Questions - Edexcel C2 - A-Level Maths

Find
1. the coordinates of the point $M$,
2. the radius of the circle $C$.
  $N$ is the point with coordinates $( 25,32 )$.
Find the length of the line $M N$. The tangent to $C$ at a point $P$ on the circle passes through point $N$.
Find the length of the line $N P$.

Edexcel C2 2013 January Q6

6. Given that $$2 \log _ { 2 } ( x + 15 ) - \log _ { 2 } x = 6$$

Show that $$x ^ { 2 } - 34 x + 225 = 0$$
Hence, or otherwise, solve the equation $$2 \log _ { 2 } ( x + 15 ) - \log _ { 2 } x = 6$$

Edexcel C2 2013 January Q7

7. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{6f31b6f1-33b5-4bca-9030-cf93760b454d-09_432_656_210_644} \captionsetup{labelformat=empty} \caption{Figure 1}

\end{figure} The triangle $X Y Z$ in Figure 1 has $X Y = 6 \mathrm {~cm} , Y Z = 9 \mathrm {~cm} , Z X = 4 \mathrm {~cm}$ and angle $Z X Y = \alpha$. The point $W$ lies on the line $X Y$. The circular arc $Z W$, in Figure 1 is a major arc of the circle with centre $X$ and radius 4 cm .

Show that, to 3 significant figures, $\alpha = 2.22$ radians.
Find the area, in $\mathrm { cm } ^ { 2 }$, of the major sector $X Z W X$. The region enclosed by the major arc $Z W$ of the circle and the lines $W Y$ and $Y Z$ is shown shaded in Figure 1. Calculate
the area of this shaded region,
the perimeter $Z W Y Z$ of this shaded region.

Edexcel C2 2013 January Q8

8. The curve $C$ has equation $y = 6 - 3 x - \frac { 4 } { x ^ { 3 } } , x \neq 0$

Use calculus to show that the curve has a turning point $P$ when $x = \sqrt { } 2$
Find the $x$-coordinate of the other turning point $Q$ on the curve.
Find $\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }$.
Hence or otherwise, state with justification, the nature of each of these turning points $P$ and $Q$.

Edexcel C2 2013 January Q9

9. $y$ \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{6f31b6f1-33b5-4bca-9030-cf93760b454d-13_895_1308_207_294} \captionsetup{labelformat=empty} \caption{Figure 2}

\end{figure} The finite region $R$, as shown in Figure 2, is bounded by the $x$-axis and the curve with equation $$y = 27 - 2 x - 9 \sqrt { } x - \frac { 16 } { x ^ { 2 } } , \quad x > 0$$ The curve crosses the $x$-axis at the points $( 1,0 )$ and $( 4,0 )$.

Complete the table below, by giving your values of $y$ to 3 decimal places.
$x$ 1 1.5 2 2.5 3 3.5 4
$y$ 0 5.866 5.210 1.856 0
Use the trapezium rule with all the values in the completed table to find an approximate value for the area of $R$, giving your answer to 2 decimal places.
Use integration to find the exact value for the area of $R$.

Edexcel C2 2014 January Q1

The first three terms in ascending powers of $x$ in the binomial expansion of $( 1 + p x ) ^ { 12 }$ are given by

$$1 + 18 x + q x ^ { 2 }$$ where $p$ and $q$ are constants.
Find the value of $p$ and the value of $q$.

Edexcel C2 2014 January Q2

2. $\mathrm { f } ( x ) = 2 x ^ { 3 } + x ^ { 2 } + a x + b$, where $a$ and $b$ are constants. Given that when $\mathrm { f } ( x )$ is divided by ( $x - 2$ ) the remainder is 25 ,

show that $2 a + b = 5$ Given also that $( x + 3 )$ is a factor of $\mathrm { f } ( x )$,
find the value of $a$ and the value of $b$.
\includegraphics[max width=\textwidth, alt={}, center]{e7043e7a-2c8f-425a-8471-f647828cc297-05_90_97_2613_1784}
\includegraphics[max width=\textwidth, alt={}, center]{e7043e7a-2c8f-425a-8471-f647828cc297-05_52_169_2709_1765}

Edexcel C2 2014 January Q3

3. The curve $C$ has equation $$y = 2 \sqrt { } x + \frac { 18 } { \sqrt { } x } - 1 , \quad x > 0$$

Find
1. $\frac { \mathrm { d } y } { \mathrm {~d} x }$
2. $\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }$
Use calculus to find the coordinates of the stationary point of $C$.
Determine whether the stationary point is a maximum or minimum, giving a reason for your answer.
\includegraphics[max width=\textwidth, alt={}, center]{e7043e7a-2c8f-425a-8471-f647828cc297-09_138_154_2597_1804}

Edexcel C2 2014 January Q4

4. The first term of a geometric series is 5 and the common ratio is 1.2 For this series find, to 1 decimal place,

1. the $20 ^ { \text {th } }$ term,
2. the sum of the first 20 terms. The sum of the first $n$ terms of the series is greater than 3000
Calculate the smallest possible value of $n$.

Edexcel C2 2014 January Q5

5. The height of water, $H$ metres, in a harbour on a particular day is given by the equation $$H = 10 + 5 \sin \left( \frac { \pi t } { 6 } \right) , \quad 0 \leqslant t < 24$$ where $t$ is the number of hours after midnight.

Show that the height of the water 1 hour after midnight is 12.5 metres.
Find, to the nearest minute, the times before midday when the height of the water is 9 metres.

Edexcel C2 2014 January Q6

6. Given that $$\log _ { x } ( 7 y + 1 ) - \log _ { x } ( 2 y ) = 1 , \quad x > 4 , \quad 0 < y < 1$$ express $y$ in terms of $x$.

Edexcel C2 2014 January Q7

7. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{e7043e7a-2c8f-425a-8471-f647828cc297-18_1109_958_214_502} \captionsetup{labelformat=empty} \caption{Figure 1}

\end{figure} Figure 1 shows a sketch of part of the curve $C$ with equation $$y = x ^ { 3 } - 6 x ^ { 2 } + 9 x + 5$$ The point $P ( 4,9 )$ lies on $C$.

Show that the normal to $C$ at the point $P$ has equation $$x + 9 y = 85$$ The region $R$, shown shaded in Figure 1, is bounded by the curve $C$, the $y$-axis and the normal to $C$ at $P$.
Showing all your working, calculate the exact area of $R$.

Edexcel C2 2014 January Q8

8. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{e7043e7a-2c8f-425a-8471-f647828cc297-22_1015_1542_267_185} \captionsetup{labelformat=empty} \caption{Figure 2}

\end{figure} Figure 2 shows a circle $C$ with centre $O$ and radius 5

Write down the cartesian equation of $C$. The points $P ( - 3 , - 4 )$ and $Q ( 3 , - 4 )$ lie on $C$.
Show that the tangent to $C$ at the point $Q$ has equation $$3 x - 4 y = 25$$
Show that, to 3 decimal places, angle $P O Q$ is 1.287 radians. The tangent to $C$ at $P$ and the tangent to $C$ at $Q$ intersect on the $y$-axis at the point $R$.
Find the area of the shaded region $P Q R$ shown in Figure 2.
\includegraphics[max width=\textwidth, alt={}, center]{e7043e7a-2c8f-425a-8471-f647828cc297-25_177_154_2576_1804}

Edexcel C2 2014 January Q9

9. (a) Show that the equation $$5 \sin x - \cos ^ { 2 } x + 2 \sin ^ { 2 } x = 1$$ can be written in the form $$3 \sin ^ { 2 } x + 5 \sin x - 2 = 0$$ (b) Hence solve, for $- 180 ^ { \circ } \leqslant \theta < 180 ^ { \circ }$, the equation $$5 \sin 2 \theta - \cos ^ { 2 } 2 \theta + 2 \sin ^ { 2 } 2 \theta = 1$$ giving your answers to 2 decimal places.

Edexcel C2 2005 June Q1

Find the coordinates of the stationary point on the curve with equation $y = 2 x ^ { 2 } - 12 x$.

Edexcel C2 2005 June Q5

5. Solve, for $0 \leqslant x \leqslant 180 ^ { \circ }$, the equation

$\quad \sin \left( x + 10 ^ { \circ } \right) = \frac { \sqrt { } 3 } { 2 }$,
$\cos 2 x = - 0.9$, giving your answers to 1 decimal place.

Edexcel C2 2005 June Q6

6. A river, running between parallel banks, is 20 m wide. The depth, $y$ metres, of the river measured at a point $x$ metres from one bank is given by the formula $$y = \frac { 1 } { 10 } x \sqrt { } ( 20 - x ) , \quad 0 \leqslant x \leqslant 20$$

Complete the table below, giving values of $y$ to 3 decimal places.
$x$ 0 4 8 12 16 20
$y$ 0 2.771 0
Use the trapezium rule with all the values in the table to estimate the cross-sectional area of the river. Given that the cross-sectional area is constant and that the river is flowing uniformly at $2 \mathrm {~ms} ^ { - 1 }$,
estimate, in $\mathrm { m } ^ { 3 }$, the volume of water flowing per minute, giving your answer to 3 significant figures.

Edexcel C2 2005 June Q7

7. In the triangle $A B C , A B = 8 \mathrm {~cm} , A C = 7 \mathrm {~cm} , \angle A B C = 0.5$ radians and $\angle A C B = x$ radians.

Use the sine rule to find the value of $\sin x$, giving your answer to 3 decimal places. Given that there are two possible values of $x$,
find these values of $x$, giving your answers to 2 decimal places.

Edexcel C2 2005 June Q8

8. The circle $C$, with centre at the point $A$, has equation $x ^ { 2 } + y ^ { 2 } - 10 x + 9 = 0$. Find

the coordinates of $A$,
the radius of $C$,
the coordinates of the points at which $C$ crosses the $x$-axis. Given that the line $l$ with gradient $\frac { 7 } { 2 }$ is a tangent to $C$, and that $l$ touches $C$ at the point $T$,
find an equation of the line which passes through $A$ and $T$.