Questions C4 - A-Level Maths

Edexcel C4 Q3

A curve has the equation

$$2 \sin 2 x - \tan y = 0$$

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 $\left( \frac { \pi } { 6 } , \frac { \pi } { 3 } \right)$ has the equation $$y = \frac { 1 } { 2 } x + \frac { \pi } { 4 }$$
1. continued
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{3cf64017-e982-4165-9885-8524aaabdf84-06_433_812_246_479} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 shows the curve with parametric equations $$x = a \sqrt { t } , \quad y = a t ( 1 - t ) , \quad t \geq 0$$ where $a$ is a positive constant.
Find $\frac { \mathrm { d } y } { \mathrm {~d} x }$ in terms of $t$. The curve meets the $x$-axis at the origin, $O$, and at the point $A$. The tangent to the curve at $A$ meets the $y$-axis at the point $B$ as shown.
Show that the area of triangle $O A B$ is $a ^ { 2 }$.

Edexcel C4 Q5

5. The gradient at any point $( x , y )$ on a curve is proportional to $\sqrt { y }$. Given that the curve passes through the point with coordinates $( 0,4 )$,

show that the equation of the curve can be written in the form $$2 \sqrt { y } = k x + 4$$ where $k$ is a positive constant. Given also that the curve passes through the point with coordinates ( 2,9 ),
find the equation of the curve in the form $y = \mathrm { f } ( x )$.
5. continued

Edexcel C4 Q6

6. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{3cf64017-e982-4165-9885-8524aaabdf84-10_456_553_264_571} \captionsetup{labelformat=empty} \caption{Figure 2}

\end{figure} Figure 2 shows a vertical cross-section of a vase.
The inside of the vase is in the shape of a right-circular cone with the angle between the sides in the cross-section being $60 ^ { \circ }$. When the depth of water in the vase is $h \mathrm {~cm}$, the volume of water in the vase is $V \mathrm {~cm} ^ { 3 }$.

Show that $V = \frac { 1 } { 9 } \pi h ^ { 3 }$. The vase is initially empty and water is poured in at a constant rate of $120 \mathrm {~cm} ^ { 3 } \mathrm {~s} ^ { - 1 }$.
Find, to 2 decimal places, the rate at which $h$ is increasing
1. when $h = 6$,
2. after water has been poured in for 8 seconds.
  6. continued

Edexcel C4 Q7

7. Relative to a fixed origin, the points $A$ and $B$ have position vectors $\left( \begin{array} { c } - 4
1
3 \end{array} \right)$ and $\left( \begin{array} { c } - 3
6
1 \end{array} \right)$ respectively.

Find a vector equation for the line $l _ { 1 }$ which passes through $A$ and $B$. The line $l _ { 2 }$ has vector equation $$\mathbf { r } = \left( \begin{array} { c } 3
- 7
9 \end{array} \right) + \mu \left( \begin{array} { c } 2
- 3
1 \end{array} \right)$$
Show that lines $l _ { 1 }$ and $l _ { 2 }$ do not intersect.
Find the position vector of the point $C$ on $l _ { 2 }$ such that $\angle A B C = 90 ^ { \circ }$.
7. continued

Edexcel C4 Q9

9 \end{array} \right) + \mu \left( \begin{array} { c } 2
- 3
1 \end{array} \right)$$ (b) Show that lines $l _ { 1 }$ and $l _ { 2 }$ do not intersect.
(c) Find the position vector of the point $C$ on $l _ { 2 }$ such that $\angle A B C = 90 ^ { \circ }$.
7. continued
8. $$f ( x ) = \frac { x ( 3 x - 7 ) } { ( 1 - x ) ( 1 - 3 x ) } , | x | < \frac { 1 } { 3 }$$ (a) Find the values of the constants $A , B$ and $C$ such that $$\mathrm { f } ( x ) = A + \frac { B } { 1 - x } + \frac { C } { 1 - 3 x }$$ (b) Evaluate $$\int _ { 0 } ^ { \frac { 1 } { 4 } } f ( x ) d x$$ giving your answer in the form $p + \ln q$, where $p$ and $q$ are rational.
(c) 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.
8. continued
8. continued

Edexcel C4 Q1

1. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{80bef9d4-b84c-4d3a-a093-67a466c6d1b9-02_615_791_146_532} \captionsetup{labelformat=empty} \caption{Figure 1}

\end{figure} Figure 1 shows the curve with equation $y = \frac { 3 x + 1 } { \sqrt { x } } , x > 0$.
The shaded region is bounded by the curve, the $x$-axis and the lines $x = 1$ and $x = 3$.
Find the volume of the solid formed when the shaded region is rotated through $2 \pi$ radians about the $x$-axis, giving your answer in the form $\pi ( a + \ln b )$, where $a$ and $b$ are integers.
□

Edexcel C4 Q2

2. (a) Expand $( 1 - 3 x ) ^ { - 2 } , | x | < \frac { 1 } { 3 }$, in ascending powers of $x$ up to and including the term in $x ^ { 3 }$, simplifying each coefficient.
(b) Hence, or otherwise, show that for small $x$, $$\left( \frac { 2 - x } { 1 - 3 x } \right) ^ { 2 } \approx 4 + 20 x + 85 x ^ { 2 } + 330 x ^ { 3 }$$

Edexcel C4 Q3

3. $$f ( x ) = \frac { 7 + 3 x + 2 x ^ { 2 } } { ( 1 - 2 x ) ( 1 + x ) ^ { 2 } } , \quad | x | > \frac { 1 } { 2 }$$

Express $\mathrm { f } ( x )$ in partial fractions.
Show that $$\int _ { 1 } ^ { 2 } \mathrm { f } ( x ) \mathrm { d } x = p - \ln q$$ where $p$ is rational and $q$ is an integer.
3. continued

Edexcel C4 Q4

4. Relative to a fixed origin, two lines have the equations $$\begin{aligned} & \mathbf { r } = \left( \begin{array} { c } 7
0
- 3 \end{array} \right) + \lambda \left( \begin{array} { c }

Edexcel C4 Q5

5
4
- 2 \end{array} \right)
& \mathbf { r } = \left( \begin{array} { l } a

Edexcel C4 Q7

7
0
- 3 \end{array} \right) + \lambda \left( \begin{array} { c } 5
4
- 2 \end{array} \right)
& \mathbf { r } = \left( \begin{array} { l } a
6
3 \end{array} \right) + \mu \left( \begin{array} { c } - 5

Edexcel C4 Q14

14
2 \end{array} \right) , \end{aligned}$$ and
where $a$ is a constant and $\lambda$ and $\mu$ are scalar parameters.
Given that the two lines intersect,

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 $R$.

AQA C4 2014 June Q7

1. Find an expression for $\frac { \mathrm { d } y } { \mathrm {~d} x }$.
2. Hence find the exact value of the gradient of the curve at $A$.
The normal at $A$ crosses the $y$-axis at the point $B$. Find the exact value of the $y$-coordinate of $B$.
[0pt] [2 marks]

AQA C4 2016 June Q7

Find the exact value of $\frac { \mathrm { d } y } { \mathrm {~d} x }$ at the point on $C$ where $t = \frac { 2 } { 3 }$.
Show that $x = \frac { 4 - \mathrm { e } ^ { 2 - 6 t } } { 4 }$ can be rearranged into the form $\mathrm { e } ^ { 3 t } = \frac { \mathrm { e } } { 2 \sqrt { ( 1 - x ) } }$.
Hence find the Cartesian equation of $C$, giving your answer in the form $$y = \frac { \mathrm { e } } { \mathrm { f } ( x ) [ 1 - \ln ( \mathrm { f } ( x ) ) ] }$$

OCR C4 2008 January Q7

Given that $$A ( \sin \theta + \cos \theta ) + B ( \cos \theta - \sin \theta ) \equiv 4 \sin \theta$$ find the values of the constants $A$ and $B$.
Hence find the exact value of $$\int _ { 0 } ^ { \frac { 1 } { 4 } \pi } \frac { 4 \sin \theta } { \sin \theta + \cos \theta } \mathrm { d } \theta$$ giving your answer in the form $a \pi - \ln b$.

OCR C4 2008 June Q7

Show that, if $y = \operatorname { cosec } x$, then $\frac { \mathrm { d } y } { \mathrm {~d} x }$ can be expressed as $- \operatorname { cosec } x \cot x$.
Solve the differential equation $$\frac { \mathrm { d } x } { \mathrm {~d} t } = - \sin x \tan x \cot t$$ given that $x = \frac { 1 } { 6 } \pi$ when $t = \frac { 1 } { 2 } \pi$.

OCR C4 2009 June Q7

The vector $\mathbf { u } = \frac { 3 } { 13 } \mathbf { i } + b \mathbf { j } + c \mathbf { k }$ is perpendicular to the vector $4 \mathbf { i } + \mathbf { k }$ and to the vector $4 \mathbf { i } + 3 \mathbf { j } + 2 \mathbf { k }$. Find the values of $b$ and $c$, and show that $\mathbf { u }$ is a unit vector.
Calculate, to the nearest degree, the angle between the vectors $4 \mathbf { i } + \mathbf { k }$ and $4 \mathbf { i } + 3 \mathbf { j } + 2 \mathbf { k }$.

OCR MEI C4 2010 June Q5

Verify that $\overrightarrow { \mathrm { AB } } = \left( \begin{array} { l } 300
100
100 \end{array} \right)$ and find the length of the pipeline.
Write down a vector equation of the line AB , and calculate the angle it makes with the vertical. A thin flat layer of hard rock runs through the mountain. The equation of the plane containing this layer is $x + 2 y + 3 z = 320$.
Find the coordinates of the point where the pipeline meets the layer of rock.
By calculating the angle between the line AB and the normal to the plane of the layer, find the angle at which the pipeline cuts through the layer. 8 Part of the track of a roller-coaster is modelled by a curve with the parametric equations $$x = 2 \theta - \sin \theta , \quad y = 4 \cos \theta \quad \text { for } 0 \leqslant \theta \leqslant 2 \pi$$ This is shown in Fig. 8. B is a minimum point, and BC is vertical. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{5c149cb5-7392-4219-b285-486f4694aa6f-4_602_1447_488_351} \captionsetup{labelformat=empty} \caption{Fig. 8}
\end{figure}
Find the values of the parameter at A and B . Hence show that the ratio of the lengths OA and AC is $( \pi - 1 ) : ( \pi + 1 )$.
Find $\frac { \mathrm { d } y } { \mathrm {~d} x }$ in terms of $\theta$. Find the gradient of the track at A .
Show that, when the gradient of the track is $1 , \theta$ satisfies the equation $$\cos \theta - 4 \sin \theta = 2 .$$
Express $\cos \theta - 4 \sin \theta$ in the form $R \cos ( \theta + \alpha )$. Hence solve the equation $\cos \theta - 4 \sin \theta = 2$ for $0 \leqslant \theta \leqslant 2 \pi$. {www.ocr.org.uk}) after the live examination series.
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Wednesday 9 June 2010 Afternoon
\includegraphics[max width=\textwidth, alt={}, center]{5c149cb5-7392-4219-b285-486f4694aa6f-5_264_456_881_1361} 1 The train journey from Swansea to London is 307 km and that by road is 300 km . Carry out the calculations performed on the First Great Western website to estimate how much lower the carbon dioxide emissions are when travelling by rail rather than road.
2 The equation of the curve in Fig. 3 is $$y = \frac { 1 } { 10 ^ { 4 } } \left( x ^ { 3 } - 100 x ^ { 2 } - 10000 x + 2100100 \right)$$ Calculate the speed at which the car has its lowest carbon dioxide emissions and the value of its emissions at that speed.
[0pt] [An answer obtained from the graph will be given no marks.]
3
In line 109 the carbon dioxide emissions for a particular train journey from Exeter to London are estimated to be 3.7 tonnes. Obtain this figure.
The text then goes on to state that the emissions per extra passenger on this journey are less than $\frac { 1 } { 2 } \mathrm {~kg}$. Justify this figure.
$\_\_\_\_$
$\_\_\_\_$
4 The daily number of trains, $n$, on a line in another country may be modelled by the function defined below, where $P$ is the annual number of passengers. $$\begin{aligned} & n = 10 \text { for } 0 \leqslant P < 10 ^ { 6 }
& n = 11 \text { for } 10 ^ { 6 } \leqslant P < 1.5 \times 10 ^ { 6 }
& n = 12 \text { for } 1.5 \times 10 ^ { 6 } \leqslant P < 2 \times 10 ^ { 6 }
& n = 13 \text { for } 2 \times 10 ^ { 6 } \leqslant P < 2.5 \times 10 ^ { 6 }
& n = 14 \text { for } 2.5 \times 10 ^ { 6 } \leqslant P < 3 \times 10 ^ { 6 }
& \ldots \text { and so on } \ldots \end{aligned}$$
Sketch the graph of $n$ against $P$.
Describe, in words, the relationship between the daily number of trains and the annual number of passengers.
\includegraphics[max width=\textwidth, alt={}, center]{5c149cb5-7392-4219-b285-486f4694aa6f-7_716_1249_1011_440}
$\_\_\_\_$

Questions C4 (1162 questions)

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Edexcel C4 Q3

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Edexcel C4 2013 January Q5

AQA C4 2014 June Q7

AQA C4 2016 June Q7

OCR C4 2008 January Q7

OCR C4 2008 June Q7

OCR C4 2009 June Q7

OCR MEI C4 2010 June Q5