Questions - Edexcel - A-Level Maths

Use the factor theorem to show that $( x + 2 )$ is a factor of $\mathrm { f } ( x )$.
Factorise $\mathrm { f } ( x )$ completely.
Write down all the solutions to the equation $$x ^ { 3 } + 4 x ^ { 2 } + x - 6 = 0$$

Edexcel C2 2007 January Q6

6 marks Standard +0.3

6. Find all the solutions, in the interval $0 \leqslant x < 2 \pi$, of the equation $$2 \cos ^ { 2 } x + 1 = 5 \sin x$$ giving each solution in terms of $\pi$.

Edexcel C2 2007 January Q7

9 marks Standard +0.3

7. \begin{figure}[h]

\captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{872356ab-68d3-43ee-8b76-650a2697d80e-08_1052_1116_351_413}

\end{figure} Figure 1 shows a sketch of part of the curve $C$ with equation $$y = x ( x - 1 ) ( x - 5 )$$ Use calculus to find the total area of the finite region, shown shaded in Figure 1, that is between $x = 0$ and $x = 2$ and is bounded by $C$, the $x$-axis and the line $x = 2$.
(9)

Edexcel C2 2007 January Q8

9 marks Moderate -0.3

A diesel lorry is driven from Birmingham to Bury at a steady speed of v kilometres per hour. The total cost of the journey, $\pounds C$, is given by

$$C = \frac { 1400 } { v } + \frac { 2 v } { 7 } .$$

Find the value of $v$ for which $C$ is a minimum.
Find $\frac { \mathrm { d } ^ { 2 } C } { \mathrm {~d} v ^ { 2 } }$ and hence verify that $C$ is a minimum for this value of $v$.
Calculate the minimum total cost of the journey.

Edexcel C2 2007 January Q9

11 marks Standard +0.3

9. \begin{figure}[h]

\captionsetup{labelformat=empty} \caption{Figure 2} \includegraphics[alt={},max width=\textwidth]{872356ab-68d3-43ee-8b76-650a2697d80e-11_627_965_338_502}

\end{figure} Figure 2 shows a plan of a patio. The patio $P Q R S$ is in the shape of a sector of a circle with centre $Q$ and radius 6 m . Given that the length of the straight line $P R$ is $6 \sqrt { } 3 \mathrm {~m}$,

find the exact size of angle $P Q R$ in radians.
Show that the area of the patio $P Q R S$ is $12 \pi \mathrm {~m} ^ { 2 }$.
Find the exact area of the triangle $P Q R$.
Find, in $\mathrm { m } ^ { 2 }$ to 1 decimal place, the area of the segment $P R S$.
Find, in $m$ to 1 decimal place, the perimeter of the patio $P Q R S$.

Edexcel C2 2007 January Q10

11 marks Moderate -0.8

A geometric series is $a + a r + a r ^ { 2 } + \ldots$
1. Prove that the sum of the first $n$ terms of this series is given by
$$S _ { n } = \frac { a \left( 1 - r ^ { n } \right) } { 1 - r }$$
Find $$\sum _ { k = 1 } ^ { 10 } 100 \left( 2 ^ { k } \right)$$
Find the sum to infinity of the geometric series $$\frac { 5 } { 6 } + \frac { 5 } { 18 } + \frac { 5 } { 54 } + \ldots$$
State the condition for an infinite geometric series with common ratio $r$ to be convergent.

Edexcel C2 2008 January Q1

7 marks Moderate -0.8

(a) Find the remainder when

$$x ^ { 3 } - 2 x ^ { 2 } - 4 x + 8$$ is divided by

$x - 3$,
$x + 2$.
(b) Hence, or otherwise, find all the solutions to the equation $$x ^ { 3 } - 2 x ^ { 2 } - 4 x + 8 = 0$$

Edexcel C2 2008 January Q2

6 marks Moderate -0.3

2. The fourth term of a geometric series is 10 and the seventh term of the series is 80 . For this series, find

the common ratio,
the first term,
the sum of the first 20 terms, giving your answer to the nearest whole number.

Edexcel C2 2008 January Q3

7 marks Moderate -0.8

3. (a) Find the first 4 terms of the expansion of $\left( 1 + \frac { x } { 2 } \right) ^ { 10 }$ in ascending powers of $x$, giving
each term in its simplest form. each term in its simplest form.
(b) Use your expansion to estimate the value of $( 1.005 ) ^ { 10 }$, giving your answer to 5 decimal places.

Edexcel C2 2008 January Q4

9 marks Moderate -0.8

4. (a) Show that the equation $$3 \sin ^ { 2 } \theta - 2 \cos ^ { 2 } \theta = 1$$ can be written as $$5 \sin ^ { 2 } \theta = 3$$ (b) Hence solve, for $0 ^ { \circ } \leqslant \theta < 360 ^ { \circ }$, the equation $$3 \sin ^ { 2 } \theta - 2 \cos ^ { 2 } \theta = 1$$ giving your answers to 1 decimal place.

Edexcel C2 2008 January Q5

6 marks Moderate -0.8

Given that $a$ and $b$ are positive constants, solve the simultaneous equations

$$\begin{gathered} a = 3 b , \\ \log _ { 3 } a + \log _ { 3 } b = 2 . \end{gathered}$$ Give your answers as exact numbers. \section*{6.} \begin{figure}[h]

\captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{13c5a854-baea-4875-82bc-86a19c3be09c-08_687_454_294_703}

\end{figure} Figure 1 shows 3 yachts $A , B$ and $C$ which are assumed to be in the same horizontal plane. Yacht $B$ is 500 m due north of yacht $A$ and yacht $C$ is 700 m from $A$. The bearing of $C$ from $A$ is $015 ^ { \circ }$.

Calculate the distance between yacht $B$ and yacht $C$, in metres to 3 significant figures. The bearing of yacht $C$ from yacht $B$ is $\theta ^ { \circ }$, as shown in Figure 1.
Calculate the value of $\theta$.

Edexcel C2 2008 January Q7

10 marks Moderate -0.8

7. \begin{figure}[h]

\captionsetup{labelformat=empty} \caption{Figure 2} \includegraphics[alt={},max width=\textwidth]{13c5a854-baea-4875-82bc-86a19c3be09c-10_691_995_267_477}

\end{figure} In Figure 2 the curve $C$ has equation $y = 6 x - x ^ { 2 }$ and the line $L$ has equation $y = 2 x$.

Show that the curve $C$ intersects the $x$-axis at $x = 0$ and $x = 6$.
Show that the line $L$ intersects the curve $C$ at the points $( 0,0 )$ and $( 4,8 )$. The region $R$, bounded by the curve $C$ and the line $L$, is shown shaded in Figure 2.
Use calculus to find the area of $R$.

Edexcel C2 2008 January Q8

11 marks Moderate -0.3

A circle $C$ has centre $M ( 6,4 )$ and radius 3 .
1. Write down the equation of the circle in the form
$$( x - a ) ^ { 2 } + ( y - b ) ^ { 2 } = r ^ { 2 }$$ \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 3} \includegraphics[alt={},max width=\textwidth]{13c5a854-baea-4875-82bc-86a19c3be09c-12_833_1276_605_322}
\end{figure} Figure 3 shows the circle $C$. The point $T$ lies on the circle and the tangent at $T$ passes through the point $P ( 12,6 )$. The line $M P$ cuts the circle at $Q$.
Show that the angle $T M Q$ is 1.0766 radians to 4 decimal places. The shaded region $T P Q$ is bounded by the straight lines $T P , Q P$ and the arc $T Q$, as shown in Figure 3.
Find the area of the shaded region $T P Q$. Give your answer to 3 decimal places. \section*{9.} \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 4} \includegraphics[alt={},max width=\textwidth]{13c5a854-baea-4875-82bc-86a19c3be09c-14_675_844_283_534}
\end{figure} Figure 4 shows an open-topped water tank, in the shape of a cuboid, which is made of sheet metal. The base of the tank is a rectangle $x$ metres by $y$ metres. The height of the tank is $x$ metres. The capacity of the tank is $100 \mathrm {~m} ^ { 3 }$.
Show that the area $A \mathrm {~m} ^ { 2 }$ of the sheet metal used to make the tank is given by $$A = \frac { 300 } { x } + 2 x ^ { 2 }$$
Use calculus to find the value of $x$ for which $A$ is stationary.
Prove that this value of $x$ gives a minimum value of $A$.
Calculate the minimum area of sheet metal needed to make the tank.

Edexcel C2 2009 January Q1

4 marks Easy -1.2

Find the first 3 terms, in ascending powers of $x$, of the binomial expansion of $( 3 - 2 x ) ^ { 5 }$, giving each term in its simplest form.
(4)

Edexcel C2 2009 January Q3

6 marks Moderate -0.8

3. $y = \sqrt { } \left( 10 x - x ^ { 2 } \right)$.

Complete the table below, giving the values of $y$ to 2 decimal places.
$x$ 1 1.4 1.8 2.2 2.6 3
$y$ 3 3.47 4.39
Use the trapezium rule, with all the values of $y$ from your table, to find an approximation for the value of $\int _ { 1 } ^ { 3 } \sqrt { } \left( 10 x - x ^ { 2 } \right) \mathrm { d } x$.