Questions C2 - A-Level Maths

Edexcel C2 Q1

4 marks Easy -1.2

A circle $C$ has equation

$$x ^ { 2 } + y ^ { 2 } - 10 x + 6 y - 15 = 0 .$$

Find the coordinates of the centre of $C$.
Find the radius of $C$.

Edexcel C2 Q2

5 marks Moderate -0.5

2. Express $\frac { y + 3 } { ( y + 1 ) ( y + 2 ) } - \frac { y + 1 } { ( y + 2 ) ( y + 3 ) }$ as a single fraction in its simplest form.

Edexcel C2 Q7

10 marks Moderate -0.3

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A circle $C$ has equation

$$x ^ { 2 } + y ^ { 2 } - 10 x + 6 y - 15 = 0 .$$

Find the coordinates of the centre of $C$.
Find the radius of $C$.
2. Express $\frac { y + 3 } { ( y + 1 ) ( y + 2 ) } - \frac { y + 1 } { ( y + 2 ) ( y + 3 ) }$ as a single fraction in its simplest form.
3. Given that $2 \sin 2 \theta = \cos 2 \theta$,
show that $\tan 2 \theta = 0.5$.
Hence find the values of $\theta$, to one decimal place, in the interval $0 \leq \theta < 360$ for which $2 \sin 2 \theta ^ { \circ } = \cos 2 \theta ^ { \circ }$.
4. $\mathrm { f } ( x ) = x ^ { 3 } - x ^ { 2 } - 7 x + c$, 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.
(c Hence show that, apart from $x = 4$, there are no real values of $x$ for which $\mathrm { f } ( x ) = 0$.
5. \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{13c2bf9f-f87a-420c-8cdc-9deb688112ae-3_538_618_283_749}
\end{figure} Figure 1 shows the sector $O A B$ of a circle of radius $r \mathrm {~cm}$. The area of the sector is $15 \mathrm {~cm} ^ { 2 }$ and $\angle A O B = 1.5$ radians.
Prove that $r = 2 \sqrt { } 5$.
Find, in cm , the perimeter of the sector $O A B$. The segment $R$, shaded in Fig 1, is enclosed by the arc $A B$ and the straight line $A B$.
Calculate, to 3 decimal places, the area of $R$.
6. The third and fourth terms of a geometric series are 6.4 and 5.12 respectively. Find
the common ratio of the series,
the first term of the series,
the sum to infinity of the series.
Calculate the difference between the sum to infinity of the series and the sum of the first 25 terms of the series.
7. $$\mathrm { f } ( x ) = 5 \sin 3 x ^ { \circ } , \quad 0 \leq x \leq 180 .$$
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 $\mathrm { f } ( x ) = 2.5$

Edexcel C2 Q8

13 marks Standard +0.3

8. (i) Solve, for $0 ^ { \circ } < x < 180 ^ { \circ }$, the equation $$\sin \left( 2 x + 50 ^ { \circ } \right) = 0.6$$ giving your answers to 1 decimal place.
(ii) In the triangle $A B C , A C = 18 \mathrm {~cm} , \angle A B C = 60 ^ { \circ }$ and $\sin A = \frac { 1 } { 3 }$.
(a Use the sine rule to show that $B C = 4 \sqrt { } 3$.
(b) Find the exact value of $\cos A$.

Edexcel C2 Q9

12 marks Moderate -0.3

9. \begin{figure}[h]

\captionsetup{labelformat=empty} \caption{Figure 2} \includegraphics[alt={},max width=\textwidth]{13c2bf9f-f87a-420c-8cdc-9deb688112ae-5_965_1120_324_356}

\end{figure} Figure 2 shows the line with equation $y = 9 - x$ and the curve with equation $y = x ^ { 2 } - 2 x + 3$. 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 Q1

3 marks Moderate -0.8

1. $$f ( x ) = 4 x ^ { 3 } + 3 x ^ { 2 } - 2 x - 6$$ Find the remainder when $\mathrm { f } ( x )$ is divided by ( $2 x + 1$ ).

Edexcel C2 Q2

4 marks Easy -1.2

2. The point $A$ has coordinates $( 2,5 )$ and the point $B$ has coordinates $( - 2,8 )$. Find, in cartesian form, an equation of the circle with diameter $A B$.

Edexcel C2 Q7

12 marks Easy -1.3

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\hline \end{tabular} \end{center} 1. $$f ( x ) = 4 x ^ { 3 } + 3 x ^ { 2 } - 2 x - 6$$ Find the remainder when $\mathrm { f } ( x )$ is divided by ( $2 x + 1$ ).
2. The point $A$ has coordinates $( 2,5 )$ and the point $B$ has coordinates $( - 2,8 )$. Find, in cartesian form, an equation of the circle with diameter $A B$.
3. $$f ( x ) = x ^ { 3 } - 19 x - 30$$

Show that $( x + 2 )$ is a factor of $\mathrm { f } ( x )$.
Factorise $\mathrm { f } ( x )$ completely.
4. Express $\frac { 3 } { x ^ { 2 } + 2 x } + \frac { x - 4 } { x ^ { 2 } - 4 }$ as a single fraction in its simplest form.
5. Find, in degrees, the value of $\theta$ in the interval $0 \leq \theta < 360 ^ { \circ }$ for which $$2 \cos ^ { 2 } \theta - \cos \theta - 1 = \sin ^ { 2 } \theta$$ Give your answers to 1 decimal place where appropriate.
6. 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.
7. . \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 2} \includegraphics[alt={},max width=\textwidth]{8d60bedd-6496-4fc9-abc8-16fc5ac52e01-3_1141_1297_280_360}
\end{figure} 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. 2.
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. \section*{8.} \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{8d60bedd-6496-4fc9-abc8-16fc5ac52e01-4_556_554_276_840}
\end{figure} Figure 1 shows the sector $A O B$ of a circle, with centre $O$ and radius 6.5 cm , and $\angle A O B = 0.8$ radians.
Calculate, in $\mathrm { cm } ^ { 2 }$, the area of the sector $A O B$.
Show that the length of the chord $A B$ is 5.06 cm , to 3 significant figures. The segment $R$, shaded in Fig. 1, is enclosed by the arc $A B$ and the straight line $A B$.
Calculate, in cm , the perimeter of $R$. \section*{9.} \section*{Figure 2}
\includegraphics[max width=\textwidth, alt={}]{8d60bedd-6496-4fc9-abc8-16fc5ac52e01-5_529_1205_324_269}
Figure 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 Q1

4 marks Easy -1.2

A circle $C$ has equation $x ^ { 2 } + y ^ { 2 } - 10 x + 6 y - 15 = 0$.
1. Find the coordinates of the centre of $C$.
2. Find the radius of $C$.
  [0pt] [P3 June 2001 Question 1]
3. $\mathrm { f } ( x ) \equiv a x ^ { 3 } + b x ^ { 2 } - 7 x + 14$, where $a$ and $b$ are constants.
Given that when $\mathrm { f } ( x )$ is divided by $( x - 1 )$ the remainder is 9 ,
write down an equation connecting $a$ and $b$. Given also that $( x + 2 )$ is a factor of $\mathrm { f } ( x )$,
find the values of $a$ and $b$.
[0pt] [P3 June 2001 Question 2]

Edexcel C2 Q3

8 marks Moderate -0.8

3. Find all values of $\theta$ in the interval $0 \leq \theta < 360$ for which

$\cos ( \theta + 75 ) ^ { \circ } = 0$.
$\sin 2 \theta ^ { \circ } = 0.7$, giving your answers to one decima1 place.

Edexcel C2 Q4

9 marks Moderate -0.5

4.
\includegraphics[max width=\textwidth, alt={}, center]{ffa0b566-6448-491b-96d7-d3806bcfe063-2_639_1408_1315_212} Fig. 1 shows the curve with equation $y = 5 + 2 x - x ^ { 2 }$ and the line with equation $y = 2$. The curve and the line intersect at the points $A$ and $B$.

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

Edexcel C2 Q5

10 marks Moderate -0.8

5. The third and fourth terms of a geometric series are 6.4 and 5.12 respectively. Find

the common ratio of the series,
the first term of the series,
the sum to infinity of the series.
Calculate the difference between the sum to infinity of the series and the sum of the first 25 terms of the series.

Edexcel C2 Q6

12 marks Moderate -0.3

6.
\includegraphics[max width=\textwidth, alt={}, center]{ffa0b566-6448-491b-96d7-d3806bcfe063-3_684_1237_685_239} Triangle $A B C$ has $A B = 9 \mathrm {~cm} , B C 10 \mathrm {~cm}$ and $C A = 5 \mathrm {~cm}$. A circle, centre $A$ and radius 3 cm , intersects $A B$ and $A C$ at $P$ and $Q$ respectively, as shown in Fig. 2.

Show that, to 3 decimal places, $\angle B A C = 1.504$ radians. Calculate,
the area, in $\mathrm { cm } ^ { 2 }$, of the sector $A P Q$,
the area, in $\mathrm { cm } ^ { 2 }$, of the shaded region $B P Q C$,
the perimeter, in cm , of the shaded region $B P Q C$.

Edexcel C2 Q7

12 marks Moderate -0.3

7. \begin{figure}[h]

\captionsetup{labelformat=empty} \caption{Figure 3} \includegraphics[alt={},max width=\textwidth]{ffa0b566-6448-491b-96d7-d3806bcfe063-4_556_497_294_342}

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

\includegraphics[alt={},max width=\textwidth]{ffa0b566-6448-491b-96d7-d3806bcfe063-4_549_471_251_1021} \captionsetup{labelformat=empty} \caption{Shape $Y$}

\end{figure} Fig. 3 shows the cross-sections of two drawer handles. Shape $X$ is a rectangle $A B C D$ joined to a semicircle with $B C$ as diameter. The length $A B = d \mathrm {~cm}$ and $B C = 2 d \mathrm {~cm}$. Shape $Y$ is a sector $O P Q$ of a circle with centre $O$ and radius $2 d \mathrm {~cm}$. Angle $P O Q$ is $\theta$ radians. Given that the areas of the shapes $X$ and $Y$ are equal,

prove that $\theta = 1 + \frac { 1 } { 4 } \pi$. Using this value of $\theta$, and given that $d = 3$, find in terms of $\pi$,
the perimeter of shape $X$,
the perimeter of shape $Y$.
Hence find the difference, in mm , between the perimeters of shapes $X$ and $Y$.

Edexcel C2 Q8

14 marks Standard +0.3

8. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{ffa0b566-6448-491b-96d7-d3806bcfe063-4_483_453_1503_623} \captionsetup{labelformat=empty} \caption{Figure 3}

\end{figure} A manufacturer produces cartons for fruit juice. Each carton is in the shape of a closed cuboid with base dimensions $2 x \mathrm {~cm}$ by $x \mathrm {~cm}$ and height $h \mathrm {~cm}$, as shown in Fig. 3.
Given that the capacity of a carton has to be $1030 \mathrm {~cm} ^ { 3 }$,

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

6 marks Moderate -0.8

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

8 marks Moderate -0.3

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

9 marks Challenging +1.2

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

11 marks Standard +0.3

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

10 marks Standard +0.3

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

12 marks Moderate -0.8

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

3 marks Easy -1.2

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

9 marks Moderate -0.3

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

9 marks Standard +0.3

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

12 marks Moderate -0.3

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 $R$.

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