Questions - Edexcel C2 - A-Level Maths

Edexcel C2 Q9

9 \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{f01026a9-e5fe-4c19-b096-2bb4ad22c389-5_686_1240_178_312} \captionsetup{labelformat=empty} \caption{Fig. 3}

\end{figure} 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. 3.

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 Q1

$\mathrm { f } ( x ) = x ^ { 3 } + a x ^ { 2 } + b x - 10$, 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 $\mathrm { f } ( x )$.

Edexcel C2 Q2

2. (a) Write down the first four terms of the binomial expansion, in ascending powers of $x$, of $( 1 + a x ) ^ { n }$, where $n > 2$. Given that, in this expansion, the coefficient of $x$ is 8 and the coefficient of $x ^ { 2 }$ is 30 ,
(b) find the value of $n$ and the value of $a$,
(c) find the coefficient of $x ^ { 3 }$.

Edexcel C2 Q7

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$\mathrm { f } ( x ) = x ^ { 3 } + a x ^ { 2 } + b x - 10$, 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 $\mathrm { f } ( x )$.
2. (a) Write down the first four terms of the binomial expansion, in ascending powers of $x$, of $( 1 + a x ) ^ { n }$, where $n > 2$. Given that, in this expansion, the coefficient of $x$ is 8 and the coefficient of $x ^ { 2 }$ is 30 ,
find the value of $n$ and the value of $a$,
find the coefficient of $x ^ { 3 }$.
3. 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.
4. Given that $\mathrm { f } ( x ) = \left( 2 x ^ { \frac { 3 } { 2 } } - 3 x ^ { - \frac { 3 } { 2 } } \right) ^ { 2 } + 5 , \quad x > 0$,
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 $\int _ { 1 } ^ { 2 } f ( x ) d x$.
5. \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{5e4e4cae-d4c6-465d-8cb7-84712e6e55fe-3_736_1266_276_404}
\end{figure} Figure 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.
6. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{5e4e4cae-d4c6-465d-8cb7-84712e6e55fe-4_641_1406_196_287} \captionsetup{labelformat=empty} \caption{Fig. 1}
\end{figure} Figure 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 $R$.
7. Find all the values of $\theta$ in the interval $0 \leq \theta < 360 ^ { \circ }$ for which
$\cos \left( \theta - 10 ^ { \circ } \right) = \cos 15 ^ { \circ }$,
$\tan 2 \theta = 0.4$,
$2 \sin \theta \tan \theta = 3$.

Edexcel C2 Q1

(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.

Edexcel C2 Q2

2. $$f ( n ) = n ^ { 3 } + p n ^ { 2 } + 11 n + 9 , \text { where } p \text { is a constant. }$$

Given that $\mathrm { f } ( n )$ has a remainder of 3 when it is divided by ( $n + 2$ ), prove that $p = 6$.
Show that $\mathrm { f } ( n )$ can be written in the form $( n + 2 ) ( n + q ) ( n + r ) + 3$, where $q$ and $r$ are integers to be found.
Hence show that $\mathrm { f } ( n )$ is divisible by 3 for all positive integer values of $n$.

Edexcel C2 Q7

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(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.
2. $$f ( n ) = n ^ { 3 } + p n ^ { 2 } + 11 n + 9 , \text { where } p \text { is a constant. }$$ (a) Given that $\mathrm { f } ( n )$ has a remainder of 3 when it is divided by ( $n + 2$ ), prove that $p = 6$.
(b) Show that $\mathrm { f } ( n )$ can be written in the form $( n + 2 ) ( n + q ) ( n + r ) + 3$, where $q$ and $r$ are integers to be found.
(c) Hence show that $\mathrm { f } ( n )$ is divisible by 3 for all positive integer values of $n$.
3. Find the values of $\theta$, to 1 decimal place, in the interval $- 180 \leq \theta < 180$ for which $$2 \sin ^ { 2 } \theta ^ { \circ } - 2 \sin \theta ^ { \circ } = \cos ^ { 2 } \theta ^ { \circ } .$$

Every $\pounds 1$ of money invested in a savings scheme continuously gains interest at a rate of $4 \%$ per year. Hence, after $x$ years, the total value of an initial $\pounds 1$ investment is $\pounds y$, where

$$y = 1.04 ^ { x }$$ (a) Sketch the graph of $y = 1.04 ^ { x } , x \geq 0$.
(b) Calculate, to the nearest $\pounds$, the total value of an initial $\pounds 800$ investment after 10 years.
(c) Use logarithms to find the number of years it takes to double the total value of any initial investment.
5. The curve $C$ with equation $y = p + q e ^ { x }$, where $p$ and $q$ are constants, passes through the point $( 0,2 )$. At the point $P ( \ln 2 , p + 2 q$ on $C$, the gradient is 5 .
(a) Find the value of p and the value of $q$. The normal to $C$ at $P$ crosses the $x$-axis at $L$ and the $y$-axis at $M$.
(b) Show that the area of $\triangle O L M$, where $O$ is the origin, is approximately 53.8
6. \begin{figure}[h]

\captionsetup{labelformat=empty} \caption{Figure 3} \includegraphics[alt={},max width=\textwidth]{04f9d12c-3423-441a-b1ff-fe39ef6e51ed-3_517_1300_760_370}

\end{figure} Figure 3 shows part of the curve $C$ with equation $$y = \frac { 3 } { 2 } x ^ { 2 } - \frac { 1 } { 4 } x ^ { 3 } .$$ The curve $C$ touches the $x$-axis at the origin and passes through the point $A ( p , 0 )$.
(a) Show that $p = 6$.
(b) Find an equation of the tangent to $C$ at $A$. The curve $C$ has a maximum at the point $P$.
(c) Find the $x$-coordinate of $P$. The shaded region $R$, in Fig. 3, is bounded by $C$ and the $x$-axis.
(d) Find the area of $R$.
7. Figure 1 \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{04f9d12c-3423-441a-b1ff-fe39ef6e51ed-4_499_492_319_420} \captionsetup{labelformat=empty} \caption{Shape X}

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

\includegraphics[alt={},max width=\textwidth]{04f9d12c-3423-441a-b1ff-fe39ef6e51ed-4_597_474_274_1096} \captionsetup{labelformat=empty} \caption{Shape $Y$}

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

Edexcel C2 Q8

8. $\quad \mathrm { f } ( x ) = \left( 1 + \frac { x } { k } \right) ^ { n } , \quad k , n \in \mathbb { N } , \quad n > 2$. 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 END

Edexcel C2 Q1

1. $$f ( x ) \equiv a x ^ { 3 } + b x ^ { 2 } - 7 x + 14 , \text { where } a \text { and } b \text { are constants. }$$ 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 $b$.

Edexcel C2 Q2

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

Edexcel C2 Q7

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\hline \end{tabular} \end{center} 1. $$f ( x ) \equiv a x ^ { 3 } + b x ^ { 2 } - 7 x + 14 , \text { where } a \text { and } b \text { are constants. }$$ 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 $b$.
2. (i) Differentiate with respect to $x$ $$2 x ^ { 3 } + \sqrt { } x + \frac { x ^ { 2 } + 2 x } { x ^ { 2 } }$$ (ii) Evaluate $$\int _ { 1 } ^ { 4 } \left( \frac { x } { 2 } + \frac { 1 } { x ^ { 2 } } \right) \mathrm { d } x$$
1. (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.
Find the value of $d$. Using your value of $d$,
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 $\pounds 54000$,
find the predicted profit for the year 2011.
(3 marks)
4. (a) Write down formulae for $\sin ( \mathrm { A } + \mathrm { B } )$ and $\sin ( \mathrm { A } - \mathrm { B } )$. Using $\mathrm { X } = \mathrm { A } + \mathrm { B }$ and $\mathrm { Y } = \mathrm { A } - \mathrm { B }$, prove that $$\operatorname { Sin } X + \sin Y = 2 \sin \frac { X + Y } { 2 } \cos \frac { X - Y } { 2 }$$
Hence, or otherwise, solve, for $0 \leq \theta < 360$, $$\sin 40 ^ { \circ } + \sin 20 ^ { \circ } = 0$$ \section*{5.} \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{2561e267-6495-453d-ac50-0b1542215e0a-4_833_965_303_466}
\end{figure} Figure 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 m .
$B C$ is an arc of a circle with centre $O$ and radius 7 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.
6. (a) Given that $$( 2 + x ) ^ { 5 } + ( 2 - x ) ^ { 5 } \equiv A + B x ^ { 2 } + C x ^ { 4 }$$ Find the values of the constants $A , B$ and $C$.
Using the substitution $y = x ^ { 2 }$ and your answers to part (a), solve, $$( 2 + x ) ^ { 5 } + ( 2 - x ) ^ { 5 } = 349$$ 7. \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 2} \includegraphics[alt={},max width=\textwidth]{2561e267-6495-453d-ac50-0b1542215e0a-5_780_974_922_630}
\end{figure} Figure 2 shows part of the curve $C$ with equation $y = \mathrm { f } ( x )$, where $$f ( x ) = x ^ { 3 } - 6 x ^ { 2 } + 5 x$$ The curve crosses the $x$-axis at the origin $O$ and at the points $A$ and $B$.
Factorise $\mathrm { f } ( x )$ completely
Write down the $x$-coordinates of the points $A$ and $B$.
Find the gradient of $C$ at $A$. The region $R$ is bounded by $C$ and the line $O A$, and the region $S$ is bounded by $C$ and the line $A B$.
Use integration to find the area of the combined regions $R$ and $S$, shown shaded in Fig. 2.

Edexcel C2 Q1

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

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

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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

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

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

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

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

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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

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

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

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

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

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

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 $Y$.

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