { C }$. The rate of increase of the temperature of the water at any time $t$ is modelled by the differential equation $$\frac { \mathrm { d } \theta } { \mathrm {~d} t } = \lambda ( 120 - \theta ) , \quad \theta \leqslant 100$$ where $\lambda$ is a positive constant. Given that $\theta = 20$ when $t = 0$,

solve this differential equation to show that $$\theta = 120 - 100 \mathrm { e } ^ { - \lambda t }$$ When the temperature of the water reaches $100 ^ { \circ } \mathrm { C }$, the kettle switches off.
Given that $\lambda = 0.01$, find the time, to the nearest second, when the kettle switches off.

Edexcel C4 2013 June Q7

7. A curve is described by the equation $$x ^ { 2 } + 4 x y + y ^ { 2 } + 27 = 0$$

Find $\frac { \mathrm { d } y } { \mathrm {~d} x }$ in terms of $x$ and $y$. A point $Q$ lies on the curve.
The tangent to the curve at $Q$ is parallel to the $y$-axis.
Given that the $x$ coordinate of $Q$ is negative,
use your answer to part (a) to find the coordinates of $Q$.

Edexcel C4 2013 June Q8

With respect to a fixed origin $O$, the line $l$ has equation

$$\mathbf { r } = \left( \begin{array} { c }

Edexcel C4 2013 June Q13

13
8
1 \end{array} \right) + \lambda \left( \begin{array} { r } 2
2
- 1 \end{array} \right) \text {, where } \lambda \text { is a scalar parameter. }$$ The point $A$ lies on $l$ and has coordinates ( $3 , - 2,6$ ).
The point $P$ has position vector ( $- p \mathbf { i } + 2 p \mathbf { k }$ ) relative to $O$, where $p$ is a constant.
Given that vector $\overrightarrow { P A }$ is perpendicular to $l$,

find the value of $p$. Given also that $B$ is a point on $l$ such that $\angle B P A = 45 ^ { \circ }$,
find the coordinates of the two possible positions of $B$.

Edexcel C4 2014 June Q1

(a) Find the binomial expansion of

$$\frac { 1 } { \sqrt { } ( 9 - 10 x ) } , \quad | x | < \frac { 9 } { 10 }$$ in ascending powers of $x$ up to and including the term in $x ^ { 2 }$.
Give each coefficient as a simplified fraction.
(b) Hence, or otherwise, find the expansion of $$\frac { 3 + x } { \sqrt { } ( 9 - 10 x ) } , \quad | x | < \frac { 9 } { 10 }$$ in ascending powers of $x$, up to and including the term in $x ^ { 2 }$.
Give each coefficient as a simplified fraction.

Edexcel C4 2014 June Q2

2. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{e14881c1-5ba5-4868-92ee-8bc58d4884dc-03_606_1070_251_445} \captionsetup{labelformat=empty} \caption{Figure 1}

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

$x$	0	0.5	1	1.5	2
$y$	2	4.077	7.389	10.043	0

Use the trapezium rule with all the values of $y$ in the table, to obtain an approximation for the area of $R$, giving your answer to 2 decimal places.
Explain how the trapezium rule can be used to give a more accurate approximation for the area of $R$.
Use calculus, showing each step in your working, to obtain an exact value for the area of $R$. Give your answer in its simplest form.

Edexcel C4 2014 June Q3

3. $$x ^ { 2 } + y ^ { 2 } + 10 x + 2 y - 4 x y = 10$$

Find $\frac { \mathrm { d } y } { \mathrm {~d} x }$ in terms of $x$ and $y$, fully simplifying your answer.
Find the values of $y$ for which $\frac { \mathrm { d } y } { \mathrm {~d} x } = 0$

Edexcel C4 2014 June Q4

4. (a) Express $\frac { 25 } { x ^ { 2 } ( 2 x + 1 ) }$ in partial fractions. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{e14881c1-5ba5-4868-92ee-8bc58d4884dc-06_623_849_408_561} \captionsetup{labelformat=empty} \caption{Figure 2}

\end{figure} Figure 2 shows a sketch of part of the curve $C$ with equation $y = \frac { 5 } { x \sqrt { } ( 2 x + 1 ) } , x > 0$
The finite region $R$ is bounded by the curve $C$, the $x$-axis, the line with equation $x = 1$ and the line with equation $x = 4$ This region is shown shaded in Figure 2 The region $R$ is rotated through $360 ^ { \circ }$ about the $x$-axis.
(b) Use calculus to find the exact volume of the solid of revolution generated, giving your answer in the form $a + b \ln c$, where $a , b$ and $c$ are constants.

Edexcel C4 2014 June Q5

5. At time $t$ seconds the radius of a sphere is $r \mathrm {~cm}$, its volume is $V \mathrm {~cm} ^ { 3 }$ and its surface area is $S \mathrm {~cm} ^ { 2 }$. [You are given that $V = \frac { 4 } { 3 } \pi r ^ { 3 }$ and that $S = 4 \pi r ^ { 2 }$ ] The volume of the sphere is increasing uniformly at a constant rate of $3 \mathrm {~cm} ^ { 3 } \mathrm {~s} ^ { - 1 }$.

Find $\frac { \mathrm { d } r } { \mathrm {~d} t }$ when the radius of the sphere is 4 cm , giving your answer to 3 significant figures.
Find the rate at which the surface area of the sphere is increasing when the radius is 4 cm .

Edexcel C4 2014 June Q6

6. With respect to a fixed origin, the point $A$ with position vector $\mathbf { i } + 2 \mathbf { j } + 3 \mathbf { k }$ lies on the line $l _ { 1 }$ with equation $$\mathbf { r } = \left( \begin{array} { l } 1
2
3 \end{array} \right) + \lambda \left( \begin{array} { r } 0
2
- 1 \end{array} \right) , \quad \text { where } \lambda \text { is a scalar parameter, }$$ and the point $B$ with position vector $4 \mathbf { i } + p \mathbf { j } + 3 \mathbf { k }$, where $p$ is a constant, lies on the line $l _ { 2 }$ with equation $$\mathbf { r } = \left( \begin{array} { l }

Edexcel C4 2014 June Q7

7
0
7 \end{array} \right) + \mu \left( \begin{array} { r } 3
- 5
4 \end{array} \right) , \quad \text { where } \mu \text { is a scalar parameter. }$$

Find the value of the constant $p$.
Show that $l _ { 1 }$ and $l _ { 2 }$ intersect and find the position vector of their point of intersection, $C$.
Find the size of the angle $A C B$, giving your answer in degrees to 3 significant figures.
Find the area of the triangle $A B C$, giving your answer to 3 significant figures.
7. The rate of increase of the number, $N$, of fish in a lake is modelled by the differential equation $$\frac { \mathrm { d } N } { \mathrm {~d} t } = \frac { ( k t - 1 ) ( 5000 - N ) } { t } \quad t > 0 , \quad 0 < N < 5000$$ In the given equation, the time $t$ is measured in years from the start of January 2000 and $k$ is a positive constant.
By solving the differential equation, show that $$N = 5000 - A t \mathrm { e } ^ { - k t }$$ where $A$ is a positive constant. After one year, at the start of January 2001, there are 1200 fish in the lake. After two years, at the start of January 2002, there are 1800 fish in the lake.
Find the exact value of the constant $A$ and the exact value of the constant $k$.
Hence find the number of fish in the lake after five years. Give your answer to the nearest hundred fish.

Edexcel C4 2014 June Q8

8. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{e14881c1-5ba5-4868-92ee-8bc58d4884dc-13_808_965_248_502} \captionsetup{labelformat=empty} \caption{Figure 3}

\end{figure} The curve shown in Figure 3 has parametric equations $$x = t - 4 \sin t , y = 1 - 2 \cos t , \quad - \frac { 2 \pi } { 3 } \leqslant t \leqslant \frac { 2 \pi } { 3 }$$ The point $A$, with coordinates ( $k , 1$ ), lies on the curve. Given that $k > 0$

find the exact value of $k$,
find the gradient of the curve at the point $A$. There is one point on the curve where the gradient is equal to $- \frac { 1 } { 2 }$
Find the value of $t$ at this point, showing each step in your working and giving your answer to 4 decimal places.
[0pt] [Solutions based entirely on graphical or numerical methods are not acceptable.]

Edexcel C4 2014 June Q1

A curve $C$ has the equation

$$x ^ { 3 } + 2 x y - x - y ^ { 3 } - 20 = 0$$

Find $\frac { \mathrm { d } y } { \mathrm {~d} x }$ in terms of $x$ and $y$.
Find an equation of the tangent to $C$ at the point $( 3 , - 2 )$, giving your answer in the form $a x + b y + c = 0$, where $a , b$ and $c$ are integers.

Edexcel C4 2014 June Q3

3
3
\end{tabular}}
\hline \multicolumn{2}{|l|}{\begin{tabular}{l}

Edexcel C4 2014 June Q4

4
4
\end{tabular}}
\hline 5 &
\hline \multicolumn{2}{|l|}{\begin{tabular}{l}

Edexcel C4 2014 June Q6

6
6
\end{tabular}}
\hline 7 &
\hline 8 &
\hline &
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\hline Total &
\hline \end{tabular} \end{center} Turn over

A curve $C$ has the equation

$$x ^ { 3 } + 2 x y - x - y ^ { 3 } - 20 = 0$$

Find $\frac { \mathrm { d } y } { \mathrm {~d} x }$ in terms of $x$ and $y$.
Find an equation of the tangent to $C$ at the point $( 3 , - 2 )$, giving your answer in the form $a x + b y + c = 0$, where $a , b$ and $c$ are integers.
2. Given that the binomial expansion of $( 1 + k x ) ^ { - 4 } , | k x | < 1$, is $$1 - 6 x + A x ^ { 2 } + \ldots$$
find the value of the constant $k$,
find the value of the constant $A$, giving your answer in its simplest form.
3. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{a9963b13-7db4-4a1d-8c75-829f4aade994-05_659_865_269_550} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 shows a sketch of part of the curve with equation $y = \frac { 10 } { 2 x + 5 \sqrt { } x } , x > 0$
The finite region $R$, shown shaded in Figure 1, is bounded by the curve, the $x$-axis, and the lines with equations $x = 1$ and $x = 4$ The table below shows corresponding values of $x$ and $y$ for $y = \frac { 10 } { 2 x + 5 \sqrt { } x }$
$x$ 1 2 3 4
$y$ 1.42857 0.90326 0.55556
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.
By reference to the curve in Figure 1, state, giving a reason, whether your estimate in part (b) is an overestimate or an underestimate for the area of $R$.
Use the substitution $u = \sqrt { } x$, or otherwise, to find the exact value of $$\int _ { 1 } ^ { 4 } \frac { 10 } { 2 x + 5 \sqrt { x } } d x$$ 4. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{a9963b13-7db4-4a1d-8c75-829f4aade994-07_618_703_246_625} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} A vase with a circular cross-section is shown in Figure 2. Water is flowing into the vase. When the depth of the water is $h \mathrm {~cm}$, the volume of water $V \mathrm {~cm} ^ { 3 }$ is given by $$V = 4 \pi h ( h + 4 ) , \quad 0 \leqslant h \leqslant 25$$ Water flows into the vase at a constant rate of $80 \pi \mathrm {~cm} ^ { 3 } \mathrm {~s} ^ { - 1 }$ Find the rate of change of the depth of the water, in $\mathrm { cms } ^ { - 1 }$, when $h = 6$
5. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{a9963b13-7db4-4a1d-8c75-829f4aade994-08_675_1262_267_340} \captionsetup{labelformat=empty} \caption{Figure 3}
\end{figure} Figure 3 shows a sketch of the curve $C$ with parametric equations $$x = 4 \cos \left( t + \frac { \pi } { 6 } \right) , \quad y = 2 \sin t , \quad 0 \leqslant t < 2 \pi$$
Show that $$x + y = 2 \sqrt { 3 } \cos t$$
Show that a cartesian equation of $C$ is $$( x + y ) ^ { 2 } + a y ^ { 2 } = b$$ where $a$ and $b$ are integers to be determined.
\includegraphics[max width=\textwidth, alt={}, center]{a9963b13-7db4-4a1d-8c75-829f4aade994-09_104_51_2617_1900}
6. (i) Find $$\int x \mathrm { e } ^ { 4 x } \mathrm {~d} x$$ (ii) Find $$\int \frac { 8 } { ( 2 x - 1 ) ^ { 3 } } \mathrm {~d} x , \quad x > \frac { 1 } { 2 }$$ (iii) Given that $y = \frac { \pi } { 6 }$ at $x = 0$, solve the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \mathrm { e } ^ { x } \operatorname { cosec } 2 y \operatorname { cosec } y$$

Edexcel C4 2014 June Q7

7. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{a9963b13-7db4-4a1d-8c75-829f4aade994-12_681_1203_258_376} \captionsetup{labelformat=empty} \caption{Figure 4}

\end{figure} Figure 4 shows a sketch of part of the curve $C$ with parametric equations $$x = 3 \tan \theta , \quad y = 4 \cos ^ { 2 } \theta , \quad 0 \leqslant \theta < \frac { \pi } { 2 }$$ The point $P$ lies on $C$ and has coordinates ( 3,2 ). The line $l$ is the normal to $C$ at $P$. The normal cuts the $x$-axis at the point $Q$.

Find the $x$ coordinate of the point $Q$. The finite region $S$, shown shaded in Figure 4, is bounded by the curve $C$, the $x$-axis, the $y$-axis and the line $l$. This shaded region is rotated $2 \pi$ radians about the $x$-axis to form a solid of revolution.
Find the exact value of the volume of the solid of revolution, giving your answer in the form $p \pi + q \pi ^ { 2 }$, where $p$ and $q$ are rational numbers to be determined.
[0pt] [You may use the formula $V = \frac { 1 } { 3 } \pi r ^ { 2 } h$ for the volume of a cone.]

Edexcel C4 2014 June Q8

8. Relative to a fixed origin $O$, the point $A$ has position vector $\left( \begin{array} { r } - 2
4
7 \end{array} \right)$ and the point $B$ has position vector $\left( \begin{array} { r } - 1
3
8 \end{array} \right)$ The line $l _ { 1 }$ passes through the points $A$ and $B$.

Find the vector $\overrightarrow { A B }$.
Hence find a vector equation for the line $l _ { 1 }$ The point $P$ has position vector $\left( \begin{array} { l } 0
2
3 \end{array} \right)$
Given that angle $P B A$ is $\theta$,
show that $\cos \theta = \frac { 1 } { 3 }$ 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 points $C$ and $D$ both lie on the line $l _ { 2 }$
Given that $A B = P C = D P$ and the $x$ coordinate of $C$ is positive,
find the coordinates of $C$ and the coordinates of $D$.
find the exact area of the trapezium $A B C D$, giving your answer as a simplified surd.

Edexcel C4 2015 June Q1

(a) Find the binomial expansion of

$$( 4 + 5 x ) ^ { \frac { 1 } { 2 } } , \quad | x | < \frac { 4 } { 5 }$$ in ascending powers of $x$, up to and including the term in $x ^ { 2 }$.
Give each coefficient in its simplest form.
(b) Find the exact value of $( 4 + 5 x ) ^ { \frac { 1 } { 2 } }$ when $x = \frac { 1 } { 10 }$ Give your answer in the form $k \sqrt { 2 }$, where $k$ is a constant to be determined.
(c) Substitute $x = \frac { 1 } { 10 }$ into your binomial expansion from part (a) and hence find an approximate value for $\sqrt { 2 }$ Give your answer in the form $\frac { p } { q }$ where $p$ and $q$ are integers.

Edexcel C4 2015 June Q2

2. The curve $C$ has equation $$x ^ { 2 } - 3 x y - 4 y ^ { 2 } + 64 = 0$$

Find $\frac { \mathrm { d } y } { \mathrm {~d} x }$ in terms of $x$ and $y$.
Find the coordinates of the points on $C$ where $\frac { \mathrm { d } y } { \mathrm {~d} x } = 0$
(Solutions based entirely on graphical or numerical methods are not acceptable.)

Edexcel C4 2015 June Q3

3. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{89d4a7a5-3f4f-4d16-b14e-a27243cedd78-05_620_867_301_536} \captionsetup{labelformat=empty} \caption{Figure 1}

\end{figure} Figure 1 shows a sketch of part of the curve with equation $y = 4 x - x \mathrm { e } ^ { \frac { 1 } { 2 } x } , x \geqslant 0$
The curve meets the $x$-axis at the origin $O$ and cuts the $x$-axis at the point $A$.

Find, in terms of $\ln 2$, the $x$ coordinate of the point $A$.
Find $$\int x \mathrm { e } ^ { \frac { 1 } { 2 } x } \mathrm {~d} x$$ The finite region $R$, shown shaded in Figure 1, is bounded by the $x$-axis and the curve with equation $$y = 4 x - x \mathrm { e } ^ { \frac { 1 } { 2 } x } , x \geqslant 0$$
Find, by integration, the exact value for the area of $R$. Give your answer in terms of $\ln 2$

Edexcel C4 2015 June Q4

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 }

Edexcel C4 2015 June Q5

5
- 3
p \end{array} \right) + \lambda \left( \begin{array} { r } 0
1
- 3 \end{array} \right) , \quad l _ { 2 } : \mathbf { r } = \left( \begin{array} { r }

Edexcel C4 2015 June Q6

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 $R$.

Questions — Edexcel C4 (360 questions)

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\(x\)	1	2	3	4
\(y\)	1.42857	0.90326		0.55556