Questions - Edexcel C2 - A-Level Maths

express $h$ in terms of $x$,
show that the surface area, $A \mathrm {~cm} ^ { 2 }$, of a carton is given by $A = 4 x ^ { 2 } + \frac { 3090 } { x }$. The manufacturer needs to minimise the surface area of a carton.
Use calculus to find the value of $x$ for which $A$ is a minimum.
Calculate the minimum value of $A$.
Prove that this value of $A$ is a minimum.

Edexcel C2 Q1

Given that $p = \log _ { q } 16$, express in terms of $p$,
1. $\log _ { q } 2$,
2. $\log _ { q } ( 8 q )$.
  [0pt] [P2 January 2002 Question 2]
3. $\mathrm { f } ( x ) = x ^ { 3 } - x ^ { 2 } - 7 x + c$, where $c$ is a constant.
Given that $\mathrm { f } ( 4 ) = 0$,
find the value of $c$,
factorise $\mathrm { f } ( x )$ as the product of a linear factor and a quadratic factor.
Hence show that, apart from $x = 4$, there are no real values of $x$ for which $\mathrm { f } ( x ) = 0$.

Edexcel C2 Q3

3. Find the values of $\theta$, to 1 decimal place, in the interval $- 180 \leq \theta < 180$ for which $$2 \sin ^ { 2 } \theta ^ { \circ } - 2 \sin \theta ^ { \circ } = \cos ^ { 2 } \theta ^ { \circ }$$ [P1 January 2002 Question 3]

Edexcel C2 Q4

4. A population of deer is introduced into a park. The population $P$ at $t$ years after the deer have been introduced is modelled by $P = \frac { 2000 a ^ { t } } { 4 + a ^ { t } }$, where $a$ is a constant. Given that there are 800 deer in the park after 6 years,

calculate, to 4 decimal places, the value of $a$,
use the model to predict the number of years needed for the population of deer to increase from 800 to 1800.
With reference to this model, give a reason why the population of deer cannot exceed 2000.

Edexcel C2 Q5

5. (a) Given that $( 2 + x ) ^ { 5 } + ( 2 - x ) ^ { 5 } = A + B x ^ { 2 } + C x ^ { 4 }$, find the values of the constants $A , B$ and $C$.
(b) Using the substitution $y = x ^ { 2 }$ and your answers to part (a), solve, $$( 2 + x ) ^ { 5 } + ( 2 - x ) ^ { 5 } = 349$$

Edexcel C2 Q6

6. \begin{figure}[h]

\captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{1033051d-18bf-4734-a556-4c8e1c789992-3_842_963_280_392}

\end{figure} Fig. 1 shows a gardener's design for the shape of a flower bed with perimeter $A B C D . A D$ is an arc of a circle with centre $O$ and radius $5 \mathrm {~m} . B C$ is an arc of a circle with centre $O$ and radius $7 \mathrm {~m} . O A B$ and $O D C$ are straight lines and the size of $\angle A O D$ is $\theta$ radians.

Find, in terms of $\theta$, an expression for the area of the flower bed. Given that the area of the flower bed is $15 \mathrm {~m} ^ { 2 }$,
show that $\theta = 1.25$,
calculate, in m , the perimeter of the flower bed. The gardener now decides to replace arc $A D$ with the straight line $A D$.
Find, to the nearest cm , the reduction in the perimeter of the flower bed.

Edexcel C2 Q7

7. A geometric series is $a + a r + a r ^ { 2 } + \ldots$

Prove that the sum of the first $n$ terms of this series is given by $S _ { n } = \frac { a \left( 1 - r ^ { n } \right) } { 1 - r }$. The second and fourth terms of the series are 3 and 1.08 respectively.
Given that all terms in the series are positive, find
the value of $r$ and the value of $a$,
the sum to infinity of the series.
\includegraphics[max width=\textwidth, alt={}, center]{1033051d-18bf-4734-a556-4c8e1c789992-4_764_1159_294_299} Fig. 2 shows part of the curve with equation $y = x ^ { 3 } - 6 x ^ { 2 } + 9 x$. The curve touches the $x$-axis at $A$ and has a maximum turning point at $B$.
Show that the equation of the curve may be written as $y = x ( x - 3 ) ^ { 2 }$, and hence write down the coordinates of $A$.
Find the coordinates of $B$. The shaded region $R$ is bounded by the curve and the $x$-axis.
Find the area of $R$.

Edexcel C2 Q1

Find the remainder when $\mathrm { f } ( x ) = 4 x ^ { 3 } + 3 x ^ { 2 } - 2 x - 6$ is divided by $( 2 x + 1 )$.
Given that $2 \sin 2 \theta = \cos 2 \theta$,
1. show that $\tan 2 \theta = 0.5$.
2. Hence find the values of $\theta$, to one decimal place, in the interval $0 \leq \theta < 360$ for which $2 \sin 2 \theta ^ { \circ } = \cos 2 \theta ^ { \circ }$.
  [0pt] [P1 June 2001 Question 2]
3. (a) Using the substitution $u = 2 ^ { x }$, show that the equation $4 ^ { x } - 2 ^ { ( x + 1 ) } - 15 = 0$ can be written in the form $u ^ { 2 } - 2 u - 15 = 0$.
4. Hence solve the equation $4 ^ { x } - 2 ^ { ( x + 1 ) } - 15 = 0$, giving your answers to 2 d . p.
  [0pt] [P2 November 2002 Question 2]
\begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{1425d933-47e3-4a12-bcab-fd2ca41827e2-2_558_643_1101_609}
\end{figure} The shape of a badge is a sector $A B C$ of a circle with centre $A$ and radius $A B$, as shown in Fig 1. The triangle $A B C$ is equilateral and has a perpendicular height 3 cm .

Edexcel C2 Q5

5. A circle $C$ has centre $( 3,4 )$ and radius $3 \sqrt { } 2$. A straight line $l$ has equation $y = x + 3$.

Write down an equation of the circle $C$.
Calculate the exact coordinates of the two points where the line $l$ intersects $C$, giving your answers in surds.
Find the distance between these two points.

Edexcel C2 Q6

6. The sequence $u _ { 1 } , u _ { 2 } , u _ { 3 } , \ldots , u _ { n }$ is defined by the recurrence relation $$u _ { n + 1 } = p u _ { n } + 5 , u _ { 1 } = 2 , \text { where } p \text { is a constant. }$$ Given that $u _ { 3 } = 8$,

show that one possible value of $p$ is $\frac { 1 } { 2 }$ and find the other value of $p$. Using $p = \frac { 1 } { 2 }$,
write down the value of $\log _ { 2 } p$. Given also that $\log _ { 2 } q = t$,
express $\log _ { 2 } \left( \frac { p ^ { 3 } } { \sqrt { q } } \right)$ in terms of $t$.
[0pt] [P2 November 2002 Question 4]

Edexcel C2 Q7

7. \begin{figure}[h]

\captionsetup{labelformat=empty} \caption{Figure 2} \includegraphics[alt={},max width=\textwidth]{1425d933-47e3-4a12-bcab-fd2ca41827e2-3_531_1216_998_189}

\end{figure} Fig. 2 shows the line with equation $y = x + 1$ and the curve with equation $y = 6 x - x ^ { 2 } - 3$.
The line and the curve intersect at the points $A$ and $B$, and $O$ is the origin.

Calculate the coordinates of $A$ and the coordinates of $B$. The shaded region $R$ is bounded by the line and the curve.
Calculate the area of $R$.

Edexcel C2 Q8

8. $$\mathrm { f } ( x ) = \left( 1 + \frac { x } { k } \right) ^ { n } , \quad k , n \in \mathbb { N } , \quad n > 2 .$$ Given that the coefficient of $x ^ { 3 }$ is twice the coefficient of $x ^ { 2 }$ in the binomial expansion of $\mathrm { f } ( x )$,

prove that $n = 6 k + 2$. Given also that the coefficients of $x ^ { 4 }$ and $x ^ { 5 }$ are equal and non-zero,
form another equation in $n$ and $k$ and hence show that $k = 2$ and $n = 14$. Using these values of $k$ and $n$,
expand $\mathrm { f } ( x )$ in ascending powers of $x$, up to and including the term in $x ^ { 5 }$. Give each coefficient as an exact fraction in its lowest terms

Edexcel C2 Q9

9. \begin{figure}[h]

\captionsetup{labelformat=empty} \caption{Figure 3} \includegraphics[alt={},max width=\textwidth]{1425d933-47e3-4a12-bcab-fd2ca41827e2-4_799_1299_303_285}

\end{figure}

\includegraphics[max width=\textwidth, alt={}]{1425d933-47e3-4a12-bcab-fd2ca41827e2-4_303_1127_1144_338}

A rectangular sheet of metal measures 50 cm by 40 cm . Squares of side $x \mathrm {~cm}$ are cut from each corner of the sheet and the remainder is folded along the dotted lines to make an open tray, as shown in Fig. 3.

Show that the volume, $V \mathrm {~cm} ^ { 3 }$, of the tray is given by $V = 4 x \left( x ^ { 2 } - 45 x + 500 \right)$.
State the range of possible values of $x$.
Find the value of $x$ for which $V$ is a maximum.
Hence find the maximum value of $V$.
Justify that the value of $V$ you found in part (d) is a maximum.

Edexcel C2 Q1

(a) Write down the first four terms of the binomial expansion, in ascending powers of $x$, of $( 1 + 3 x ) ^ { n }$, where $n > 2$.

Given that the coefficient of $x ^ { 3 }$ in this expansion is ten times the coefficient of $x ^ { 2 }$,
(b) find the value of $n$,
(c) find the coefficient of $x ^ { 4 }$ in the expansion.

Edexcel C2 Q2

2. $\mathrm { f } ( x ) = x ^ { 3 } + a x ^ { 2 } + b x - 10$, where $a$ and $b$ are constants. When $\mathrm { f } ( x )$ is divided by $( x - 3 )$, the remainder is 14 . When $\mathrm { f } ( x )$ is divided by $( x + 1 )$, the remainder is - 18 .

Find the value of $a$ and the value of $b$.
Show that $( x - 2 )$ is a factor of $\mathrm { f } ( x )$.
[0pt] [P3 June 2002 Question 1]

Edexcel C2 Q3

3. Given that $\mathrm { f } ( x ) = 15 - 7 x - 2 x ^ { 2 }$,

find the coordinates of all points at which the graph of $y = \mathrm { f } ( x )$ crosses the coordinate axes.
Sketch the graph of $y = \mathrm { f } ( x )$.
Calculate the coordinates of the stationary point of $\mathrm { f } ( x )$.
[0pt] [P1 June 2002 Question 3]

Edexcel C2 Q4

4. $$\mathrm { f } ( x ) = 5 \sin 3 x ^ { \circ } , \quad 0 \leq x \leq 180$$

Sketch the graph of $\mathrm { f } ( x )$, indicating the value of $x$ at each point where the graph intersects the $x$ axis.
Write down the coordinates of all the maximum and minimum points of $\mathrm { f } ( x )$.
Calculate the values of $x$ for which $\mathrm { f } ( x ) = 2.5$
[0pt] [P1 June 2002 Question 5]

Edexcel C2 Q5

5. (a) Given that $3 + 2 \log _ { 2 } x = \log _ { 2 } y$, show that $y = 8 x ^ { 2 }$.
(b) Hence, or otherwise, find the roots $\alpha$ and $\beta$, where $\alpha < \beta$, of the equation $$3 + 2 \log _ { 2 } x = \log _ { 2 } ( 14 x - 3 )$$ (c) Show that $\log _ { 2 } \alpha = - 2$.
(d) Calculate $\log _ { 2 } \beta$, giving your answer to 3 significant figures.

Edexcel C2 Q6

6. Given that $\mathrm { f } ( x ) = \left( 2 x ^ { \frac { 3 } { 2 } } - 3 x ^ { - \frac { 3 } { 2 } } \right) ^ { 2 } + 5 , x > 0$,

find, to 3 significant figures, the value of $x$ for which $\mathrm { f } ( x ) = 5$.
Show that $\mathrm { f } ( x )$ may be written in the form $A x ^ { 3 } + \frac { B } { x ^ { 3 } } + C$, where $A , B$ and $C$ are constants to be found.
Hence evaluate $\int _ { 1 } ^ { 2 } f ( x ) d x$.

Edexcel C2 Q7

7. \begin{figure}[h]

\captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{ba5cb933-dedd-4ad9-9e66-49636870b3de-3_739_1272_826_328}

\end{figure} Fig. 1 shows the cross-section $A B C D$ of a chocolate bar, where $A B , C D$ and $A D$ are straight lines and $M$ is the mid-point of $A D$. The length $A D$ is 28 mm , and $B C$ is an arc of a circle with centre $M$. Taking $A$ as the origin, $B , C$ and $D$ have coordinates (7,24), (21,24) and (28,0) respectively.

Show that the length of $B M$ is 25 mm .
Show that, to 3 significant figures, $\angle B M C = 0.568$ radians.
Hence calculate, in $\mathrm { mm } ^ { 2 }$, the area of the cross-section of the chocolate bar. Given that this chocolate bar has length 85 mm ,
calculate, to the nearest $\mathrm { cm } ^ { 3 }$, the volume of the bar.

Edexcel C2 Q8

8. (a) An arithmetic series has first term $a$ and common difference $d$. Prove that the sum of the first $n$ terms of the series is $\frac { 1 } { 2 } n [ 2 a + ( n - 1 ) d ]$. A company made a profit of $\pounds 54000$ in the year 2001. A model for future performance assumes that yearly profits will increase in an arithmetic sequence with common difference $\pounds d$. This model predicts total profits of $\pounds 619200$ for the 9 years 2001 to 2009 inclusive.
(b) Find the value of $d$. Using your value of $d$,
(c) find the predicted profit for the year 2011. An alternative model assumes that the company's yearly profits will increase in a geometric sequence with common ratio 1.06 . Using this alternative model and again taking the profit in 2001 to be $\pounds 54000$,
(d) find the predicted profit for the year 2011.

Edexcel C2 Q1

\begin{enumerate} \item (a) Using the factor theorem, show that $( x + 3 )$ is a factor of $x ^ { 3 } - 3 x ^ { 2 } - 10 x + 24$.
(b) Factorise $x ^ { 3 } - 3 x ^ { 2 } - 10 x + 24$ completely. \item (a) Expand $( 2 \sqrt { } x + 3 ) ^ { 2 }$.
(b) Hence evaluate $\int _ { 1 } ^ { 2 } ( 2 \sqrt { } x + 3 ) ^ { 2 } \mathrm {~d} x$, giving your answer in the form $a + b \sqrt { } 2$, where $a$ and $b$ are integers. \item The first three terms in the expansion, in ascending powers of $x$, of $( 1 + p x ) ^ { n }$, are $1 - 18 x + 36 p ^ { 2 } x ^ { 2 }$. Given that $n$ is a positive integer, find the value of $n$ and the value of $p$.
[0pt] [P2 January 2003 Question 2] \item A circle $C$ has equation $x ^ { 2 } + y ^ { 2 } - 6 x + 8 y - 75 = 0$.

Edexcel C2 Q5

5. (i) Differentiate $2 x ^ { 3 } + \sqrt { } x + \frac { x ^ { 2 } + 2 x } { x ^ { 2 } }$ with respect to $x$
(ii) Evaluate $\int _ { 1 } ^ { 4 } \left( \frac { x } { 2 } + \frac { 1 } { x ^ { 2 } } \right) \mathrm { d } x$.

Edexcel C2 Q6

6. A geometric series has first term 1200. Its sum to infinity is 960 .

Show that the common ratio of the series is $- \frac { 1 } { 4 }$.
Find, to 3 decimal places, the difference between the ninth and tenth terms of the series.
Write down an expression for the sum of the first $n$ terms of the series. Given that $n$ is odd,
prove that the sum of the first $n$ terms of the series is $960 \left( 1 + 0.25 ^ { n } \right)$.

Edexcel C2 Q7

7. On a journey, the average speed of a car is $v \mathrm {~m} \mathrm {~s} ^ { - 1 }$. For $v \geq 5$, the cost per kilometre, $C$ pence, of the journey is modelled by $C = \frac { 160 } { v } + \frac { v ^ { 2 } } { 100 }$.
Using this model,

show, by calculus, that there is a value of $v$ for which $C$ has a stationary value, and find this value of $v$.
Justify that this value of $v$ gives a minimum value of $C$.
Find the minimum value of $C$ and hence find the minimum cost of a 250 km car journey.

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