Questions - Edexcel C12 - A-Level Maths

1. State the gradient of $l _ { 1 }$
2. Write down the $x$ coordinate of point $A$. Another straight line $l _ { 2 }$ intersects $l _ { 1 }$ at the point $B$ with $x$ coordinate 1 and crosses the $x$-axis at the point $C$, as shown in Figure 2. Given that $l _ { 2 }$ is perpendicular to $l _ { 1 }$
find an equation for $l _ { 2 }$ in the form $a x + b y + c = 0$, where $a$, b and $c$ are integers,
find the exact area of triangle $A B C$.

Edexcel C12 2017 January Q7

7. (i) Find $$\int \frac { 2 + 4 x ^ { 3 } } { x ^ { 2 } } \mathrm {~d} x$$ giving each term in its simplest form.
(ii) Given that $k$ is a constant and $$\int _ { 2 } ^ { 4 } \left( \frac { 4 } { \sqrt { x } } + k \right) \mathrm { d } x = 30$$ find the exact value of $k$.

Edexcel C12 2017 January Q8

8. $$f ( x ) = 2 x ^ { 3 } - 5 x ^ { 2 } - 23 x - 10$$

Find the remainder when $\mathrm { f } ( x )$ is divided by ( $x - 3$ ).
Show that ( $x + 2$ ) is a factor of $\mathrm { f } ( x )$.
Hence fully factorise $\mathrm { f } ( x )$.
Hence solve $$2 \left( 3 ^ { 3 t } \right) - 5 \left( 3 ^ { 2 t } \right) - 23 \left( 3 ^ { t } \right) = 10$$ giving your answer to 3 decimal places.

Edexcel C12 2017 January Q9

9. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{f39ade34-32e2-4b5c-b80a-9663c6a65c87-14_609_744_223_593} \captionsetup{labelformat=empty} \caption{Figure 3}

\end{figure} Figure 3 shows a sketch of the curve with equation $y = \mathrm { f } ( x )$ where $$f ( x ) = \frac { 8 } { x } + \frac { 1 } { 2 } x - 5 , \quad 0 < x \leqslant 12$$ The curve crosses the $x$-axis at $( 2,0 )$ and $( 8,0 )$ and has a minimum point at $A$.

Use calculus to find the coordinates of point $A$.
State
1. the roots of the equation $2 \mathrm { f } ( x ) = 0$
2. the coordinates of the turning point on the curve $y = \mathrm { f } ( x ) + 2$
3. the roots of the equation $\mathrm { f } ( 4 x ) = 0$

Edexcel C12 2017 January Q10

10. The first 3 terms, in ascending powers of $x$, in the binomial expansion of $( 1 + a x ) ^ { 20 }$ are given by $$1 + 4 x + p x ^ { 2 }$$ where $a$ and $p$ are constants.

Find the value of $a$.
Find the value of $p$. One of the terms in the binomial expansion of $( 1 + a x ) ^ { 20 }$ is $q x ^ { 4 }$, where $q$ is a constant.
Find the value of $q$.

Edexcel C12 2017 January Q11

11. In this question solutions based entirely on graphical or numerical methods are not acceptable.

Solve, for $0 \leqslant x < 2 \pi$, $$3 \cos ^ { 2 } x + 1 = 4 \sin ^ { 2 } x$$ giving your answers in radians to 2 decimal places.
Solve, for $0 \leqslant \theta < 360 ^ { \circ }$, $$5 \sin \left( \theta + 10 ^ { \circ } \right) = \cos \left( \theta + 10 ^ { \circ } \right)$$ giving your answers in degrees to one decimal place.

Edexcel C12 2017 January Q12

12. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{f39ade34-32e2-4b5c-b80a-9663c6a65c87-20_775_1015_260_459} \captionsetup{labelformat=empty} \caption{Figure 4}

\end{figure} Figure 4 shows a sketch of part of the curve $C$ with equation $$y = \frac { 3 } { 4 } x ^ { 2 } - 4 \sqrt { x } + 7 , \quad x > 0$$ The point $P$ lies on $C$ and has coordinates $( 4,11 )$.
Line $l$ is the tangent to $C$ at the point $P$.

Use calculus to show that $l$ has equation $y = 5 x - 9$ The finite region $R$, shown shaded in Figure 4, is bounded by the curve $C$, the line $x = 1$, the $x$-axis and the line $l$.
Find, by using calculus, the area of $R$, giving your answer to 2 decimal places.
(Solutions based entirely on graphical or numerical methods are not acceptable.)

Edexcel C12 2017 January Q13

13. (a) On separate axes sketch the graphs of

$y = c ^ { 2 } - x ^ { 2 }$
$y = x ^ { 2 } ( x - 3 c )$
where $c$ is a positive constant.
Show clearly the coordinates of the points where each graph crosses or meets the $x$-axis and the $y$-axis.
(b) Prove that the $x$ coordinate of any point of intersection of $$y = c ^ { 2 } - x ^ { 2 } \text { and } y = x ^ { 2 } ( x - 3 c )$$ where $c$ is a positive constant, is given by a solution of the equation $$x ^ { 3 } + ( 1 - 3 c ) x ^ { 2 } - c ^ { 2 } = 0$$ Given that the graphs meet when $x = 2$
(c) find the exact value of $c$, writing your answer as a fully simplified surd.

Edexcel C12 2017 January Q14

14. A geometric series has a first term $a$ and a common ratio $r$.

Prove that the sum of the first $n$ terms of this series is given by $$S _ { n } = \frac { a \left( 1 - r ^ { n } \right) } { 1 - r }$$ A liquid is to be stored in a barrel. Due to evaporation, the volume of the liquid in a barrel at the end of a year is $7 \%$ less than the volume at the start of the year. At the start of the first year, a barrel is filled with 180 litres of the liquid.
Show that the amount of the liquid in this barrel at the end of 5 years is approximately 125.2 litres. At the start of each year a new identical barrel is filled with 180 litres of the liquid so that, at the end of 20 years, there are 20 barrels containing varying amounts of the liquid.
Calculate the total amount of the liquid, to the nearest litre, in the 20 barrels at the end of 20 years.

Edexcel C12 2017 January Q15

15. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{f39ade34-32e2-4b5c-b80a-9663c6a65c87-26_780_871_242_539} \captionsetup{labelformat=empty} \caption{Figure 5}

\end{figure} Figure 5 shows the design for a logo.
The logo is in the shape of an equilateral triangle $A B C$ of side length $2 r \mathrm {~cm}$, where $r$ is a constant. The points $L , M$ and $N$ are the midpoints of sides $A C , A B$ and $B C$ respectively.
The shaded section $R$, of the logo, is bounded by three curves $M N , N L$ and $L M$. The curve $M N$ is the arc of a circle centre $L$, radius $r \mathrm {~cm}$.
The curve $N L$ is the arc of a circle centre $M$, radius $r \mathrm {~cm}$.
The curve $L M$ is the arc of a circle centre $N$, radius $r \mathrm {~cm}$. Find, in $\mathrm { cm } ^ { 2 }$, the area of $R$. Give your answer in the form $k r ^ { 2 }$, where $k$ is an exact constant to be determined.

Edexcel C12 2018 January Q1

Given that

$$y = \frac { 2 x ^ { \frac { 2 } { 3 } } + 3 } { 6 } , \quad x > 0$$ find, in the simplest form,

$\frac { \mathrm { d } y } { \mathrm {~d} x }$
$\int y \mathrm {~d} x$

Edexcel C12 2018 January Q2

2. A sequence is defined by $$\begin{aligned} u _ { 1 } & = 1
u _ { n + 1 } & = 2 - 3 u _ { n } \quad n \geqslant 1 \end{aligned}$$

Find the value of $u _ { 2 }$ and the value of $u _ { 3 }$
Calculate the value of $\sum _ { r = 1 } ^ { 4 } \left( r - u _ { r } \right)$
□

Edexcel C12 2018 January Q3

3. Simplify fully

$\left( 3 x ^ { \frac { 1 } { 2 } } \right) ^ { 4 }$
$\frac { 2 y ^ { 7 } \times ( 4 y ) ^ { - 2 } } { 3 y }$

Edexcel C12 2018 January Q4

4. The equation
$( p - 2 ) x ^ { 2 } + 8 x + ( p + 4 ) = 0 , \quad$ where $p$ is a constant has no real roots.

Show that $p$ satisfies $p ^ { 2 } + 2 p - 24 > 0$
Hence find the set of possible values of $p$.

Edexcel C12 2018 January Q5

5. (In this question, solutions based entirely on graphical or numerical methods are not acceptable.)

Solve, for $0 < \theta < \frac { \pi } { 2 }$ $$5 \sin 3 \theta - 7 \cos 3 \theta = 0$$ Give each solution, in radians, to 3 significant figures.
Solve, for $0 < x < 360 ^ { \circ }$ $$9 \cos ^ { 2 } x + 5 \cos x = 3 \sin ^ { 2 } x$$ Give each solution, in degrees, to one decimal place.

Edexcel C12 2018 January Q6

6. $$f ( x ) = a x ^ { 3 } - 8 x ^ { 2 } + b x + 6$$ where $a$ and $b$ are constants. When $\mathrm { f } ( x )$ is divided by ( $x + 1$ ) there is no remainder. When $\mathrm { f } ( x )$ is divided by $( x - 2 )$ the remainder is - 12

Find the value of $a$ and the value of $b$.
Factorise $\mathrm { f } ( x )$ completely.

Edexcel C12 2018 January Q7

7. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{6f9ace43-747b-419f-a9d1-d30165d77379-18_675_1408_292_358} \captionsetup{labelformat=empty} \caption{Figure 1}

\end{figure} Figure 1 shows a rectangular sheet of metal of negligible thickness, which measures 25 cm by 15 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 cuboid box, as shown in Figure 2.

Show that the volume, $V \mathrm {~cm} ^ { 3 }$, of the box is given by $$V = 4 x ^ { 3 } - 80 x ^ { 2 } + 375 x$$
Use calculus to find the value of $x$, to 3 significant figures, for which the volume of the box is a maximum.
Justify that this value of $x$ gives a maximum value for $V$.
Find, to 3 significant figures, the maximum volume of the box.
\section*{8.} \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{6f9ace43-747b-419f-a9d1-d30165d77379-22_670_1004_292_392} \captionsetup{labelformat=empty} \caption{Figure 3}
\end{figure} Figure 3 shows a sketch of the curve with equation $y = \mathrm { f } ( x ) , x \in \mathbb { R }$. The curve crosses the $y$-axis at the point $( 0,5 )$ and crosses the $x$-axis at the point $( 6,0 )$. The curve has a minimum point at $( 1,3 )$ and a maximum point at $( 4,7 )$. On separate diagrams, sketch the curve with equation

Edexcel C12 2018 January Q9

The first term of a geometric series is 20 and the common ratio is 0.9
1. Find the value of the fifth term.
2. Find the sum of the first 8 terms, giving your answer to one decimal place.
Given that $S _ { \infty } - S _ { N } < 0.04$, where $S _ { N }$ is the sum of the first $N$ terms of this series, (c) show that $0.9 ^ { N } < 0.0002$
Hence find the smallest possible value of $N$.

Edexcel C12 2018 January Q10

10. (i) Use the laws of logarithms to solve the equation $$3 \log _ { 8 } 2 + \log _ { 8 } ( 7 - x ) = 2 + \log _ { 8 } x$$ (ii) Using algebra, find, in terms of logarithms, the exact value of $y$ for which $$3 ^ { 2 y } + 3 ^ { y + 1 } = 10$$

Edexcel C12 2018 January Q11

11. The circle $C$ has equation $$x ^ { 2 } + y ^ { 2 } - 8 x - 10 y + 16 = 0$$ The centre of $C$ is at the point $T$.

Find
1. the coordinates of the point $T$,
2. the radius of the circle $C$. The point $M$ has coordinates $( 20,12 )$.
Find the exact length of the line $M T$. Point $P$ lies on the circle $C$ such that the tangent at $P$ passes through the point $M$.
Find the exact area of triangle $M T P$, giving your answer as a simplified surd.

Edexcel C12 2018 January Q12

The line $l _ { 1 }$ has equation $x + 3 y - 11 = 0$

The point $A$ and the point $B$ lie on $l _ { 1 }$
Given that $A$ has coordinates ( $- 1 , p$ ) and $B$ has coordinates ( $q , 2$ ), where $p$ and $q$ are integers,

find the value of $p$ and the value of $q$,
find the length of $A B$, giving your answer as a simplified surd. The line $l _ { 2 }$ is perpendicular to $l _ { 1 }$ and passes through the midpoint of $A B$.
Find an equation for $l _ { 2 }$ giving your answer in the form $y = m x + c$, where $m$ and $c$ are constants to be found.

Edexcel C12 2018 January Q13

13. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{6f9ace43-747b-419f-a9d1-d30165d77379-42_840_1010_287_571} \captionsetup{labelformat=empty} \caption{Figure 4}

\end{figure} Figure 4 shows the position of two stationary boats, $A$ and $B$, and a port $P$ which are assumed to be in the same horizontal plane. Boat $A$ is 8.7 km on a bearing of $314 ^ { \circ }$ from port $P$.
Boat $B$ is 3.5 km on a bearing of $052 ^ { \circ }$ from port $P$.

Show that angle $A P B$ is $98 ^ { \circ }$
Find the distance of boat $B$ from boat $A$, giving your answer to one decimal place.
Find the bearing of boat $B$ from boat $A$, giving your answer to the nearest degree.

Edexcel C12 2018 January Q14

14. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{6f9ace43-747b-419f-a9d1-d30165d77379-46_812_1091_292_429} \captionsetup{labelformat=empty} \caption{Figure 5}

\end{figure} Figure 5 shows a sketch of part of the line $l$ with equation $y = 8 - x$ and part of the curve $C$ with equation $y = 14 + 3 x - 2 x ^ { 2 }$ The line $l$ and the curve $C$ intersect at the point $A$ and the point $B$ as shown.

Use algebra to find the coordinates of $A$ and the coordinates of $B$. The region $R$, shown shaded in Figure 5, is bounded by the coordinate axes, the line $l$, and the curve $C$.
Use algebraic integration to calculate the exact area of $R$.

Edexcel C12 2018 January Q15

15. The binomial expansion, in ascending powers of $x$, of $( 1 + k x ) ^ { n }$ is $$1 + 36 x + 126 k x ^ { 2 } + \ldots$$ where $k$ is a non-zero constant and $n$ is a positive integer.

Show that $n k ( n - 1 ) = 252$
Find the value of $k$ and the value of $n$.
Using the values of $k$ and $n$, find the coefficient of $x ^ { 3 }$ in the binomial expansion of $( 1 + k x ) ^ { n }$