Questions - Edexcel C4 - A-Level Maths

solve the differential equation and show that she made $\pounds 72$ in the first hour of the sale. The mathematician believes that is it not worth staying at the sale once she is making sales at a rate of less than $\pounds 10$ per hour.
Verify that at 3 hours and 5 minutes after the start of the sale, she should have already left.
4. continued

Edexcel C4 Q5

5. Relative to a fixed origin, two lines have the equations $$\begin{aligned} & \mathbf { r } = \left( \begin{array} { l } 4
1
1 \end{array} \right) + s \left( \begin{array} { l } 1
4
5 \end{array} \right)
& \mathbf { r } = \left( \begin{array} { c } - 3
1
- 6 \end{array} \right) + t \left( \begin{array} { l } 3
a
b \end{array} \right) \end{aligned}$$ and
where $a$ and $b$ are constants and $s$ and $t$ are scalar parameters.
Given that the two lines are perpendicular,

find a linear relationship between $a$ and $b$. Given also that the two lines intersect,
find the values of $a$ and $b$,
find the coordinates of the point where they intersect.
5. continued

Edexcel C4 Q6

6. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{a4ac7e65-267e-45c0-bbf2-2c38608eacc3-10_581_823_146_477} \captionsetup{labelformat=empty} \caption{Figure 1}

\end{figure} Figure 1 shows the curve with equation $y = x \sqrt { 1 - x } , 0 \leq x \leq 1$.

Use the substitution $u ^ { 2 } = 1 - x$ to show that the area of the region bounded by the curve and the $x$-axis is $\frac { 4 } { 15 }$.
Find, in terms of $\pi$, the volume of the solid formed when the region bounded by the curve and the $x$-axis is rotated through $360 ^ { \circ }$ about the $x$-axis.
6. continued

Edexcel C4 Q7

7. A curve has parametric equations $$x = 3 \cos ^ { 2 } t , \quad y = \sin 2 t , \quad 0 \leq t < \pi$$

Show that $\frac { \mathrm { d } y } { \mathrm {~d} x } = - \frac { 2 } { 3 } \cot 2 t$.
Find the coordinates of the points where the tangent to the curve is parallel to the $x$-axis.
Show that the tangent to the curve at the point where $t = \frac { \pi } { 6 }$ has the equation $$2 x + 3 \sqrt { 3 } y = 9$$
Find a cartesian equation for the curve in the form $y ^ { 2 } = \mathrm { f } ( x )$.
7. continued
7. continued

Edexcel C4 Q1

Find

$$\int \cot ^ { 2 } 2 x \mathrm {~d} x$$

Edexcel C4 Q2

A curve has the equation

$$4 \cos x + 2 \sin y = 3$$

Show that $\frac { \mathrm { d } y } { \mathrm {~d} x } = 2 \sin x \sec y$.
Find an equation for the tangent to the curve at the point $\left( \frac { \pi } { 3 } , \frac { \pi } { 6 } \right)$, giving your answer in the form $a x + b y = c$, where $a$ and $b$ are integers.

Edexcel C4 Q3

3. (a) Express $\frac { 2 + 20 x } { 1 + 2 x - 8 x ^ { 2 } }$ as a sum of partial fractions.
(b) Hence find the series expansion of $\frac { 2 + 20 x } { 1 + 2 x - 8 x ^ { 2 } } , | x | < \frac { 1 } { 4 }$, in ascending powers of $x$ up to and including the term in $x ^ { 3 }$, simplifying each coefficient.
3. continued

Edexcel C4 Q4

4. The line $l _ { 1 }$ passes through the points $P$ and $Q$ with position vectors ( $- \mathbf { i } - 8 \mathbf { j } + 3 \mathbf { k }$ ) and ( $2 \mathbf { i } - 9 \mathbf { j } + \mathbf { k }$ ) respectively, relative to a fixed origin.

Find a vector equation for $l _ { 1 }$. The line $l _ { 2 }$ has the equation $$\mathbf { r } = ( 6 \mathbf { i } + a \mathbf { j } + b \mathbf { k } ) + \mu ( \mathbf { i } + 4 \mathbf { j } - \mathbf { k } )$$ and also passes through the point $Q$.
Find the values of the constants $a$ and $b$.
Find, in degrees to 1 decimal place, the acute angle between lines $l _ { 1 }$ and $l _ { 2 }$.
4. continued

Edexcel C4 Q5

5. At time $t = 0$, a tank of height 2 metres is completely filled with water. Water then leaks from a hole in the side of the tank such that the depth of water in the tank, $y$ metres, after $t$ hours satisfies the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} t } = - k \mathrm { e } ^ { - 0.2 t }$$ where $k$ is a positive constant,

Find an expression for $y$ in terms of $k$ and $t$. Given that two hours after being filled the depth of water in the tank is 1.6 metres,
find the value of $k$ to 4 significant figures. Given also that the hole in the tank is $h \mathrm {~cm}$ above the base of the tank,
show that $h = 79$ to 2 significant figures.
5. continued

Edexcel C4 Q6

6. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{922f404e-12d5-490b-9c8d-509f3a304c1e-10_438_700_255_518} \captionsetup{labelformat=empty} \caption{Figure 1}

\end{figure} Figure 1 shows the curve with parametric equations $$x = 2 - t ^ { 2 } , \quad y = t ( t + 1 ) , \quad t \geq 0$$

Find the coordinates of the points where the curve meets the coordinate axes.
Find the exact area of the region bounded by the curve and the coordinate axes.
6. continued

Edexcel C4 Q7

7. (a) Prove that $$\frac { \mathrm { d } } { \mathrm {~d} x } \left( a ^ { x } \right) = a ^ { x } \ln a .$$ A curve has the equation $y = 4 ^ { x } - 2 ^ { x - 1 } + 1$.
(b) Show that the tangent to the curve at the point where it crosses the $y$-axis has the equation $$3 x \ln 2 - 2 y + 3 = 0 .$$ (c) Find the exact coordinates of the stationary point of the curve.
7. continued

Edexcel C4 Q8

8. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{922f404e-12d5-490b-9c8d-509f3a304c1e-14_656_999_146_429} \captionsetup{labelformat=empty} \caption{Figure 2}

\end{figure} Figure 2 shows the curve with equation $y = \sqrt { \frac { x } { x + 1 } }$.
The shaded region is bounded by the curve, the $x$-axis and the line $x = 3$.

1. Use the trapezium rule with three strips to find an estimate for the area of the shaded region.
2. Use the trapezium rule with six strips to find an improved estimate for the area of the shaded region. The shaded region is rotated through $2 \pi$ radians about the $x$-axis.
Show that the volume of the solid formed is $\pi ( 3 - \ln 4 )$.
8. continued
8. continued

Edexcel C4 Q1

A curve has the equation

$$2 x ^ { 2 } + x y - y ^ { 2 } + 18 = 0$$ Find the coordinates of the points where the tangent to the curve is parallel to the $x$-axis.

Edexcel C4 Q2

2. Use the substitution $x = 2 \tan u$ to show that $$\int _ { 0 } ^ { 2 } \frac { x ^ { 2 } } { x ^ { 2 } + 4 } \mathrm {~d} x = \frac { 1 } { 2 } ( 4 - \pi )$$

Edexcel C4 Q3

(a) Show that $\left( 1 \frac { 1 } { 24 } \right) ^ { - \frac { 1 } { 2 } } = k \sqrt { 6 }$, where $k$ is rational.
(b) Expand $\left( 1 + \frac { 1 } { 2 } x \right) ^ { - \frac { 1 } { 2 } } , | x | < 2$, in ascending powers of $x$ up to and including the term in $x ^ { 3 }$, simplifying each coefficient.
(c) Use your answer to part (b) with $x = \frac { 1 } { 12 }$ to find an approximate value for $\sqrt { 6 }$, giving your answer to 5 decimal places.
continued
Relative to a fixed origin, two lines have the equations

$$\mathbf { r } = ( 7 \mathbf { j } - 4 \mathbf { k } ) + s ( 4 \mathbf { i } - 3 \mathbf { j } + \mathbf { k } )$$ and $$\mathbf { r } = ( - 7 \mathbf { i } + \mathbf { j } + 8 \mathbf { k } ) + t ( - 3 \mathbf { i } + 2 \mathbf { k } )$$ where $s$ and $t$ are scalar parameters.
(a) Show that the two lines intersect and find the position vector of the point where they meet.
(b) Find, in degrees to 1 decimal place, the acute angle between the lines.

Edexcel C4 Q5

5. A curve has parametric equations $$x = \frac { t } { 2 - t } , \quad y = \frac { 1 } { 1 + t } , \quad - 1 < t < 2$$

Show that $\frac { \mathrm { d } y } { \mathrm {~d} x } = - \frac { 1 } { 2 } \left( \frac { 2 - t } { 1 + t } \right) ^ { 2 }$.
Find an equation for the normal to the curve at the point where $t = 1$.
Show that the cartesian equation of the curve can be written in the form $$y = \frac { 1 + x } { 1 + 3 x }$$
1. continued
2. (a) Find $\int \tan ^ { 2 } x d x$.
3. Show that
$$\int \tan x \mathrm {~d} x = \ln | \sec x | + c$$ where $c$ is an arbitrary constant. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{fe01157a-7617-43d3-900c-8d043bcbe784-10_566_789_648_504} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 shows part of the curve with equation $y = x ^ { \frac { 1 } { 2 } } \tan x$.
The shaded region bounded by the curve, the $x$-axis and the line $x = \frac { \pi } { 3 }$ is rotated through $2 \pi$ radians about the $x$-axis.
Show that the volume of the solid formed is $\frac { 1 } { 18 } \pi ^ { 2 } ( 6 \sqrt { 3 } - \pi ) - \pi \ln 2$.

Edexcel C4 Q7

7. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{fe01157a-7617-43d3-900c-8d043bcbe784-12_252_757_267_484} \captionsetup{labelformat=empty} \caption{Figure 2}

\end{figure} Figure 2 shows a hemispherical bowl of radius 5 cm .
The bowl is filled with water but the water leaks from a hole at the base of the bowl. At time $t$ minutes, the depth of water is $h \mathrm {~cm}$ and the volume of water in the bowl is $V \mathrm {~cm} ^ { 3 }$, where $$V = \frac { 1 } { 3 } \pi h ^ { 2 } ( 15 - h ) .$$ In a model it is assumed that the rate at which the volume of water in the bowl decreases is proportional to $V$.

Show that $$\frac { \mathrm { d } h } { \mathrm {~d} t } = - \frac { k h ( 15 - h ) } { 3 ( 10 - h ) } ,$$ where $k$ is a positive constant.
Express $\frac { 3 ( 10 - h ) } { h ( 15 - h ) }$ in partial fractions. Given that when $t = 0 , h = 5$,
show that $$h ^ { 2 } ( 15 - h ) = 250 \mathrm { e } ^ { - k t }$$ Given also that when $t = 2 , h = 4$,
find the value of $k$ to 3 significant figures.
7. continued
7. continued

Edexcel C4 Q1

A curve has the equation

$$x ^ { 2 } + 2 x y ^ { 2 } + y = 4$$ Find an expression for $\frac { \mathrm { d } y } { \mathrm {~d} x }$ in terms of $x$ and $y$.

Edexcel C4 Q2

2. Use integration by parts to find $$\int x ^ { 2 } \mathrm { e } ^ { - x } \mathrm {~d} x$$

Edexcel C4 Q3

The first four terms in the series expansion of $( 1 + a x ) ^ { n }$ in ascending powers of $x$ are

$$1 - 4 x + 24 x ^ { 2 } + k x ^ { 3 }$$ where $a , n$ and $k$ are constants and $| a x | < 1$.

Find the values of $a$ and $n$.
Show that $k = - 160$.
3. continued

Edexcel C4 Q4

4. (a) Use the trapezium rule with two intervals of equal width to find an estimate for the value of the integral $$\int _ { 0 } ^ { 3 } e ^ { \cos x } d x$$ giving your answer to 3 significant figures.
(b) Use the trapezium rule with four intervals of equal width to find another estimate for the value of the integral to 3 significant figures.
(c) Given that the true value of the integral lies between the estimates made in parts (a) and (b), comment on the shape of the curve $y = \mathrm { e } ^ { \cos x }$ in the interval $0 \leq x \leq 3$ and explain your answer.
4. continued

Edexcel C4 Q5

5. A straight road passes through villages at the points $A$ and $B$ with position vectors ( $9 \mathbf { i } - 8 \mathbf { j } + 2 \mathbf { k }$ ) and ( $4 \mathbf { j } + \mathbf { k }$ ) respectively, relative to a fixed origin. The road ends at a junction at the point $C$ with another straight road which lies along the line with equation $$\mathbf { r } = ( 2 \mathbf { i } + 16 \mathbf { j } - \mathbf { k } ) + \mu ( - 5 \mathbf { i } + 3 \mathbf { j } ) ,$$ where $\mu$ is a scalar parameter.

Find the position vector of $C$. Given that 1 unit on each coordinate axis represents 200 metres,
find the distance, in kilometres, from the village at $A$ to the junction at $C$.
5. continued

Edexcel C4 Q6

6. A small town had a population of 9000 in the year 2001. In a model, it is assumed that the population of the town, $P$, at time $t$ years after 2001 satisfies the differential equation $$\frac { \mathrm { d } P } { \mathrm {~d} t } = 0.05 P \mathrm { e } ^ { - 0.05 t }$$

Show that, according to the model, the population of the town in 2011 will be 13300 to 3 significant figures.
Find the value which the population of the town will approach in the long term, according to the model.
6. continued

Edexcel C4 Q7

7. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{4232f6a1-00ff-4e88-b5f4-1abf3d4742c4-12_560_911_146_456} \captionsetup{labelformat=empty} \caption{Figure 1}

\end{figure} Figure 1 shows the curve with parametric equations $$x = t ^ { 3 } + 1 , \quad y = \frac { 2 } { t } , \quad t > 0 .$$ The shaded region is bounded by the curve, the $x$-axis and the lines $x = 2$ and $x = 9$.

Find the area of the shaded region.
Show that the volume of the solid formed when the shaded region is rotated through $2 \pi$ radians about the $x$-axis is $12 \pi$.
Find a cartesian equation for the curve in the form $y = \mathrm { f } ( x )$.
7. continued

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