Questions C4 - A-Level Maths

Edexcel C4 2009 January Q1

A curve $C$ has the equation $y ^ { 2 } - 3 y = x ^ { 3 } + 8$.

Find $\frac { \mathrm { d } y } { \mathrm {~d} x }$ in terms of $x$ and $y$.
Hence find the gradient of $C$ at the point where $y = 3$.

Edexcel C4 2009 January Q3

3. $$f ( x ) = \frac { 27 x ^ { 2 } + 32 x + 16 } { ( 3 x + 2 ) ^ { 2 } ( 1 - x ) } , \quad | x | < \frac { 2 } { 3 }$$ Given that $\mathrm { f } ( x )$ can be expressed in the form $$f ( x ) = \frac { A } { ( 3 x + 2 ) } + \frac { B } { ( 3 x + 2 ) ^ { 2 } } + \frac { C } { ( 1 - x ) }$$

find the values of $B$ and $C$ and show that $A = 0$.
Hence, or otherwise, find the series expansion of $\mathrm { f } ( x )$, in ascending powers of $x$, up to and including the term in $x ^ { 2 }$. Simplify each term.
Find the percentage error made in using the series expansion in part (b) to estimate the value of $\mathrm { f } ( 0.2 )$. Give your answer to 2 significant figures. \section*{LU}

Edexcel C4 2009 January Q4

4. With respect to a fixed origin $O$ the lines $l _ { 1 }$ and $l _ { 2 }$ are given by the equations $$l _ { 1 } : \quad \mathbf { r } = \left( \begin{array} { c }

Edexcel C4 2009 January Q5

5. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{a5579938-e202-4543-8513-6483ede49850-09_696_686_196_626} \captionsetup{labelformat=empty} \caption{Figure 2}

\end{figure} A container is made in the shape of a hollow inverted right circular cone. The height of the container is 24 cm and the radius is 16 cm , as shown in Figure 2. Water is flowing into the container. When the height of water is $h \mathrm {~cm}$, the surface of the water has radius $r \mathrm {~cm}$ and the volume of water is $V \mathrm {~cm} ^ { 3 }$.

Show that $V = \frac { 4 \pi h ^ { 3 } } { 27 }$.
[0pt] [The volume $V$ of a right circular cone with vertical height $h$ and base radius $r$ is given by the formula $V = \frac { 1 } { 3 } \pi r ^ { 2 } h$.] Water flows into the container at a rate of $8 \mathrm {~cm} ^ { 3 } \mathrm {~s} ^ { - 1 }$.
Find, in terms of $\pi$, the rate of change of $h$ when $h = 12$.

Edexcel C4 2009 January Q6

6. (a) Find $\int \tan ^ { 2 } x \mathrm {~d} x$.
(b) Use integration by parts to find $\int \frac { 1 } { x ^ { 3 } } \ln x \mathrm {~d} x$.
(c) Use the substitution $u = 1 + e ^ { x }$ to show that $$\int \frac { \mathrm { e } ^ { 3 x } } { 1 + \mathrm { e } ^ { x } } \mathrm {~d} x = \frac { 1 } { 2 } \mathrm { e } ^ { 2 x } - \mathrm { e } ^ { x } + \ln \left( 1 + \mathrm { e } ^ { x } \right) + k$$ where $k$ is a constant.

Edexcel C4 2009 January Q17

17 \end{array} \right) + \lambda \left( \begin{array} { c } - 2
1
- 4 \end{array} \right) \quad l _ { 2 } : \quad \mathbf { r } = \left( \begin{array} { c } - 5
11
p \end{array} \right) + \mu \left( \begin{array} { l } q
2
2 \end{array} \right)$$ where $\lambda$ and $\mu$ are parameters and $p$ and $q$ are constants. Given that $l _ { 1 }$ and $l _ { 2 }$ are perpendicular,

show that $q = - 3$. Given further that $l _ { 1 }$ and $l _ { 2 }$ intersect, find
the value of $p$,
the coordinates of the point of intersection. The point $A$ lies on $l _ { 1 }$ and has position vector $\left( \begin{array} { c } 9
3
13 \end{array} \right)$. The point $C$ lies on $l _ { 2 }$.
Given that a circle, with centre $C$, cuts the line $l _ { 1 }$ at the points $A$ and $B$,
find the position vector of $B$.
5. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{a5579938-e202-4543-8513-6483ede49850-09_696_686_196_626} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} A container is made in the shape of a hollow inverted right circular cone. The height of the container is 24 cm and the radius is 16 cm , as shown in Figure 2. Water is flowing into the container. When the height of water is $h \mathrm {~cm}$, the surface of the water has radius $r \mathrm {~cm}$ and the volume of water is $V \mathrm {~cm} ^ { 3 }$.
Show that $V = \frac { 4 \pi h ^ { 3 } } { 27 }$.
[0pt] [The volume $V$ of a right circular cone with vertical height $h$ and base radius $r$ is given by the formula $V = \frac { 1 } { 3 } \pi r ^ { 2 } h$.] Water flows into the container at a rate of $8 \mathrm {~cm} ^ { 3 } \mathrm {~s} ^ { - 1 }$.
Find, in terms of $\pi$, the rate of change of $h$ when $h = 12$. 6. (a) Find $\int \tan ^ { 2 } x \mathrm {~d} x$.
Use integration by parts to find $\int \frac { 1 } { x ^ { 3 } } \ln x \mathrm {~d} x$.
Use the substitution $u = 1 + e ^ { x }$ to show that $$\int \frac { \mathrm { e } ^ { 3 x } } { 1 + \mathrm { e } ^ { x } } \mathrm {~d} x = \frac { 1 } { 2 } \mathrm { e } ^ { 2 x } - \mathrm { e } ^ { x } + \ln \left( 1 + \mathrm { e } ^ { x } \right) + k$$ where $k$ is a constant.
7. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{a5579938-e202-4543-8513-6483ede49850-13_511_714_237_612} \captionsetup{labelformat=empty} \caption{Figure 3}
\end{figure} The curve $C$ shown in Figure 3 has parametric equations $$x = t ^ { 3 } - 8 t , \quad y = t ^ { 2 }$$ where $t$ is a parameter. Given that the point $A$ has parameter $t = - 1$,
find the coordinates of $A$. The line $l$ is the tangent to $C$ at $A$.
Show that an equation for $l$ is $2 x - 5 y - 9 = 0$. The line $l$ also intersects the curve at the point $B$.
Find the coordinates of $B$.

Edexcel C4 2010 January Q1

(a) Find the binomial expansion of

$$\sqrt { } ( 1 - 8 x ) , \quad | x | < \frac { 1 } { 8 }$$ in ascending powers of $x$ up to and including the term in $x ^ { 3 }$, simplifying each term.
(b) Show that, when $x = \frac { 1 } { 100 }$, the exact value of $\sqrt { } ( 1 - 8 x )$ is $\frac { \sqrt { } 23 } { 5 }$.
(c) Substitute $x = \frac { 1 } { 100 }$ into the binomial expansion in part (a) and hence obtain an approximation to $\sqrt { } 23$. Give your answer to 5 decimal places.

Edexcel C4 2010 January Q2

2. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{5ef3ae4a-a06d-48c1-8b79-7d7c3f95d120-03_623_1176_196_374} \captionsetup{labelformat=empty} \caption{Figure 1}

\end{figure} Figure 1 shows a sketch of the curve with equation $y = x \ln x , x \geqslant 1$. 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 $y = x \ln x$.

$x$	1	1.5	2	2.5	3	3.5	4
$y$	0	0.608			3.296	4.385	5.545

Complete the table with the values of $y$ corresponding to $x = 2$ and $x = 2.5$, giving your answers to 3 decimal places.
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. Use integration by parts to find $\int x \ln x \mathrm {~d} x$.
2. Hence find the exact area of $R$, giving your answer in the form $\frac { 1 } { 4 } ( a \ln 2 + b )$, where $a$ and $b$ are integers.

Edexcel C4 2010 January Q3

The curve $C$ has the equation

$$\cos 2 x + \cos 3 y = 1 , \quad - \frac { \pi } { 4 } \leqslant x \leqslant \frac { \pi } { 4 } , \quad 0 \leqslant y \leqslant \frac { \pi } { 6 }$$

Find $\frac { \mathrm { d } y } { \mathrm {~d} x }$ in terms of $x$ and $y$. The point $P$ lies on $C$ where $x = \frac { \pi } { 6 }$.
Find the value of $y$ at $P$.
Find the equation of the tangent to $C$ at $P$, giving your answer in the form $a x + b y + c \pi = 0$, where $a , b$ and $c$ are integers. \section*{LU}

Edexcel C4 2010 January Q4

4. The line $l _ { 1 }$ has vector equation $$\mathbf { r } = \left( \begin{array} { c } - 6
4
- 1 \end{array} \right) + \lambda \left( \begin{array} { c } 4
- 1
3 \end{array} \right)$$ and the line $l _ { 2 }$ has vector equation $$\mathbf { r } = \left( \begin{array} { c } - 6
4
- 1 \end{array} \right) + \mu \left( \begin{array} { c } 3
- 4
1 \end{array} \right)$$ where $\lambda$ and $\mu$ are parameters.
The lines $l _ { 1 }$ and $l _ { 2 }$ intersect at the point $A$ and the acute angle between $l _ { 1 }$ and $l _ { 2 }$ is $\theta$.

Write down the coordinates of $A$.
Find the value of $\cos \theta$. The point $X$ lies on $l _ { 1 }$ where $\lambda = 4$.
Find the coordinates of $X$.
Find the vector $\overrightarrow { A X }$.
Hence, or otherwise, show that $| \overrightarrow { A X } | = 4 \sqrt { } 26$. The point $Y$ lies on $l _ { 2 }$. Given that the vector $\overrightarrow { Y X }$ is perpendicular to $l _ { 1 }$,
find the length of $A Y$, giving your answer to 3 significant figures. \section*{LU}

Edexcel C4 2010 January Q5

5. (a) Find $\int \frac { 9 x + 6 } { x } \mathrm {~d} x , x > 0$.
(b) Given that $y = 8$ at $x = 1$, solve the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { ( 9 x + 6 ) y ^ { \frac { 1 } { 3 } } } { x }$$ giving your answer in the form $y ^ { 2 } = \mathrm { g } ( x )$. \section*{LU}

Edexcel C4 2010 January Q6

6. The area $A$ of a circle is increasing at a constant rate of $1.5 \mathrm {~cm} ^ { 2 } \mathrm {~s} ^ { - 1 }$. Find, to 3 significant figures, the rate at which the radius $r$ of the circle is increasing when the area of the circle is $2 \mathrm {~cm} ^ { 2 }$.
(5)

Edexcel C4 2010 January Q7

7. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{5ef3ae4a-a06d-48c1-8b79-7d7c3f95d120-12_734_1395_210_249} \captionsetup{labelformat=empty} \caption{Figure 2}

\end{figure} Figure 2 shows a sketch of the curve $C$ with parametric equations $$x = 5 t ^ { 2 } - 4 , \quad y = t \left( 9 - t ^ { 2 } \right)$$ The curve $C$ cuts the $x$-axis at the points $A$ and $B$.

Find the $x$-coordinate at the point $A$ and the $x$-coordinate at the point $B$. The region $R$, as shown shaded in Figure 2, is enclosed by the loop of the curve.
Use integration to find the area of $R$.
\section*{LU}

Edexcel C4 2010 January Q8

8. (a) Using the substitution $x = 2 \cos u$, or otherwise, find the exact value of $$\int _ { 1 } ^ { \sqrt { 2 } } \frac { 1 } { x ^ { 2 } \sqrt { } \left( 4 - x ^ { 2 } \right) } d x$$ \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{5ef3ae4a-a06d-48c1-8b79-7d7c3f95d120-14_680_1264_502_338} \captionsetup{labelformat=empty} \caption{Figure 3}

\end{figure} Figure 3 shows a sketch of part of the curve with equation $y = \frac { 4 } { x \left( 4 - x ^ { 2 } \right) ^ { \frac { 1 } { 4 } } } , \quad 0 < x < 2$. The shaded region $S$, shown in Figure 3, is bounded by the curve, the $x$-axis and the lines with equations $x = 1$ and $x = \sqrt { } 2$. The shaded region $S$ is rotated through $2 \pi$ radians about the $x$-axis to form a solid of revolution.
(b) Using your answer to part (a), find the exact volume of the solid of revolution formed.

Edexcel C4 2011 January Q1

Use integration to find the exact value of

$$\int _ { 0 } ^ { \frac { \pi } { 2 } } x \sin 2 x \mathrm {~d} x$$

Edexcel C4 2011 January Q2

2. The current, $I$ amps, in an electric circuit at time $t$ seconds is given by $$I = 16 - 16 ( 0.5 ) ^ { t } , \quad t \geqslant 0$$ Use differentiation to find the value of $\frac { \mathrm { d } I } { \mathrm {~d} t }$ when $t = 3$.
Give your answer in the form $\ln a$, where $a$ is a constant.

Edexcel C4 2011 January Q3

3. (a) Express $\frac { 5 } { ( x - 1 ) ( 3 x + 2 ) }$ in partial fractions.
(b) Hence find $\int \frac { 5 } { ( x - 1 ) ( 3 x + 2 ) } \mathrm { d } x$, where $x > 1$.
(c) Find the particular solution of the differential equation $$( x - 1 ) ( 3 x + 2 ) \frac { \mathrm { d } y } { \mathrm {~d} x } = 5 y , \quad x > 1$$ for which $y = 8$ at $x = 2$. Give your answer in the form $y = \mathrm { f } ( x )$.

Edexcel C4 2011 January Q4

Relative to a fixed origin $O$, the point $A$ has position vector $\mathbf { i } - 3 \mathbf { j } + 2 \mathbf { k }$ and the point $B$ has position vector $- 2 \mathbf { i } + 2 \mathbf { j } - \mathbf { k }$. The points $A$ and $B$ lie on a straight line $l$.
1. Find $\overrightarrow { A B }$.
2. Find a vector equation of $l$.
The point $C$ has position vector $2 \mathbf { i } + p \mathbf { j } - 4 \mathbf { k }$ with respect to $O$, where $p$ is a constant. Given that $A C$ is perpendicular to $l$, find
the value of $p$,
the distance $A C$.

Edexcel C4 2011 January Q5

(a) Use the binomial theorem to expand

$$( 2 - 3 x ) ^ { - 2 } , \quad | x | < \frac { 2 } { 3 }$$ in ascending powers of $x$, up to and including the term in $x ^ { 3 }$. Give each coefficient as a simplified fraction. $$\mathrm { f } ( x ) = \frac { a + b x } { ( 2 - 3 x ) ^ { 2 } } , \quad | x | < \frac { 2 } { 3 } , \quad \text { where } a \text { and } b \text { are constants. }$$ In the binomial expansion of $\mathrm { f } ( x )$, in ascending powers of $x$, the coefficient of $x$ is 0 and the coefficient of $x ^ { 2 }$ is $\frac { 9 } { 16 }$. Find
(b) the value of $a$ and the value of $b$,
(c) the coefficient of $x ^ { 3 }$, giving your answer as a simplified fraction.

Edexcel C4 2011 January Q6

The curve $C$ has parametric equations

$$x = \ln t , \quad y = t ^ { 2 } - 2 , \quad t > 0$$ Find

an equation of the normal to $C$ at the point where $t = 3$,
a cartesian equation of $C$. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{a3ece8a8-8107-4c3a-a6a9-c19b5e35ec5a-10_579_759_740_571} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} The finite area $R$, shown in Figure 1, is bounded by $C$, the $x$-axis, the line $x = \ln 2$ and the line $x = \ln 4$. The area $R$ is rotated through $360 ^ { \circ }$ about the $x$-axis.
Use calculus to find the exact volume of the solid generated.

Edexcel C4 2011 January Q7

7. $$I = \int _ { 2 } ^ { 5 } \frac { 1 } { 4 + \sqrt { } ( x - 1 ) } \mathrm { d } x$$

Given that $y = \frac { 1 } { 4 + \sqrt { } ( x - 1 ) }$, complete the table below with values of $y$ corresponding to $x = 3$ and $x = 5$. Give your values to 4 decimal places.
$x$ 2 3 4 5
$y$ 0.2 0.1745
Use the trapezium rule, with all of the values of $y$ in the completed table, to obtain an estimate of $I$, giving your answer to 3 decimal places.
Using the substitution $x = ( u - 4 ) ^ { 2 } + 1$, or otherwise, and integrating, find the exact value of $I$.

Edexcel C4 2012 January Q1

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

The point $P$ on the curve has coordinates $( - 1,1 )$.

Find the gradient of the curve at $P$.
Hence find the equation of the normal to $C$ at $P$, giving your answer in the form $a x + b y + c = 0$, where $a , b$ and $c$ are integers.

Edexcel C4 2012 January Q2

2. (a) Use integration by parts to find $\int x \sin 3 x \mathrm {~d} x$.
(b) Using your answer to part (a), find $\int x ^ { 2 } \cos 3 x \mathrm {~d} x$.

Edexcel C4 2012 January Q3

3. (a) Expand $$\frac { 1 } { ( 2 - 5 x ) ^ { 2 } } , \quad | x | < \frac { 2 } { 5 }$$ in ascending powers of $x$, up to and including the term in $x ^ { 2 }$, giving each term as a simplified fraction. Given that the binomial expansion of $\frac { 2 + k x } { ( 2 - 5 x ) ^ { 2 } } , | x | < \frac { 2 } { 5 }$, is $$\frac { 1 } { 2 } + \frac { 7 } { 4 } x + A x ^ { 2 } + \ldots$$ (b) find the value of the constant $k$,
(c) find the value of the constant $A$.

Edexcel C4 2012 January Q4

4. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{8c963567-d751-4898-b7a7-7095d90514f0-06_606_1185_237_383} \captionsetup{labelformat=empty} \caption{Figure 1}

\end{figure} Figure 1 shows the curve with equation $$y = \sqrt { } \left( \frac { 2 x } { 3 x ^ { 2 } + 4 } \right) , x \geqslant 0$$ The finite region $S$, shown shaded in Figure 1, is bounded by the curve, the $x$-axis and the line $x = 2$ The region $S$ is rotated $360 ^ { \circ }$ about the $x$-axis.
Use integration to find the exact value of the volume of the solid generated, giving your answer in the form $k \ln a$, where $k$ and $a$ are constants.

Questions C4 (1162 questions)

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