Questions - Edexcel - A-Level Maths

Edexcel C4 2013 June Q2

7 marks Standard +0.3

2. The curve $C$ has equation $$3 ^ { x - 1 } + x y - y ^ { 2 } + 5 = 0$$ Show that $\frac { \mathrm { d } y } { \mathrm {~d} x }$ at the point $( 1,3 )$ on the curve $C$ can be written in the form $\frac { 1 } { \lambda } \ln \left( \mu \mathrm { e } ^ { 3 } \right)$, where $\lambda$ and $\mu$ are integers to be found.

Edexcel C4 2013 June Q3

8 marks Standard +0.8

3. Using the substitution $u = 2 + \sqrt { } ( 2 x + 1 )$, or other suitable substitutions, find the exact value of $$\int _ { 0 } ^ { 4 } \frac { 1 } { 2 + \sqrt { } ( 2 x + 1 ) } d x$$ giving your answer in the form $A + 2 \ln B$, where $A$ is an integer and $B$ is a positive constant.

Edexcel C4 2013 June Q4

9 marks Standard +0.3

4. (a) Find the binomial expansion of $$\sqrt [ 3 ] { ( 8 - 9 x ) , \quad } \quad | x | < \frac { 8 } { 9 }$$ in ascending powers of $x$, up to and including the term in $x ^ { 3 }$. Give each coefficient as a simplified fraction.
(b) Use your expansion to estimate an approximate value for $\sqrt [ 3 ] { 7100 }$, giving your answer to 4 decimal places. State the value of $x$, which you use in your expansion, and show all your working.

Edexcel C4 2013 June Q5

11 marks Moderate -0.3

5. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{08f62966-2e63-4542-a10a-c6453a3215e7-06_689_992_118_484} \captionsetup{labelformat=empty} \caption{Figure 1}

\end{figure} Figure 1 shows part of the curve with equation $x = 4 t \mathrm { e } ^ { - \frac { 1 } { 3 } t } + 3$. The finite region $R$ shown shaded in Figure 1 is bounded by the curve, the $x$-axis, the $t$-axis and the line $t = 8$.

Complete the table with the value of $x$ corresponding to $t = 6$, giving your answer to 3 decimal places.
$t$ 0 2 4 6 8
$x$ 3 7.107 7.218 5.223
Use the trapezium rule with all the values of $x$ in the completed table to obtain an estimate for the area of the region $R$, giving your answer to 2 decimal places.
Use calculus to find the exact value for the area of $R$.
Find the difference between the values obtained in part (b) and part (c), giving your answer to 2 decimal places.

Edexcel C4 2013 June Q6

12 marks Standard +0.3

6. Relative to a fixed origin $O$, the point $A$ has position vector $21 \mathbf { i } - 17 \mathbf { j } + 6 \mathbf { k }$ and the point $B$ has position vector $25 \mathbf { i } - 14 \mathbf { j } + 18 \mathbf { k }$. The line $l$ has vector equation $$\mathbf { r } = \left( \begin{array} { r } a
b

Edexcel C4 2013 June Q7

12 marks

7. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{08f62966-2e63-4542-a10a-c6453a3215e7-10_542_1164_251_477} \captionsetup{labelformat=empty} \caption{Figure 2}

\end{figure} Figure 2 shows a sketch of the curve $C$ with parametric equations $$x = 27 \sec ^ { 3 } t , y = 3 \tan t , \quad 0 \leqslant t \leqslant \frac { \pi } { 3 }$$

Find the gradient of the curve $C$ at the point where $t = \frac { \pi } { 6 }$
Show that the cartesian equation of $C$ may be written in the form $$y = \left( x ^ { \frac { 2 } { 3 } } - 9 \right) ^ { \frac { 1 } { 2 } } , \quad a \leqslant x \leqslant b$$ stating the values of $a$ and $b$.
(3) \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{08f62966-2e63-4542-a10a-c6453a3215e7-10_581_1173_1628_475} \captionsetup{labelformat=empty} \caption{Figure 3}
\end{figure} The finite region $R$ which is bounded by the curve $C$, the $x$-axis and the line $x = 125$ is shown shaded in Figure 3. This region is rotated through $2 \pi$ radians about the $x$-axis to form a solid of revolution.
Use calculus to find the exact value of the volume of the solid of revolution.

Edexcel C4 2013 June Q10

Challenging +1.2

10 \end{array} \right) + \lambda \left( \begin{array} { r } 6
c
- 1 \end{array} \right)$$ where $a , b$ and $c$ are constants and $\lambda$ is a parameter.\\ Given that the point $A$ lies on the line $l$,

find the value of $a$. Given also that the vector $\overrightarrow { A B }$ is perpendicular to $l$,
find the values of $b$ and $c$,
find the distance $A B$. The image of the point $B$ after reflection in the line $l$ is the point $B ^ { \prime }$.
Find the position vector of the point $B ^ { \prime }$.\\ 7. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{08f62966-2e63-4542-a10a-c6453a3215e7-10_542_1164_251_477} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} Figure 2 shows a sketch of the curve $C$ with parametric equations $$x = 27 \sec ^ { 3 } t , y = 3 \tan t , \quad 0 \leqslant t \leqslant \frac { \pi } { 3 }$$
Find the gradient of the curve $C$ at the point where $t = \frac { \pi } { 6 }$
Show that the cartesian equation of $C$ may be written in the form $$y = \left( x ^ { \frac { 2 } { 3 } } - 9 \right) ^ { \frac { 1 } { 2 } } , \quad a \leqslant x \leqslant b$$ stating the values of $a$ and $b$.\\ (3) \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{08f62966-2e63-4542-a10a-c6453a3215e7-10_581_1173_1628_475} \captionsetup{labelformat=empty} \caption{Figure 3}
\end{figure} The finite region $R$ which is bounded by the curve $C$, the $x$-axis and the line $x = 125$ is shown shaded in Figure 3. This region is rotated through $2 \pi$ radians about the $x$-axis to form a solid of revolution.
Use calculus to find the exact value of the volume of the solid of revolution. 8. In an experiment testing solid rocket fuel, some fuel is burned and the waste products are collected. Throughout the experiment the sum of the masses of the unburned fuel and waste products remains constant. Let $x$ be the mass of waste products, in kg , at time $t$ minutes after the start of the experiment. It is known that at time $t$ minutes, the rate of increase of the mass of waste products, in kg per minute, is $k$ times the mass of unburned fuel remaining, where $k$ is a positive constant. The differential equation connecting $x$ and $t$ may be written in the form $$\frac { \mathrm { d } x } { \mathrm {~d} t } = k ( M - x ) , \text { where } M \text { is a constant. }$$
Explain, in the context of the problem, what $\frac { \mathrm { d } x } { \mathrm {~d} t }$ and $M$ represent. Given that initially the mass of waste products is zero,
solve the differential equation, expressing $x$ in terms of $k , M$ and $t$. Given also that $x = \frac { 1 } { 2 } M$ when $t = \ln 4$,
find the value of $x$ when $t = \ln 9$, expressing $x$ in terms of $M$, in its simplest form.

Edexcel C4 2013 June Q1

8 marks Standard +0.3

(a) Find the binomial expansion of

$$\sqrt { } ( 9 + 8 x ) , \quad | x | < \frac { 9 } { 8 }$$ in ascending powers of $x$, up to and including the term in $x ^ { 2 }$.
Give each coefficient as a simplified fraction.
(b) Use your expansion to estimate the value of $\sqrt { } ( 11 )$, giving your answer as a single fraction.

Edexcel C4 2013 June Q2

11 marks Moderate -0.3

2. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{b0d21f0f-f5f6-4ca5-8e3e-98aee0d9db7a-03_735_1171_360_490} \captionsetup{labelformat=empty} \caption{Figure 1}

\end{figure} Figure 1 shows a sketch of part of the curve with equation $y = x \mathrm { e } ^ { - \frac { 1 } { 2 } x } , x \geqslant 0$.
The finite region $R$, shown shaded in Figure 1, is bounded by the curve, the $x$-axis, and the line $x = 4$. The table shows corresponding values of $x$ and $y$ for $y = x e ^ { - \frac { 1 } { 2 } x }$.

$x$	0	1	2	3	4
$y$	0	$\mathrm { e } ^ { - \frac { 1 } { 2 } }$		$3 \mathrm { e } ^ { - \frac { 3 } { 2 } }$	$4 \mathrm { e } ^ { - 2 }$

Complete the table with the value of $y$ corresponding to $x = 2$
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 2 decimal places.
1. Find $\int x \mathrm { e } ^ { - \frac { 1 } { 2 } x } \mathrm {~d} x$.
2. Hence find the exact area of $R$, giving your answer in the form $a + b \mathrm { e } ^ { - 2 }$, where $a$ and $b$ are integers.

Edexcel C4 2013 June Q3

7 marks Moderate -0.3

A curve $C$ has parametric equations

$$x = 2 t + 5 , \quad y = 3 + \frac { 4 } { t } , \quad t \neq 0$$

Find the value of $\frac { \mathrm { d } y } { \mathrm {~d} x }$ at the point on $C$ with coordinates $( 9,5 )$.
Find a cartesian equation of the curve in the form $$y = \frac { a x + b } { c x + d }$$ where $a$, $b$, $c$ and $d$ are integers.

Edexcel C4 2013 June Q4

10 marks Moderate -0.3

4. With respect to a fixed origin $O$, the line $l _ { 1 }$ has vector equation $$\mathbf { r } = \left( \begin{array} { r } - 9

Edexcel C4 2013 June Q5

9 marks Standard +0.3

5 \end{array} \right) + \mu \left( \begin{array} { r } 5
- 4
- 3 \end{array} \right)$$ where $\mu$ is a scalar parameter.\\ The point $A$ is on $l _ { 1 }$ where $\mu = 2$.

Write down the coordinates of $A$. The acute angle between $O A$ and $l _ { 1 }$ is $\theta$, where $O$ is the origin.
Find the value of $\cos \theta$. The point $B$ is such that $\overrightarrow { O B } = 3 \overrightarrow { O A }$.\\ The line $l _ { 2 }$ passes through the point $B$ and is parallel to the line $l _ { 1 }$.
Find a vector equation of $l _ { 2 }$.
Find the length of $O B$, giving your answer as a simplified surd. The point $X$ lies on $l _ { 2 }$. Given that the vector $\overrightarrow { O X }$ is perpendicular to $l _ { 2 }$,
find the length of $O X$, giving your answer to 3 significant figures.\\ 5. The curve $C$ has the equation $$\sin ( \pi y ) - y - x ^ { 2 } y = - 5 , \quad x > 0$$
Find $\frac { \mathrm { d } y } { \mathrm {~d} x }$ in terms of $x$ and $y$. The point $P$ with coordinates $( 2,1 )$ lies on $C$.
The tangent to $C$ at $P$ meets the $x$-axis at the point $A$.
Find the exact value of the $x$-coordinate of $A$.

Edexcel C4 2013 June Q6

11 marks Standard +0.3

6. (i) (a) Express $\frac { 7 x } { ( x + 3 ) ( 2 x - 1 ) }$ in partial fractions.
(b) Given that $x > \frac { 1 } { 2 }$, find $$\int \frac { 7 x } { ( x + 3 ) ( 2 x - 1 ) } d x$$ (ii) Using the substitution $u ^ { 3 } = x$, or otherwise, find $$\int \frac { 1 } { x + x ^ { \frac { 1 } { 3 } } } d x , \quad x > 0$$

Edexcel C4 2013 June Q8

9 marks Standard +0.3

8
5 \end{array} \right) + \mu \left( \begin{array} { r } 5
- 4
- 3 \end{array} \right)$$ where $\mu$ is a scalar parameter.\\ The point $A$ is on $l _ { 1 }$ where $\mu = 2$.

Write down the coordinates of $A$. The acute angle between $O A$ and $l _ { 1 }$ is $\theta$, where $O$ is the origin.
Find the value of $\cos \theta$. The point $B$ is such that $\overrightarrow { O B } = 3 \overrightarrow { O A }$.\\ The line $l _ { 2 }$ passes through the point $B$ and is parallel to the line $l _ { 1 }$.
Find a vector equation of $l _ { 2 }$.
Find the length of $O B$, giving your answer as a simplified surd. The point $X$ lies on $l _ { 2 }$. Given that the vector $\overrightarrow { O X }$ is perpendicular to $l _ { 2 }$,
find the length of $O X$, giving your answer to 3 significant figures.\\ 5. The curve $C$ has the equation $$\sin ( \pi y ) - y - x ^ { 2 } y = - 5 , \quad x > 0$$
Find $\frac { \mathrm { d } y } { \mathrm {~d} x }$ in terms of $x$ and $y$. The point $P$ with coordinates $( 2,1 )$ lies on $C$.\\ The tangent to $C$ at $P$ meets the $x$-axis at the point $A$.
Find the exact value of the $x$-coordinate of $A$.\\ 6. (i) (a) Express $\frac { 7 x } { ( x + 3 ) ( 2 x - 1 ) }$ in partial fractions.
Given that $x > \frac { 1 } { 2 }$, find $$\int \frac { 7 x } { ( x + 3 ) ( 2 x - 1 ) } d x$$ (ii) Using the substitution $u ^ { 3 } = x$, or otherwise, find $$\int \frac { 1 } { x + x ^ { \frac { 1 } { 3 } } } d x , \quad x > 0$$ 7. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{b0d21f0f-f5f6-4ca5-8e3e-98aee0d9db7a-11_703_1164_373_492} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} Figure 2 shows a sketch of part of the curve $C$ with parametric equations $$x = \tan \theta , \quad y = 1 + 2 \cos 2 \theta , \quad 0 \leqslant \theta < \frac { \pi } { 2 }$$ The curve $C$ crosses the $x$-axis at $( \sqrt { } 3,0 )$. The finite shaded region $S$ shown in Figure 2 is bounded by $C$, the line $x = 1$ and the $x$-axis. This shaded region is rotated through $2 \pi$ radians about the $x$-axis to form a solid of revolution.
Show that the volume of the solid of revolution formed is given by the integral $$k \int _ { \frac { \pi } { 4 } } ^ { \frac { \pi } { 3 } } \left( 16 \cos ^ { 2 } \theta - 8 + \sec ^ { 2 } \theta \right) d \theta$$ where $k$ is a constant.
Hence, use integration to find the exact value for this volume.\\ 8. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{b0d21f0f-f5f6-4ca5-8e3e-98aee0d9db7a-13_869_545_312_811} \captionsetup{labelformat=empty} \caption{Figure 3}
\end{figure} Figure 3 shows a large vertical cylindrical tank containing a liquid. The radius of the circular cross-section of the tank is 40 cm . At time $t$ minutes, the depth of liquid in the tank is $h$ centimetres. The liquid leaks from a hole $P$ at the bottom of the tank. The liquid leaks from the tank at a rate of $32 \pi \sqrt { } h \mathrm {~cm} ^ { 3 } \mathrm {~min} ^ { - 1 }$.
Show that at time $t$ minutes, the height $h \mathrm {~cm}$ of liquid in the tank satisfies the differential equation $$\frac { \mathrm { d } h } { \mathrm {~d} t } = - 0.02 \sqrt { } h$$
Find the time taken, to the nearest minute, for the depth of liquid in the tank to decrease from 100 cm to 50 cm .

\includegraphics[max width=\textwidth, alt={}]{b0d21f0f-f5f6-4ca5-8e3e-98aee0d9db7a-14_2639_1834_214_217}

Edexcel C4 2013 June Q1

7 marks Standard +0.3

Find $\int x ^ { 2 } e ^ { x } d x$.
Hence find the exact value of $\int _ { 0 } ^ { 1 } x ^ { 2 } \mathrm { e } ^ { x } \mathrm {~d} x$.

Edexcel C4 2013 June Q3

8 marks Standard +0.2

3. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{5c9f77f0-9f7c-4125-9da7-20fb8d79b05e-04_814_882_258_539} \captionsetup{labelformat=empty} \caption{Figure 1}

\end{figure} Figure 1 shows the finite region $R$ bounded by the $x$-axis, the $y$-axis, the line $x = \frac { \pi } { 2 }$ and the curve with equation $$y = \sec \left( \frac { 1 } { 2 } x \right) , \quad 0 \leqslant x \leqslant \frac { \pi } { 2 }$$ The table shows corresponding values of $x$ and $y$ for $y = \sec \left( \frac { 1 } { 2 } x \right)$.

$x$	0	$\frac { \pi } { 6 }$	$\frac { \pi } { 3 }$	$\frac { \pi } { 2 }$
$y$	1	1.035276		1.414214

Complete the table above giving the missing value of $y$ to 6 decimal places.
Using the trapezium rule, with all of the values of $y$ from the completed table, find an approximation for the area of $R$, giving your answer to 4 decimal places. Region $R$ is rotated through $2 \pi$ radians about the $x$-axis.
Use calculus to find the exact volume of the solid formed.

Edexcel C4 2013 June Q4

9 marks Standard +0.3

A curve $C$ has parametric equations

$$x = 2 \sin t , \quad y = 1 - \cos 2 t , \quad - \frac { \pi } { 2 } \leqslant t \leqslant \frac { \pi } { 2 }$$

Find $\frac { \mathrm { d } y } { \mathrm {~d} x }$ at the point where $t = \frac { \pi } { 6 }$
Find a cartesian equation for $C$ in the form $$y = \mathrm { f } ( x ) , \quad - k \leqslant x \leqslant k$$ stating the value of the constant $k$.
Write down the range of $\mathrm { f } ( x )$.

Edexcel C4 2013 June Q5

10 marks Standard +0.3

(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

11 marks Standard +0.3

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

12 marks Standard +0.8

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

9 marks Standard +0.3

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

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

Edexcel C4 2013 June Q13

Standard +0.8

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

8 marks Standard +0.3

(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

9 marks Moderate -0.3

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

10 marks Standard +0.3

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 $\frac { \mathrm { d } y } { \mathrm {~d} x } = 0$

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

\(x\)	0	\(\frac { \pi } { 6 }\)	\(\frac { \pi } { 3 }\)	\(\frac { \pi } { 2 }\)
\(y\)	1	1.035276		1.414214

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\(t\)	0	2	4	6	8
\(x\)	3	7.107	7.218		5.223