Questions C34 - A-Level Maths

Edexcel C34 2016 January Q2

(a) Show that

$$\cot ^ { 2 } x - \operatorname { cosec } x - 11 = 0$$ may be expressed in the form $\operatorname { cosec } ^ { 2 } x - \operatorname { cosec } x + k = 0$, where $k$ is a constant.
(b) Hence solve for $0 \leqslant x < 360 ^ { \circ }$ $$\cot ^ { 2 } x - \operatorname { cosec } x - 11 = 0$$ Give each solution in degrees to one decimal place.
(Solutions based entirely on graphical or numerical methods are not acceptable.)

Edexcel C34 2016 January Q3

3. A curve $C$ has equation $$3 ^ { x } + 6 y = \frac { 3 } { 2 } x y ^ { 2 }$$ Find the exact value of $\frac { \mathrm { d } y } { \mathrm {~d} x }$ at the point on $C$ with coordinates (2, 3). Give your answer in the form $\frac { a + \ln b } { 8 }$, where $a$ and $b$ are integers.

Edexcel C34 2016 January Q4

4. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{101ec3c2-699e-4c74-bfdc-d8c610646571-05_504_844_255_543} \captionsetup{labelformat=empty} \caption{Figure 1}

\end{figure} The curve $C$ with equation $y = \frac { 2 } { ( 4 + 3 x ) } , x > - \frac { 4 } { 3 }$ is shown in Figure 1
The region bounded by the curve, the $x$-axis and the lines $x = - 1$ and $x = \frac { 2 } { 3 }$, is shown shaded in Figure 1 This region is rotated through 360 degrees about the $x$-axis.

Use calculus to find the exact value of the volume of the solid generated. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{101ec3c2-699e-4c74-bfdc-d8c610646571-05_583_433_1398_753} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} Figure 2 shows a candle with axis of symmetry $A B$ where $A B = 15 \mathrm {~cm}$. $A$ is a point at the centre of the top surface of the candle and $B$ is a point at the centre of the base of the candle. The candle is geometrically similar to the solid generated in part (a).
Find the volume of this candle.

Edexcel C34 2016 January Q5

5. $$f ( x ) = - x ^ { 3 } + 4 x ^ { 2 } - 6$$

Show that the equation $\mathrm { f } ( x ) = 0$ has a root between $x = 1$ and $x = 2$
Show that the equation $\mathrm { f } ( x ) = 0$ can be rewritten as $$x = \sqrt { \left( \frac { 6 } { 4 - x } \right) }$$
Starting with $x _ { 1 } = 1.5$ use the iteration $x _ { n + 1 } = \sqrt { \left( \frac { 6 } { 4 - x _ { n } } \right) }$ to calculate the values of $x _ { 2 }$, $x _ { 3 }$ and $x _ { 4 }$ giving all your answers to 4 decimal places.
Using a suitable interval, show that 1.572 is a root of $\mathrm { f } ( x ) = 0$ correct to 3 decimal places.

Edexcel C34 2016 January Q6

6. A hot piece of metal is dropped into a cool liquid. As the metal cools, its temperature $T$ degrees Celsius, $t$ minutes after it enters the liquid, is modelled by $$T = 300 \mathrm { e } ^ { - 0.04 t } + 20 , \quad t \geqslant 0$$

Find the temperature of the piece of metal as it enters the liquid.
Find the value of $t$ for which $T = 180$, giving your answer to 3 significant figures. (Solutions based entirely on graphical or numerical methods are not acceptable.)
Show, by differentiation, that the rate, in degrees Celsius per minute, at which the temperature of the metal is changing, is given by the expression $$\frac { 20 - T } { 25 }$$
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Edexcel C34 2016 January Q7

7. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{101ec3c2-699e-4c74-bfdc-d8c610646571-10_543_817_278_584} \captionsetup{labelformat=empty} \caption{Figure 3}

\end{figure} Figure 3 shows part of the curve $C$ with equation $$y = \frac { 3 \ln \left( x ^ { 2 } + 1 \right) } { \left( x ^ { 2 } + 1 \right) } , \quad x \in \mathbb { R }$$

Find $\frac { \mathrm { d } y } { \mathrm {~d} x }$
Using your answer to (a), find the exact coordinates of the stationary point on the curve $C$ for which $x > 0$. Write each coordinate in its simplest form.
(5) The finite region $R$, shown shaded in Figure 3, is bounded by the curve $C$, the $x$-axis and the line $x = 3$
Complete the table below with the value of $y$ corresponding to $x = 1$
$x$ 0 1 2 3
$y$ 0 $\frac { 3 } { 5 } \ln 5$ $\frac { 3 } { 10 } \ln 10$
Use the trapezium rule with all the $y$ values in the completed table to find an approximate value for the area of $R$, giving your answer to 4 significant figures.

Edexcel C34 2016 January Q8

8. $$f ( \theta ) = 9 \cos ^ { 2 } \theta + \sin ^ { 2 } \theta$$

Show that $\mathrm { f } ( \theta ) = a + b \cos 2 \theta$, where $a$ and $b$ are integers which should be found.
Using your answer to part (a) and integration by parts, find the exact value of $$\int _ { 0 } ^ { \frac { \pi } { 2 } } \theta ^ { 2 } \mathrm { f } ( \theta ) \mathrm { d } \theta$$

Edexcel C34 2016 January Q9

(a) Express $\frac { 3 x ^ { 2 } - 4 } { x ^ { 2 } ( 3 x - 2 ) }$ in partial fractions.
(b) Given that $x > \frac { 2 } { 3 }$, find the general solution of the differential equation

$$x ^ { 2 } ( 3 x - 2 ) \frac { \mathrm { d } y } { \mathrm {~d} x } = y \left( 3 x ^ { 2 } - 4 \right)$$ Give your answer in the form $y = \mathrm { f } ( x )$.

Edexcel C34 2016 January Q10

10. (a) Express $3 \sin 2 x + 5 \cos 2 x$ in the form $R \sin ( 2 x + \alpha )$, where $R > 0$ and $0 < \alpha < \frac { \pi } { 2 }$ Give the exact value of $R$ and give the value of $\alpha$ to 3 significant figures.
(b) Solve, for $0 < x < \pi$, $$3 \sin 2 x + 5 \cos 2 x = 4$$ (Solutions based entirely on graphical or numerical methods are not acceptable.) $$g ( x ) = 4 ( 3 \sin 2 x + 5 \cos 2 x ) ^ { 2 } + 3$$ (c) Using your answer to part (a) and showing your working,

find the greatest value of $\mathrm { g } ( x )$,
find the least value of $\mathrm { g } ( x )$.

Edexcel C34 2016 January Q11

11. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{101ec3c2-699e-4c74-bfdc-d8c610646571-16_572_1338_278_239} \captionsetup{labelformat=empty} \caption{Figure 4}

\end{figure} Figure 4 shows a sketch of part of the curve with equation $y = \mathrm { f } ( x ) , \quad x \in \mathbb { R }$
The curve meets the coordinate axes at the points $A ( 0 , - 3 )$ and $B \left( - \frac { 1 } { 3 } \ln 4,0 \right)$ and the curve has an asymptote with equation $y = - 4$ In separate diagrams, sketch the graph with equation

$y = | f ( x ) |$
$y = 2 \mathrm { f } ( x ) + 6$ On each sketch, give the exact coordinates of the points where the curve crosses or meets the coordinate axes and the equation of any asymptote. Given that $$\begin{array} { l l } \mathrm { f } ( x ) = \mathrm { e } ^ { - 3 x } - 4 , & x \in \mathbb { R }
\mathrm {~g} ( x ) = \ln \left( \frac { 1 } { x + 2 } \right) , & x > - 2 \end{array}$$
state the range of f,
find $\mathrm { f } ^ { - 1 } ( x )$,
express $f g ( x )$ as a polynomial in $x$.

Edexcel C34 2016 January Q12

With respect to a fixed origin $O$, the lines $l _ { 1 }$ and $l _ { 2 }$ are given by the equations

$$l _ { 1 } : \mathbf { r } = \left( \begin{array} { r } 12
- 4
5 \end{array} \right) + \lambda \left( \begin{array} { r } 5
- 4
2 \end{array} \right) , \quad l _ { 2 } : \mathbf { r } = \left( \begin{array} { l } 2
2
0 \end{array} \right) + \mu \left( \begin{array} { l } 0
6
3 \end{array} \right)$$ where $\lambda$ and $\mu$ are scalar parameters.

Show that $l _ { 1 }$ and $l _ { 2 }$ meet, and find the position vector of their point of intersection $A$.
Find, to the nearest $0.1 ^ { \circ }$, the acute angle between $l _ { 1 }$ and $l _ { 2 }$ The point $B$ has position vector $\left( \begin{array} { l } 7
0
3 \end{array} \right)$.
Show that $B$ lies on $l _ { 1 }$
Find the shortest distance from $B$ to the line $l _ { 2 }$, giving your answer to 3 significant figures.

Edexcel C34 2016 January Q13

13. A curve $C$ has parametric equations $$x = 6 \cos 2 t , \quad y = 2 \sin t , \quad - \frac { \pi } { 2 } < t < \frac { \pi } { 2 }$$

Show that $\frac { \mathrm { d } y } { \mathrm {~d} x } = \lambda \operatorname { cosec } t$, giving the exact value of the constant $\lambda$.
Find an equation of the normal to $C$ at the point where $t = \frac { \pi } { 3 }$ Give your answer in the form $y = m x + c$, where $m$ and $c$ are simplified surds. The cartesian equation for the curve $C$ can be written in the form $$x = f ( y ) , \quad - k < y < k$$ where $\mathrm { f } ( y )$ is a polynomial in $y$ and $k$ is a constant.
Find $\mathrm { f } ( y )$.
State the value of $k$.

Edexcel C34 2017 January Q1

Find an equation of the tangent to the curve

$$x ^ { 3 } + 3 x ^ { 2 } y + y ^ { 3 } = 37$$ at the point $( 1,3 )$. Give your answer in the form $a x + b y + c = 0$, where $a , b$ and $c$ are integers.
(6)

Edexcel C34 2017 January Q2

2. $$f ( x ) = x ^ { 3 } - 5 x + 16$$

Show that the equation $\mathrm { f } ( x ) = 0$ can be rewritten as $$x = ( a x + b ) ^ { \frac { 1 } { 3 } }$$ giving the values of the constants $a$ and $b$. The equation $\mathrm { f } ( x ) = 0$ has exactly one real root $\alpha$, where $\alpha = - 3$ to one significant figure.
Starting with $x _ { 1 } = - 3$, use the iteration $$x _ { n + 1 } = \left( a x _ { n } + b \right) ^ { \frac { 1 } { 3 } }$$ with the values of $a$ and $b$ found in part (a), to calculate the values of $x _ { 2 } , x _ { 3 }$ and $x _ { 4 }$, giving all your answers to 3 decimal places.
Using a suitable interval, show that $\alpha = - 3.17$ correct to 2 decimal places.

Edexcel C34 2017 January Q3

3. (a) Express $\frac { 9 + 11 x } { ( 1 - x ) ( 3 + 2 x ) }$ in partial fractions.
(b) Hence, or otherwise, find the series expansion of $$\frac { 9 + 11 x } { ( 1 - x ) ( 3 + 2 x ) } , \quad | x | < 1$$ in ascending powers of $x$, up to and including the term in $x ^ { 3 }$.
Give each coefficient as a simplified fraction.

Edexcel C34 2017 January Q4

Given that

$$\begin{array} { l l } \mathrm { f } ( x ) = \frac { 4 } { 3 x + 5 } , & x > 0
\mathrm {~g} ( x ) = \frac { 1 } { x } , & x > 0 \end{array}$$

state the range of f,
find $\mathrm { f } ^ { - 1 } ( x )$,
find $\mathrm { fg } ( x )$.
Show that the equation $\mathrm { fg } ( x ) = \mathrm { gf } ( x )$ has no real solutions.

Edexcel C34 2017 January Q5

5. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{e30f0c28-1695-40a1-8e9a-6ea7e29042bf-08_579_1038_258_452} \captionsetup{labelformat=empty} \caption{Figure 1}

\end{figure} Figure 1 shows a sketch of part of the curve $C$ with equation $$y = x \cos x , \quad x \in \mathbb { R }$$ The finite region $R$, shown shaded in Figure 1, is bounded by the curve $C$ and the $x$-axis for $\frac { 3 \pi } { 2 } \leqslant x \leqslant \frac { 5 \pi } { 2 }$

Complete the table below with the exact value of $y$ corresponding to $x = \frac { 7 \pi } { 4 }$ and with the exact value of $y$ corresponding to $x = \frac { 9 \pi } { 4 }$
$x$ $\frac { 3 \pi } { 2 }$ $\frac { 7 \pi } { 4 }$ $2 \pi$ $\frac { 9 \pi } { 4 }$ $\frac { 5 \pi } { 2 }$
$y$ 0 $2 \pi$ 0
Use the trapezium rule, with all five $y$ values in the completed table, to find an approximate value for the area of $R$, giving your answer to 4 significant figures.
Find $$\int x \cos x d x$$
Using your answer from part (c), find the exact area of the region $R$.

Edexcel C34 2017 January Q6

(i) Differentiate $y = 5 x ^ { 2 } \ln 3 x , \quad x > 0$
(ii) Given that

$$y = \frac { x } { \sin x + \cos x } , \quad - \frac { \pi } { 4 } < x < \frac { 3 \pi } { 4 }$$ show that $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { ( 1 + x ) \sin x + ( 1 - x ) \cos x } { 1 + \sin 2 x } , \quad - \frac { \pi } { 4 } < x < \frac { 3 \pi } { 4 }$$ \includegraphics[max width=\textwidth, alt={}, center]{e30f0c28-1695-40a1-8e9a-6ea7e29042bf-11_99_104_2631_1781}

Edexcel C34 2017 January Q7

7. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{e30f0c28-1695-40a1-8e9a-6ea7e29042bf-12_458_433_264_781} \captionsetup{labelformat=empty} \caption{Figure 2}

\end{figure} Figure 2 shows a sketch of the graph of $y = \mathrm { f } ( x ) , x \in \mathbb { R }$.
The point $P \left( \frac { 1 } { 3 } , 0 \right)$ is the vertex of the graph.
The point $Q ( 0,5 )$ is the intercept with the $y$-axis. Given that $\mathrm { f } ( x ) = | a x + b |$, where $a$ and $b$ are constants,

1. find all possible values for $a$ and $b$,
2. hence find an equation for the graph.
Sketch the graph with equation $$y = \mathrm { f } \left( \frac { 1 } { 2 } x \right) + 3$$ showing the coordinates of its vertex and its intercept with the $y$-axis.

Edexcel C34 2017 January Q8

8. (a) Using the trigonometric identity for $\tan ( A + B )$, prove that $$\tan 3 x = \frac { 3 \tan x - \tan ^ { 3 } x } { 1 - 3 \tan ^ { 2 } x } , \quad x \neq ( 2 n + 1 ) 30 ^ { \circ } , \quad n \in \mathbb { Z }$$ (b) Hence solve, for $- 30 ^ { \circ } < x < 30 ^ { \circ }$, $$\tan 3 x = 11 \tan x$$ (Solutions based entirely on graphical or numerical methods are not acceptable.)

Edexcel C34 2017 January Q9

9. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{e30f0c28-1695-40a1-8e9a-6ea7e29042bf-16_727_1491_258_239} \captionsetup{labelformat=empty} \caption{Figure 3}

\end{figure}

By using the substitution $u = 2 x + 3$, show that $$\int _ { 0 } ^ { 12 } \frac { x } { ( 2 x + 3 ) ^ { 2 } } \mathrm {~d} x = \frac { 1 } { 2 } \ln 3 - \frac { 2 } { 9 }$$ The curve $C$ has equation $$y = \frac { 9 \sqrt { x } } { ( 2 x + 3 ) } , \quad x > 0$$ The finite region $R$, shown shaded in Figure 3, is bounded by the curve $C$, the $x$-axis and the line with equation $x = 12$. The region $R$ is rotated through $2 \pi$ radians about the $x$-axis to form a solid of revolution.
Use the result of part (a) to find the exact value of the volume of the solid generated.

Edexcel C34 2017 January Q10

10. A population of insects is being studied. The number of insects, $N$, in the population, is modelled by the equation $$N = \frac { 300 } { 3 + 17 \mathrm { e } ^ { - 0.2 t } } \quad t \in \mathbb { R } , t \geqslant 0$$ where $t$ is the time, in weeks, from the start of the study.
Using the model,

find the number of insects at the start of the study,
find the number of insects when $t = 10$,
find the time from the start of the study when there are 82 insects. (Solutions based entirely on graphical or numerical methods are not acceptable.)
Find, by differentiating, the rate, measured in insects per week, at which the number of insects is increasing when $t = 5$. Give your answer to the nearest whole number.

Edexcel C34 2017 January Q11

(a) Express $35 \sin x - 12 \cos x$ in the form $R \sin ( x - \alpha )$, where $R > 0$ and $0 < \alpha < \frac { \pi } { 2 }$

Give the exact value of $R$, and give the value of $\alpha$, in radians, to 4 significant figures.
(b) Hence solve, for $0 \leqslant x < 2 \pi$, $$70 \sin x - 24 \cos x = 37$$ (Solutions based entirely on graphical or numerical methods are not acceptable.) $$y = \frac { 7000 } { 31 + ( 35 \sin x - 12 \cos x ) ^ { 2 } } , \quad x > 0$$ (c) Use your answer to part (a) to calculate

the minimum value of $y$,
the smallest value of $x , x > 0$, at which this minimum value occurs.

Edexcel C34 2017 January Q12

In freezing temperatures, ice forms on the surface of the water in a barrel. At time $t$ hours after the start of freezing, the thickness of the ice formed is $x \mathrm {~mm}$. You may assume that the thickness of the ice is uniform across the surface of the water.

At 4 pm there is no ice on the surface, and freezing begins.
At 6pm, after two hours of freezing, the ice is 1.5 mm thick.
In a simple model, the rate of increase of $x$, in mm per hour, is assumed to be constant for a period of 20 hours. Using this simple model,

express $t$ in terms of $x$,
find the value of $t$ when $x = 3$ In a second model, the rate of increase of $x$, in mm per hour, is given by the differential equation $$\frac { \mathrm { d } x } { \mathrm {~d} t } = \frac { \lambda } { ( 2 x + 1 ) } \text { where } \lambda \text { is a constant and } 0 \leqslant t \leqslant 20$$ Using this second model,
solve the differential equation and express $t$ in terms of $x$ and $\lambda$,
find the exact value for $\lambda$,
find at what time the ice is predicted to be 3 mm thick.

\(x\)	0	1	2	3
\(y\)	0		\(\frac { 3 } { 5 } \ln 5\)	\(\frac { 3 } { 10 } \ln 10\)

\(x\)	\(\frac { 3 \pi } { 2 }\)	\(\frac { 7 \pi } { 4 }\)	\(2 \pi\)	\(\frac { 9 \pi } { 4 }\)	\(\frac { 5 \pi } { 2 }\)
\(y\)	0		\(2 \pi\)		0

Questions C34 (197 questions)

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