Questions C34 - A-Level Maths

Edexcel C34 2014 January Q1

1. $$\mathrm { f } ( x ) = \frac { 2 x } { x ^ { 2 } + 3 } , \quad x \in \mathbb { R }$$ Find the set of values of $x$ for which $\mathrm { f } ^ { \prime } ( x ) > 0$ You must show your working.
(Solutions based entirely on graphical or numerical methods are not acceptable.)

Edexcel C34 2014 January Q2

2. Solve, for $0 \leqslant x \leqslant 270 ^ { \circ }$, the equation $$\frac { \tan 2 x + \tan 50 ^ { \circ } } { 1 - \tan 2 x \tan 50 ^ { \circ } } = 2$$ Give your answers in degrees to 2 decimal places.
(6)
\includegraphics[max width=\textwidth, alt={}, center]{5b698944-41ac-4072-b5e1-c580b7752c39-05_104_95_2613_1786}

Edexcel C34 2014 January Q3

3. Given that $$4 x ^ { 3 } + 2 x ^ { 2 } + 17 x + 8 \equiv ( A x + B ) \left( x ^ { 2 } + 4 \right) + C x + D$$

find the values of the constants $A , B , C$ and $D$.
Hence find $$\int _ { 1 } ^ { 4 } \frac { 4 x ^ { 3 } + 2 x ^ { 2 } + 17 x + 8 } { x ^ { 2 } + 4 } d x$$ giving your answer in the form $p + \ln q$, where $p$ and $q$ are integers.

Edexcel C34 2014 January Q4

4. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{5b698944-41ac-4072-b5e1-c580b7752c39-10_606_613_285_278} \captionsetup{labelformat=empty} \caption{Figure 1}

\end{figure} \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{5b698944-41ac-4072-b5e1-c580b7752c39-10_602_608_287_1062} \captionsetup{labelformat=empty} \caption{Figure 2}

\end{figure} Figure 1 shows a sketch of part of the graph $y = \mathrm { f } ( x )$, where $$f ( x ) = 2 | 3 - x | + 5 , \quad x \geqslant 0$$ Figure 2 shows a sketch of part of the graph $y = \mathrm { g } ( x )$, where $$\operatorname { g } ( x ) = \frac { x + 9 } { 2 x + 3 } , \quad x \geqslant 0$$

Find the value of $\mathrm { fg } ( 1 )$
State the range of g
Find $\mathrm { g } ^ { - 1 } ( x )$ and state its domain. Given that the equation $\mathrm { f } ( x ) = k$, where $k$ is a constant, has exactly two roots,
state the range of possible values of $k$.

Edexcel C34 2014 January Q5

(a) Prove, by using logarithms, that

$$\frac { \mathrm { d } } { \mathrm {~d} x } \left( 2 ^ { x } \right) = 2 ^ { x } \ln 2$$ The curve $C$ has the equation $$2 x + 3 y ^ { 2 } + 3 x ^ { 2 } y + 12 = 4 \times 2 ^ { x }$$ The point $P$, with coordinates $( 2,0 )$, lies on $C$.
(b) Find an equation of the tangent to $C$ at $P$.

Edexcel C34 2014 January Q6

6. Given that the binomial expansion, in ascending powers of $x$, of $$\frac { 6 } { \sqrt { } \left( 9 + A x ^ { 2 } \right) } , \quad | x | < \frac { 3 } { \sqrt { } | A | }$$ is $\quad B - \frac { 2 } { 3 } x ^ { 2 } + C x ^ { 4 } + \ldots$

find the values of the constants $A , B$ and $C$.
Hence find the coefficient of $x ^ { 6 }$

Edexcel C34 2014 January Q7

7. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{5b698944-41ac-4072-b5e1-c580b7752c39-20_689_712_248_680} \captionsetup{labelformat=empty} \caption{Figure 3}

\end{figure} Figure 3 shows a sketch of part of the curve with equation $y = \mathrm { f } ( x )$, where $$f ( x ) = 2 x ( 1 + x ) \ln x , \quad x > 0$$ The curve has a minimum turning point at $A$.

Find f'(x)
Hence show that the $x$ coordinate of $A$ is the solution of the equation $$x = \mathrm { e } ^ { - \frac { 1 + x } { 1 + 2 x } }$$
Use the iteration formula $$x _ { n + 1 } = \mathrm { e } ^ { - \frac { 1 + x _ { n } } { 1 + 2 x _ { n } } } , \quad x _ { 0 } = 0.46$$ to find the values of $x _ { 1 } , x _ { 2 }$ and $x _ { 3 }$ to 4 decimal places.
Use your answer to part (c) to estimate the coordinates of $A$ to 2 decimal places.

Edexcel C34 2014 January Q8

8. (a) Prove that $$\text { 2cosec } 2 A - \cot A \equiv \tan A , \quad A \neq \frac { n \pi } { 2 } , n \in \mathbb { Z }$$ (b) Hence solve, for $0 \leqslant \theta \leqslant \frac { \pi } { 2 }$

$2 \operatorname { cosec } 4 \theta - \cot 2 \theta = \sqrt { } 3$
$\tan \theta + \cot \theta = 5$ Give your answers to 3 significant figures.

Edexcel C34 2014 January Q9

9. (a) Use the substitution $u = 4 - \sqrt { } x$ to find $$\int \frac { \mathrm { d } x } { 4 - \sqrt { } x }$$ A team of scientists is studying a species of slow growing tree.
The rate of change in height of a tree in this species is modelled by the differential equation $$\frac { \mathrm { d } h } { \mathrm {~d} t } = \frac { 4 - \sqrt { } h } { 20 }$$ where $h$ is the height in metres and $t$ is the time measured in years after the tree is planted.
(b) Find the range in values of $h$ for which the height of a tree in this species is increasing.
(c) Given that one of these trees is 1 metre high when it is planted, calculate the time it would take to reach a height of 10 metres. Write your answer to 3 significant figures.
\includegraphics[max width=\textwidth, alt={}, center]{5b698944-41ac-4072-b5e1-c580b7752c39-31_154_145_2599_1804}

Edexcel C34 2014 January Q10

10. With respect to a fixed origin $O$, the lines $l _ { 1 }$ and $l _ { 2 }$ are given by the equations $$\begin{aligned} & l _ { 1 } : \mathbf { r } = ( \mathbf { i } + 5 \mathbf { j } + 5 \mathbf { k } ) + \lambda ( 2 \mathbf { i } + \mathbf { j } - \mathbf { k } )
& l _ { 2 } : \mathbf { r } = ( 2 \mathbf { j } + 12 \mathbf { k } ) + \mu ( 3 \mathbf { i } - \mathbf { j } + 5 \mathbf { k } ) \end{aligned}$$ 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.
Show that $l _ { 1 }$ and $l _ { 2 }$ are perpendicular to each other. The point $A$, with position vector $5 \mathbf { i } + 7 \mathbf { j } + 3 \mathbf { k }$, lies on $l _ { 1 }$
The point $B$ is the image of $A$ after reflection in the line $l _ { 2 }$
Find the position vector of $B$.
\includegraphics[max width=\textwidth, alt={}, center]{5b698944-41ac-4072-b5e1-c580b7752c39-35_133_163_2604_1786}

Edexcel C34 2014 January Q11

11. The curve $C$ has parametric equations $$x = 10 \cos 2 t , \quad y = 6 \sin t , \quad - \frac { \pi } { 2 } \leqslant t \leqslant \frac { \pi } { 2 }$$ The point $A$ with coordinates $( 5,3 )$ lies on $C$.

Find the value of $t$ at the point $A$.
Show that an equation of the normal to $C$ at $A$ is $$3 y = 10 x - 41$$ The normal to $C$ at $A$ cuts $C$ again at the point $B$.
Find the exact coordinates of $B$.

Edexcel C34 2014 January Q12

12. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{5b698944-41ac-4072-b5e1-c580b7752c39-40_695_1212_276_420} \captionsetup{labelformat=empty} \caption{Figure 4}

\end{figure} Figure 4 shows a sketch of part of the curve with equation $$y = x ( \sin x + \cos x ) , \quad 0 \leqslant x \leqslant \frac { \pi } { 4 }$$ The finite region $R$, shown shaded in Figure 4, is bounded by the curve, the $x$-axis and the line $x = \frac { \pi } { 4 }$. This shaded region is rotated through $2 \pi$ radians about the $x$-axis to form a solid of revolution, with volume $V$.

Assuming the formula for volume of revolution show that $V = \int _ { 0 } ^ { \frac { \pi } { 4 } } \pi x ^ { 2 } ( 1 + \sin 2 x ) \mathrm { d } x$
Hence using calculus find the exact value of $V$. You must show your working.
(Solutions based entirely on graphical or numerical methods are not acceptable.)

Edexcel C34 2015 January Q1

The curve $C$ has equation

$$y = \frac { 3 x - 2 } { ( x - 2 ) ^ { 2 } } , \quad x \neq 2$$ The point $P$ on $C$ has $x$ coordinate 3
Find an equation of the normal to $C$ at the point $P$ in the form $a x + b y + c = 0$, where $a , b$ and $c$ are integers.

Edexcel C34 2015 January Q2

2. Solve, for $0 \leqslant \theta < 2 \pi$, $$2 \cos 2 \theta = 5 - 13 \sin \theta$$ Give your answers in radians to 3 decimal places.
(Solutions based entirely on graphical or numerical methods are not acceptable.)

Edexcel C34 2015 January Q3

3. The function $g$ is defined by $$\mathrm { g } : x \mapsto | 8 - 2 x | , \quad x \in \mathbb { R } , \quad x \geqslant 0$$

Sketch the graph with equation $y = \mathrm { g } ( x )$, showing the coordinates of the points where the graph cuts or meets the axes.
Solve the equation $$| 8 - 2 x | = x + 5$$ The function f is defined by $$\mathrm { f } : x \mapsto x ^ { 2 } - 3 x + 1 , \quad x \in \mathbb { R } , \quad 0 \leqslant x \leqslant 4$$
Find fg(5).
Find the range of f. You must make your method clear.

Edexcel C34 2015 January Q4

4. Use the substitution $x = 2 \sin \theta$ to find the exact value of $$\int _ { 0 } ^ { \sqrt { 3 } } \frac { 1 } { \left( 4 - x ^ { 2 } \right) ^ { \frac { 3 } { 2 } } } \mathrm {~d} x$$

Edexcel C34 2015 January Q5

(a) Use the binomial expansion, in ascending powers of $x$, of $\frac { 1 } { \sqrt { } ( 1 - 2 x ) }$ to show that

$$\frac { 2 + 3 x } { \sqrt { } ( 1 - 2 x ) } \approx 2 + 5 x + 6 x ^ { 2 } , \quad | x | < 0.5$$ (b) Substitute $x = \frac { 1 } { 20 }$ into $$\frac { 2 + 3 x } { \sqrt { } ( 1 - 2 x ) } = 2 + 5 x + 6 x ^ { 2 }$$ to obtain an approximation to $\sqrt { 10 }$
Give your answer as a fraction in its simplest form.

Edexcel C34 2015 January Q6

6. (i) Given $x = \tan ^ { 2 } 4 y , 0 < y < \frac { \pi } { 8 }$, find $\frac { \mathrm { d } y } { \mathrm {~d} x }$ as a function of $x$. Write your answer in the form $\frac { 1 } { A \left( x ^ { p } + x ^ { q } \right) }$, where $A , p$ and $q$ are constants to
be found.
(ii) The volume $V$ of a cube is increasing at a constant rate of $2 \mathrm {~cm} ^ { 3 } \mathrm {~s} ^ { - 1 }$. Find the rate at which the length of the edge of the cube is increasing when the volume of the cube is $64 \mathrm {~cm} ^ { 3 }$.

Edexcel C34 2015 January Q7

7. (a) Given that $$2 \cos ( x + 30 ) ^ { \circ } = \sin ( x - 30 ) ^ { \circ }$$ without using a calculator, show that $$\tan x ^ { \circ } = 3 \sqrt { 3 } - 4$$ (b) Hence or otherwise solve, for $0 \leqslant \theta < 180$, $$2 \cos ( 2 \theta + 40 ) ^ { \circ } = \sin ( 2 \theta - 20 ) ^ { \circ }$$ Give your answers to one decimal place.

Edexcel C34 2015 January Q8

8. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{03548211-79cb-4629-b6ca-aa9dfcc77a33-13_743_1198_219_372} \captionsetup{labelformat=empty} \caption{Figure 1}

\end{figure} The value of Lin's car is modelled by the formula $$V = 18000 \mathrm { e } ^ { - 0.2 t } + 4000 \mathrm { e } ^ { - 0.1 t } + 1000 , \quad t \geqslant 0$$ where the value of the car is $V$ pounds when the age of the car is $t$ years.
A sketch of $t$ against $V$ is shown in Figure 1.

State the range of $V$. According to this model,
find the rate at which the value of the car is decreasing when $t = 10$ Give your answer in pounds per year.
Calculate the exact value of $t$ when $V = 15000$

Edexcel C34 2015 January Q9

9. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{03548211-79cb-4629-b6ca-aa9dfcc77a33-15_618_899_262_566} \captionsetup{labelformat=empty} \caption{Figure 2}

\end{figure} The curve $C$ has parametric equations $$x = \ln ( t + 2 ) , \quad y = \frac { 4 } { t ^ { 2 } } \quad t > 0$$ The finite region $R$, shown shaded in Figure 2, is bounded by the curve $C$, the $x$-axis and the lines with equations $x = \ln 3$ and $x = \ln 5$

Show that the area of $R$ is given by the integral $$\int _ { 1 } ^ { 3 } \frac { 4 } { t ^ { 2 } ( t + 2 ) } \mathrm { d } t$$
Hence find an exact value for the area of $R$. Write your answer in the form ( $a + \ln b$ ), where $a$ and $b$ are rational numbers.
Find a cartesian equation of the curve $C$ in the form $y = \mathrm { f } ( x )$.

Edexcel C34 2015 January Q10

10. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{03548211-79cb-4629-b6ca-aa9dfcc77a33-17_598_736_223_603} \captionsetup{labelformat=empty} \caption{Figure 3}

\end{figure} Figure 3 shows a sketch of part of the curve $C$ with equation $$y = \frac { x ^ { 2 } \ln x } { 3 } - 2 x + 4 , \quad x > 0$$ Point $A$ is the minimum turning point on the curve.

Show, by using calculus, that the $x$ coordinate of point $A$ is a solution of $$x = \frac { 6 } { 1 + \ln \left( x ^ { 2 } \right) }$$
Starting with $x _ { 0 } = 2.27$, use the iteration $$x _ { n + 1 } = \frac { 6 } { 1 + \ln \left( x _ { n } ^ { 2 } \right) }$$ to calculate the values of $x _ { 1 } , x _ { 2 }$ and $x _ { 3 }$, giving your answers to 3 decimal places.
Use your answer to part (b) to deduce the coordinates of point $A$ to one decimal place.

Edexcel C34 2015 January Q11

11. 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 }

Edexcel C34 2015 January Q12

12. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{03548211-79cb-4629-b6ca-aa9dfcc77a33-21_615_732_233_605} \captionsetup{labelformat=empty} \caption{Figure 4}

\end{figure} Figure 4 shows a sketch of part of the curve $C$ with equation $$y = \frac { x ^ { 2 } \ln x } { 3 } - 2 x + 4 , \quad x > 0$$ The finite region $S$, shown shaded in Figure 4, is bounded by the curve $C$, the $x$-axis and the lines with equations $x = 1$ and $x = 3$

Complete the table below with the value of $y$ corresponding to $x = 2$. Give your answer to 4 decimal places.
$x$ 1 1.5 2 2.5 3
$y$ 2 1.3041 0.9089 1.2958
Use the trapezium rule, with all the values of $y$ in the completed table, to obtain an estimate for the area of $S$, giving your answer to 3 decimal places.
Use calculus to find the exact area of $S$. Give your answer in the form $\frac { a } { b } + \ln c$, where $a , b$ and $c$ are integers.
Hence calculate the percentage error in using your answer to part (b) to estimate the area of $S$. Give your answer to one decimal place.
Explain how the trapezium rule could be used to obtain a more accurate estimate for the area of $S$.

Edexcel C34 2015 January Q14

14
- 6
- 13 \end{array} \right) + \lambda \left( \begin{array} { r } - 2
1
4 \end{array} \right) \quad l _ { 2 } : \mathbf { r } = \left( \begin{array} { r } p
- 7
4 \end{array} \right) + \mu \left( \begin{array} { l } q
2
1 \end{array} \right)$$ where $\lambda$ and $\mu$ are scalar 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 at point $X$, find
the value of $p$,
the coordinates of $X$. The point $A$ lies on $l _ { 1 }$ and has position vector $\left( \begin{array} { r } 6
- 2
3 \end{array} \right)$
Given that point $B$ also lies on $l _ { 1 }$ and that $A B = 2 A X$
find the two possible position vectors of $B$.
12. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{03548211-79cb-4629-b6ca-aa9dfcc77a33-21_615_732_233_605} \captionsetup{labelformat=empty} \caption{Figure 4}
\end{figure} Figure 4 shows a sketch of part of the curve $C$ with equation $$y = \frac { x ^ { 2 } \ln x } { 3 } - 2 x + 4 , \quad x > 0$$ The finite region $S$, shown shaded in Figure 4, is bounded by the curve $C$, the $x$-axis and the lines with equations $x = 1$ and $x = 3$
Complete the table below with the value of $y$ corresponding to $x = 2$. Give your answer to 4 decimal places.
$x$ 1 1.5 2 2.5 3
$y$ 2 1.3041 0.9089 1.2958
Use the trapezium rule, with all the values of $y$ in the completed table, to obtain an estimate for the area of $S$, giving your answer to 3 decimal places.
Use calculus to find the exact area of $S$. Give your answer in the form $\frac { a } { b } + \ln c$, where $a , b$ and $c$ are integers.
Hence calculate the percentage error in using your answer to part (b) to estimate the area of $S$. Give your answer to one decimal place.
Explain how the trapezium rule could be used to obtain a more accurate estimate for the area of $S$. 13. (a) Express $10 \cos \theta - 3 \sin \theta$ in the form $R \cos ( \theta + \alpha )$, where $R > 0$ and $0 < \alpha < 90 ^ { \circ }$ Give the exact value of $R$ and give the value of $\alpha$ to 2 decimal places. Alana models the height above the ground of a passenger on a Ferris wheel by the equation $$H = 12 - 10 \cos ( 30 t ) ^ { \circ } + 3 \sin ( 30 t ) ^ { \circ }$$ where the height of the passenger above the ground is $H$ metres at time $t$ minutes after the wheel starts turning.
\includegraphics[max width=\textwidth, alt={}, center]{03548211-79cb-4629-b6ca-aa9dfcc77a33-23_419_567_516_1160}
Calculate
1. the maximum value of $H$ predicted by this model,
2. the value of $t$ when this maximum first occurs. Give each answer to 2 decimal places.
Calculate the value of $t$ when the passenger is 18 m above the ground for the first time. Give your answer to 2 decimal places.
Determine the time taken for the Ferris wheel to complete two revolutions.

Questions C34 (197 questions)

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Edexcel C34 2014 January Q1

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Edexcel C34 2015 January Q10

Edexcel C34 2015 January Q11

Edexcel C34 2015 January Q12

Edexcel C34 2015 January Q14

\(x\)	1	1.5	2	2.5	3
\(y\)	2	1.3041		0.9089	1.2958

\(x\)	1	1.5	2	2.5	3
\(y\)	2	1.3041		0.9089	1.2958