Questions - Edexcel C4 - A-Level Maths

find the position vector of their point of intersection,
find the value of $a$. Given also that $\theta$ is the acute angle between the lines,
find the value of $\cos \theta$ in the form $k \sqrt { 5 }$ where $k$ is rational.
4. continued
5. A curve has the equation $$x ^ { 2 } - 4 x y + 2 y ^ { 2 } = 1$$
Find an expression for $\frac { \mathrm { d } y } { \mathrm {~d} x }$ in its simplest form in terms of $x$ and $y$.
Show that the tangent to the curve at the point $P ( 1,2 )$ has the equation $$3 x - 2 y + 1 = 0$$ The tangent to the curve at the point $Q$ is parallel to the tangent at $P$.
Find the coordinates of $Q$.
5. continued
6. The rate of increase in the number of bacteria in a culture, $N$, at time $t$ hours is proportional to $N$.
Write down a differential equation connecting $N$ and $t$. Given that initially there are $N _ { 0 }$ bacteria present in a culture,
Show that $N = N _ { 0 } \mathrm { e } ^ { k t }$, where $k$ is a positive constant. Given also that the number of bacteria present doubles every six hours,
find the value of $k$,
find how long it takes for the number of bacteria to increase by a factor of ten, giving your answer to the nearest minute. of ten, giving your answer to the nearest minute.
6. continued
7. A curve has parametric equations $$x = \sec \theta + \tan \theta , \quad y = \operatorname { cosec } \theta + \cot \theta , \quad 0 < \theta < \frac { \pi } { 2 } .$$
Show that $x + \frac { 1 } { x } = 2 \sec \theta$. Given that $y + \frac { 1 } { y } = 2 \operatorname { cosec } \theta$,
find a cartesian equation for the curve.
Show that $\frac { \mathrm { d } x } { \mathrm {~d} \theta } = \frac { 1 } { 2 } \left( x ^ { 2 } + 1 \right)$.
Find an expression for $\frac { \mathrm { d } y } { \mathrm {~d} x }$ in terms of $x$ and $y$.
7. continued
7. continued

Edexcel C4 Q1

The number of people, $n$, in a queue at a Post Office $t$ minutes after it opens is modelled by the differential equation

$$\frac { \mathrm { d } n } { \mathrm {~d} t } = \mathrm { e } ^ { 0.5 t } - 5 , \quad t \geq 0$$

Find, to the nearest second, the time when the model predicts that there will be the least number of people in the queue.
Given that there are 20 people in the queue when the Post Office opens, solve the differential equation.
Explain why this model would not be appropriate for large values of $t$.

Edexcel C4 Q2

2. A curve has the equation $$3 x ^ { 2 } + x y - 2 y ^ { 2 } + 25 = 0$$ Find an equation for the normal to the curve at the point with coordinates $( 1,4 )$, giving your answer in the form $a x + b y + c = 0$, where $a , b$ and $c$ are integers.

Edexcel C4 Q3

3. (a) Use the substitution $u = 2 - x ^ { 2 }$ to find $$\int \frac { x } { 2 - x ^ { 2 } } \mathrm {~d} x$$ (b) Evaluate $$\int _ { 0 } ^ { \frac { \pi } { 4 } } \sin 3 x \cos x d x$$

continued

\begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{beeaedf6-62e8-4649-b023-1b7e2be9957e-06_636_837_146_511} \captionsetup{labelformat=empty} \caption{Figure 1}

\end{figure} Figure 1 shows the curve with equation $y = x \sqrt { \ln x } , x \geq 1$. The shaded region is bounded by the curve, the $x$-axis and the line $x = 3$.
(a) Using the trapezium rule with two intervals of equal width, estimate the area of the shaded region. The shaded region is rotated through $360 ^ { \circ }$ about the $x$-axis.
(b) Find the exact volume of the solid formed.

Edexcel C4 Q5

5. $$f ( x ) = \frac { 5 - 8 x } { ( 1 + 2 x ) ( 1 - x ) ^ { 2 } }$$

Express $\mathrm { f } ( x )$ in partial fractions.
Find the series expansion of $\mathrm { f } ( x )$ in ascending powers of $x$ up to and including the term in $x ^ { 3 }$, simplifying each coefficient.
State the set of values of $x$ for which your expansion is valid.
5. continued

Edexcel C4 Q6

6. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{beeaedf6-62e8-4649-b023-1b7e2be9957e-10_524_734_146_532} \captionsetup{labelformat=empty} \caption{Figure 2}

\end{figure} Figure 2 shows the curve with parametric equations $$x = t + \sin t , \quad y = \sin t , \quad 0 \leq t \leq \pi .$$

Find $\frac { \mathrm { d } y } { \mathrm {~d} x }$ in terms of $t$.
Find, in exact form, the coordinates of the point where the tangent to the curve is parallel to the $x$-axis.
Show that the region bounded by the curve and the $x$-axis has area 2 .
6. continued

Edexcel C4 Q7

7. The line $l _ { 1 }$ passes through the points $A$ and $B$ with position vectors ( $3 \mathbf { i } + 6 \mathbf { j } - 8 \mathbf { k }$ ) and ( $8 \mathbf { j } - 6 \mathbf { k }$ ) respectively, relative to a fixed origin.

Find a vector equation for $l _ { 1 }$. The line $l _ { 2 }$ has vector equation $$\mathbf { r } = ( - 2 \mathbf { i } + 10 \mathbf { j } + 6 \mathbf { k } ) + \mu ( 7 \mathbf { i } - 4 \mathbf { j } + 6 \mathbf { k } ) ,$$ where $\mu$ is a scalar parameter.
Show that lines $l _ { 1 }$ and $l _ { 2 }$ intersect.
Find the coordinates of the point where $l _ { 1 }$ and $l _ { 2 }$ intersect. The point $C$ lies on $l _ { 2 }$ and is such that $A C$ is perpendicular to $A B$.
Find the position vector of $C$.
7. continued
7. continued

Edexcel C4 2013 January Q5

Show that $A$ has coordinates $( 0,3 )$.
Find the $x$ coordinate of the point $B$.
Find an equation of the normal to $C$ at the point $A$. The region $R$, as shown shaded in Figure 2, is bounded by the curve $C$, the line $x = - 1$ and the $x$-axis.
Use integration to find the exact area of $R$.

Edexcel C4 Q4

Show that the volume of the solid formed is $\frac { 1 } { 4 } \pi ( \pi + 2 )$.
Find a cartesian equation for the curve.

Edexcel C4 Q1

Use the binomial theorem to expand

$$\sqrt { } ( 4 - 9 x ) , \quad | x | < \frac { 4 } { 9 } ,$$ in ascending powers of $x$, up to and including the term in $x ^ { 3 }$, simplifying each term.

Edexcel C4 Q3

3. (a) Express $\frac { 5 x + 3 } { ( 2 x - 3 ) ( x + 2 ) }$ in partial fractions.
(b) Hence find the exact value of $\int _ { 2 } ^ { 6 } \frac { 5 x + 3 } { ( 2 x - 3 ) ( x + 2 ) } \mathrm { d } x$, giving your answer as a single logarithm.

Edexcel C4 Q5

5. \begin{figure}[h]

\captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{5a77103e-73c1-4f32-af93-f2d5627e2290-02_467_783_317_303}

\end{figure} Figure 1 shows the graph of the curve with equation $$y = x \mathrm { e } ^ { 2 x } , \quad x \geq 0 .$$ The finite region $R$ bounded by the lines $x = 1$, the $x$-axis and the curve is shown shaded in Figure 1.

Use integration to find the exact value of the area for $R$.
Complete the table with the values of $y$ corresponding to $x = 0.4$ and 0.8 .
$x$ 0 0.2 0.4 0.6 0.8 1
$y = x \mathrm { e } ^ { 2 x }$ 0 0.29836 1.99207 7.38906
Use the trapezium rule with all the values in the table to find an approximate value for this area, giving your answer to 4 significant figures.

Edexcel C4 Q6

6. A curve has parametric equations $$x = 2 \cot t , \quad y = 2 \sin ^ { 2 } t , \quad 0 < t \leq \frac { \pi } { 2 } .$$

Find an expression for $\frac { \mathrm { d } y } { \mathrm {~d} x }$ in terms of the parameter $t$.
Find an equation of the tangent to the curve at the point where $t = \frac { \pi } { 4 }$.
Find a cartesian equation of the curve in the form $y = \mathrm { f } ( x )$. State the domain on which the curve is defined.

Edexcel C4 Q7

7. The line $l _ { 1 }$ has vector equation $$\mathbf { r } = \left( \begin{array} { l } 3
1
2 \end{array} \right) + \lambda \left( \begin{array} { r } 1
- 1
4 \end{array} \right)$$ and the line $l _ { 2 }$ has vector equation $$\mathbf { r } = \left( \begin{array} { r } 0
4
- 2 \end{array} \right) + \mu \left( \begin{array} { r } 1
- 1
0 \end{array} \right) ,$$ where $\lambda$ and $\mu$ are parameters.
The lines $l _ { 1 }$ and $l _ { 2 }$ intersect at the point $B$ and the acute angle between $l _ { 1 }$ and $l _ { 2 }$ is $\theta$.

Find the coordinates of $B$.
Find the value of $\cos \theta$, giving your answer as a simplified fraction. The point $A$, which lies on $l _ { 1 }$, has position vector $\mathbf { a } = 3 \mathbf { i } + \mathbf { j } + 2 \mathbf { k }$.
The point $C$, which lies on $l _ { 2 }$, has position vector $\mathbf { c } = 5 \mathbf { i } - \mathbf { j } - 2 \mathbf { k }$.
The point $D$ is such that $A B C D$ is a parallelogram.
Show that $| \overrightarrow { A B } | = | \overrightarrow { B C } |$.
Find the position vector of the point $D$.

Edexcel C4 Q8

8. Liquid is pouring into a container at a constant rate of $20 \mathrm {~cm} ^ { 3 } \mathrm {~s} ^ { - 1 }$ and is leaking out at a rate proportional to the volume of the liquid already in the container.

Explain why, at time $t$ seconds, the volume, $V \mathrm {~cm} ^ { 3 }$, of liquid in the container satisfies the differential equation $$\frac { \mathrm { d } V } { \mathrm {~d} t } = 20 - k V$$ where $k$ is a positive constant. The container is initially empty.
By solving the differential equation, show that $$V = A + B \mathrm { e } ^ { - k t }$$ giving the values of $A$ and $B$ in terms of $k$. Given also that $\frac { \mathrm { d } V } { \mathrm {~d} t } = 10$ when $t = 5$,
find the volume of liquid in the container at 10 s after the start. Materials required for examination
Mathematical Formulae (Green) Items included with question papers
Nil Paper Reference(s)
6666 \section*{Advanced Level} \section*{Monday 23 January 2006 - Afternoon} Candidates may use any calculator EXCEPT those with the facility for symbolic algebra, differentiation and/or integration. Thus candidates may NOT use calculators such as the Texas Instruments TI 89, TI 92, Casio CFX 9970G, Hewlett Packard HP 48G.

Edexcel C4 Q1

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 $M$. \begin{figure}[h]

\captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{615ec68b-3a32-4309-bb54-acf39ed09f96-01_738_1088_1316_380}

\end{figure} Figure 1 shows part of a curve $C$ with equation $y = x ^ { 2 } + 3$. The shaded region is bounded by $C$, the $x$-axis and the lines $x = 1$ and $x = 3$. The shaded region is rotated $360 ^ { \circ }$ about the $x$-axis. Using calculus, calculate the volume of the solid generated. Give your answer as an exact multiple of $\pi$.

Edexcel C4 Q3

3. (a) Given that $a ^ { x } = \mathrm { e } ^ { k x }$, where $a$ and $k$ are constants, $a > 0$ and $x \in \mathbb { R }$, prove that $k = \ln a$.
(b) Hence, using the derivative of $\mathrm { e } ^ { k x }$, prove that when $y = 2 ^ { x }$, $$\frac { \mathrm { d } y } { \mathrm { dx } } = 2 ^ { x } \ln 2 .$$ (c) Hence deduce that the gradient of the curve with equation $y = 2 ^ { x }$ at the point $( 2,4 )$ is $\ln 16$.

Edexcel C4 Q4

4. $$\mathrm { f } ( x ) = ( 1 + 3 x ) ^ { - 1 } , | x | < \frac { 1 } { 3 }$$

Expand $\mathrm { f } ( x )$ in ascending powers of $x$ up to and including the term in $x ^ { 3 }$.
Hence show that, for small $x$, $$\frac { 1 + x } { 1 + 3 x } \approx 1 - 2 x + 6 x ^ { 2 } - 18 x ^ { 3 }$$
Taking a suitable value for $x$, which should be stated, use the series expansion in part (b) to find an approximate value for $\frac { 101 } { 103 }$, giving your answer to 5 decimal places.

Edexcel C4 Q5

5. (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]{615ec68b-3a32-4309-bb54-acf39ed09f96-03_793_1138_524_354}

\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 $x = \frac { \pi } { 4 }$.

Edexcel C4 Q6

6. 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 $$\begin{aligned} \mathbf { r } & = 3 \mathbf { i } + 4 \mathbf { j } - 5 \mathbf { k } + \lambda ( \mathbf { i } - 2 \mathbf { j } + 2 \mathbf { k } )
\text { and } \mathbf { r } & = 9 \mathbf { i } + \mathbf { j } - 2 \mathbf { k } + \mu ( 4 \mathbf { i } + \mathbf { j } - \mathbf { k } ) , \end{aligned}$$ 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 Q7

7. 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. \section*{8.} \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 2} \includegraphics[alt={},max width=\textwidth]{615ec68b-3a32-4309-bb54-acf39ed09f96-05_716_1026_326_468}
\end{figure} Part of the design of a stained glass window is shown in Fig. 2. The two loops enclose an area of blue glass. The remaining area within the rectangle $A B C D$ is red glass. The loops are described by the curve with parametric equations $$x = 3 \cos t , \quad y = 9 \sin 2 t , \quad 0 \leq t < 2 \pi .$$
Find the cartesian equation of the curve in the form $y ^ { 2 } = \mathrm { f } ( x )$.
Show that the shaded area in Fig. 2, enclosed by the curve and the $x$-axis, is given by $$\int _ { 0 } ^ { \frac { \pi } { 2 } } A \sin 2 t \sin t \mathrm {~d} t , \text { stating the value of the constant } A \text {. }$$
Find the value of this integral. The sides of the rectangle $A B C D$, in Fig. 2, are the tangents to the curve that are parallel to the coordinate axes. Given that 1 unit on each axis represents 1 cm ,
find the total area of the red glass.

Edexcel C4 Q9

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

Edexcel C4 Q10

10. \begin{figure}[h]

\captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{615ec68b-3a32-4309-bb54-acf39ed09f96-07_624_828_335_584}

\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.
[0pt] [P2 January 2002 Question 4]

\(x\)	0	0.2	0.4	0.6	0.8	1
\(y = x \mathrm { e } ^ { 2 x }\)	0	0.29836		1.99207		7.38906

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