Questions - Edexcel C4 - A-Level Maths

Edexcel C4 Q1

(a) Expand $( 1 + 4 x ) ^ { \frac { 3 } { 2 } }$ in ascending powers of $x$ up to and including the term in $x ^ { 3 }$, simplifying each coefficient.
(b) State the set of values of $x$ for which your expansion is valid.
Use the substitution $u = 1 + \sin x$ to find the value of

$$\int _ { 0 } ^ { \frac { \pi } { 2 } } \cos x ( 1 + \sin x ) ^ { 3 } d x$$

Edexcel C4 Q3

(a) Express $\frac { x + 11 } { ( x + 4 ) ( x - 3 ) }$ as a sum of partial fractions.
(b) Evaluate

$$\int _ { 0 } ^ { 2 } \frac { x + 11 } { ( x + 4 ) ( x - 3 ) } d x$$ giving your answer in the form $\ln k$, where $k$ is an exact simplified fraction. (5)
3. continued

Edexcel C4 Q4

4. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{4d4aa72e-d781-409d-8401-ccb4241bb12f-06_588_886_255_513} \captionsetup{labelformat=empty} \caption{Figure 1}

\end{figure} Figure 1 shows the curve with equation $y = 2 \sin x + \operatorname { cosec } x , 0 < x < \pi$. The shaded region bounded by the curve, the $x$-axis and the lines $x = \frac { \pi } { 6 }$ and $x = \frac { \pi } { 2 }$ is rotated through $360 ^ { \circ }$ about the $x$-axis. Show that the volume of the solid formed is $\frac { 1 } { 2 } \pi ( 4 \pi + 3 \sqrt { 3 } )$.
4. continued

Edexcel C4 Q5

5. A curve has the equation $$x ^ { 2 } - 3 x y - y ^ { 2 } = 12$$

Find an expression for $\frac { \mathrm { d } y } { \mathrm {~d} x }$ in terms of $x$ and $y$.
Find an equation for the tangent to the curve at the point $( 2 , - 2 )$.
5. continued

Edexcel C4 Q6

6. Relative to a fixed origin, $O$, the points $A$ and $B$ have position vectors $\left( \begin{array} { c } 1
5
- 1 \end{array} \right)$ and $\left( \begin{array} { c } 6
3
- 6 \end{array} \right)$ respectively. Find, in exact, simplified form,

the cosine of $\angle A O B$,
the area of triangle $O A B$,
the shortest distance from $A$ to the line $O B$.
6. continued

Edexcel C4 Q7

7. A curve has parametric equations $$x = t ( t - 1 ) , \quad y = \frac { 4 t } { 1 - t } , \quad t \neq 1$$

Find $\frac { \mathrm { d } y } { \mathrm {~d} x }$ in terms of $t$. The point $P$ on the curve has parameter $t = - 1$.
Show that the tangent to the curve at $P$ has the equation $$x + 3 y + 4 = 0$$ The tangent to the curve at $P$ meets the curve again at the point $Q$.
Find the coordinates of $Q$.
7. continued

Edexcel C4 Q8

8. An entomologist is studying the population of insects in a colony. Initially there are 300 insects in the colony and in a model, the entomologist assumes that the population, $P$, at time $t$ weeks satisfies the differential equation $$\frac { \mathrm { d } P } { \mathrm {~d} t } = k P$$ where $k$ is a constant.

Find an expression for $P$ in terms of $k$ and $t$. Given that after one week there are 360 insects in the colony,
find the value of $k$ to 3 significant figures. Given also that after two and three weeks there are 440 and 600 insects respectively,
comment on suitability of the model. An alternative model assumes that $$\frac { \mathrm { d } P } { \mathrm {~d} t } = P ( 0.4 - 0.25 \cos 0.5 t )$$
Using the initial data, $P = 300$ when $t = 0$, solve this differential equation.
Compare the suitability of the two models.
8. continued
8. continued

Edexcel C4 Q1

A curve has the equation

$$x ^ { 3 } + 2 x y - y ^ { 2 } + 24 = 0$$ Show that the normal to the curve at the point $( 2 , - 4 )$ has the equation $y = 3 x - 10$. (8)

Edexcel C4 Q2

2. (a) Expand $( 4 - x ) ^ { \frac { 1 } { 2 } }$ in ascending powers of $x$ up to and including the term in $x ^ { 2 }$, simplifying each coefficient.
(b) State the set of values of $x$ for which your expansion is valid.
(c) Use your expansion with $x = 0.01$ to find the value of $\sqrt { 399 }$, giving your answer to 9 significant figures.

Edexcel C4 Q3

3. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{3056ad22-f87b-46c3-86cf-d46939927465-04_560_1059_146_406} \captionsetup{labelformat=empty} \caption{Figure 1}

\end{figure} Figure 1 shows the curve with equation $y = \ln ( 2 + \cos x ) , 0 \leq x \leq \pi$.

Complete the table below for points on the curve, giving the $y$ values to 4 decimal places.
Giving your answers to 3 decimal places, find estimates for the area of the region bounded by the curve and the coordinate axes using the trapezium rule with
1. 1 strip,
2. 2 strips,
3. 4 strips.
Making your reasoning clear, suggest a value to 2 decimal places for the actual area of the region bounded by the curve and the coordinate axes.
$x$ 0 $\frac { \pi } { 4 }$ $\frac { \pi } { 2 }$ $\frac { 3 \pi } { 4 }$ $\pi$
$y$ 1.0986 0
1. continued
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{3056ad22-f87b-46c3-86cf-d46939927465-06_563_983_146_379} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} Figure 2 shows the curve with parametric equations $$x = \tan \theta , \quad y = \cos ^ { 2 } \theta , \quad - \frac { \pi } { 2 } < \theta < \frac { \pi } { 2 }$$ The shaded region bounded by the curve, the $x$-axis and the lines $x = - 1$ and $x = 1$ is rotated through $2 \pi$ radians about the $x$-axis.

Edexcel C4 Q5

5. Relative to a fixed origin, the points $A , B$ and $C$ have position vectors ( $2 \mathbf { i } - \mathbf { j } + 6 \mathbf { k }$ ), $( 5 \mathbf { i } - 4 \mathbf { j } )$ and $( 7 \mathbf { i } - 6 \mathbf { j } - 4 \mathbf { k } )$ respectively.

Show that $A , B$ and $C$ all lie on a single straight line.
Write down the ratio $A B : B C$ The point $D$ has position vector $( 3 \mathbf { i } + \mathbf { j } + 4 \mathbf { k } )$.
Show that $A D$ is perpendicular to $B D$.
Find the exact area of triangle $A B D$.
5. continued

Edexcel C4 Q6

6. (a) Use the substitution $x = 2 \sin u$ to evaluate $$\int _ { 0 } ^ { \sqrt { 3 } } \frac { 1 } { \sqrt { 4 - x ^ { 2 } } } \mathrm {~d} x$$ (b) Use integration by parts to evaluate $$\int _ { 0 } ^ { \frac { \pi } { 2 } } x \cos x d x$$ 6. continued

Edexcel C4 Q7

7. When a plague of locusts attacks a wheat crop, the proportion of the crop destroyed after $t$ hours is denoted by $x$. In a model, it is assumed that the rate at which the crop is destroyed is proportional to $x ( 1 - x )$. A plague of locusts is discovered in a wheat crop when one quarter of the crop has been destroyed. Given that the rate of destruction at this instant is such that if it remained constant, the crop would be completely destroyed in a further six hours,

show that $\frac { \mathrm { d } x } { \mathrm {~d} t } = \frac { 2 } { 3 } x ( 1 - x )$,
find the percentage of the crop destroyed three hours after the plague of locusts is first discovered.
7. continued
7. continued

Edexcel C4 Q1

The region bounded by the curve $y = x ^ { 2 } - 2 x$ and the $x$-axis is rotated through $2 \pi$ radians about the $x$-axis.

Find the volume of the solid formed, giving your answer in terms of $\pi$.

Edexcel C4 Q2

2. Use the substitution $u = 1 - x ^ { \frac { 1 } { 2 } }$ to find $$\int \frac { 1 } { 1 - x ^ { \frac { 1 } { 2 } } } \mathrm {~d} x$$

Edexcel C4 Q3

A curve has the equation

$$2 \sin 2 x - \tan y = 0$$

Find an expression for $\frac { \mathrm { d } y } { \mathrm {~d} x }$ in its simplest form in terms of $x$ and $y$.
Show that the tangent to the curve at the point $\left( \frac { \pi } { 6 } , \frac { \pi } { 3 } \right)$ has the equation $$y = \frac { 1 } { 2 } x + \frac { \pi } { 4 }$$
1. continued
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{3cf64017-e982-4165-9885-8524aaabdf84-06_433_812_246_479} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 shows the curve with parametric equations $$x = a \sqrt { t } , \quad y = a t ( 1 - t ) , \quad t \geq 0$$ where $a$ is a positive constant.
Find $\frac { \mathrm { d } y } { \mathrm {~d} x }$ in terms of $t$. The curve meets the $x$-axis at the origin, $O$, and at the point $A$. The tangent to the curve at $A$ meets the $y$-axis at the point $B$ as shown.
Show that the area of triangle $O A B$ is $a ^ { 2 }$.

Edexcel C4 Q5

5. The gradient at any point $( x , y )$ on a curve is proportional to $\sqrt { y }$. Given that the curve passes through the point with coordinates $( 0,4 )$,

show that the equation of the curve can be written in the form $$2 \sqrt { y } = k x + 4$$ where $k$ is a positive constant. Given also that the curve passes through the point with coordinates ( 2,9 ),
find the equation of the curve in the form $y = \mathrm { f } ( x )$.
5. continued

Edexcel C4 Q6

6. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{3cf64017-e982-4165-9885-8524aaabdf84-10_456_553_264_571} \captionsetup{labelformat=empty} \caption{Figure 2}

\end{figure} Figure 2 shows a vertical cross-section of a vase.
The inside of the vase is in the shape of a right-circular cone with the angle between the sides in the cross-section being $60 ^ { \circ }$. When the depth of water in the vase is $h \mathrm {~cm}$, the volume of water in the vase is $V \mathrm {~cm} ^ { 3 }$.

Show that $V = \frac { 1 } { 9 } \pi h ^ { 3 }$. The vase is initially empty and water is poured in at a constant rate of $120 \mathrm {~cm} ^ { 3 } \mathrm {~s} ^ { - 1 }$.
Find, to 2 decimal places, the rate at which $h$ is increasing
1. when $h = 6$,
2. after water has been poured in for 8 seconds.
  6. continued

Edexcel C4 Q7

7. Relative to a fixed origin, the points $A$ and $B$ have position vectors $\left( \begin{array} { c } - 4
1
3 \end{array} \right)$ and $\left( \begin{array} { c } - 3
6
1 \end{array} \right)$ respectively.

Find a vector equation for the line $l _ { 1 }$ which passes through $A$ and $B$. The line $l _ { 2 }$ has vector equation $$\mathbf { r } = \left( \begin{array} { c } 3
- 7
9 \end{array} \right) + \mu \left( \begin{array} { c } 2
- 3
1 \end{array} \right)$$
Show that lines $l _ { 1 }$ and $l _ { 2 }$ do not intersect.
Find the position vector of the point $C$ on $l _ { 2 }$ such that $\angle A B C = 90 ^ { \circ }$.
7. continued

Edexcel C4 Q9

9 \end{array} \right) + \mu \left( \begin{array} { c } 2
- 3
1 \end{array} \right)$$ (b) Show that lines $l _ { 1 }$ and $l _ { 2 }$ do not intersect.
(c) Find the position vector of the point $C$ on $l _ { 2 }$ such that $\angle A B C = 90 ^ { \circ }$.
7. continued
8. $$f ( x ) = \frac { x ( 3 x - 7 ) } { ( 1 - x ) ( 1 - 3 x ) } , | x | < \frac { 1 } { 3 }$$ (a) Find the values of the constants $A , B$ and $C$ such that $$\mathrm { f } ( x ) = A + \frac { B } { 1 - x } + \frac { C } { 1 - 3 x }$$ (b) Evaluate $$\int _ { 0 } ^ { \frac { 1 } { 4 } } f ( x ) d x$$ giving your answer in the form $p + \ln q$, where $p$ and $q$ are rational.
(c) Find the series expansion of $\mathrm { f } ( x )$ in ascending powers of $x$ up to and including the term in $x ^ { 3 }$, simplifying each coefficient.
8. continued
8. continued

Edexcel C4 Q1

1. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{80bef9d4-b84c-4d3a-a093-67a466c6d1b9-02_615_791_146_532} \captionsetup{labelformat=empty} \caption{Figure 1}

\end{figure} Figure 1 shows the curve with equation $y = \frac { 3 x + 1 } { \sqrt { x } } , x > 0$.
The shaded region is bounded by the curve, the $x$-axis and the lines $x = 1$ and $x = 3$.
Find the volume of the solid formed when the shaded region is rotated through $2 \pi$ radians about the $x$-axis, giving your answer in the form $\pi ( a + \ln b )$, where $a$ and $b$ are integers.
□

Edexcel C4 Q2

2. (a) Expand $( 1 - 3 x ) ^ { - 2 } , | x | < \frac { 1 } { 3 }$, in ascending powers of $x$ up to and including the term in $x ^ { 3 }$, simplifying each coefficient.
(b) Hence, or otherwise, show that for small $x$, $$\left( \frac { 2 - x } { 1 - 3 x } \right) ^ { 2 } \approx 4 + 20 x + 85 x ^ { 2 } + 330 x ^ { 3 }$$

Edexcel C4 Q3

3. $$f ( x ) = \frac { 7 + 3 x + 2 x ^ { 2 } } { ( 1 - 2 x ) ( 1 + x ) ^ { 2 } } , \quad | x | > \frac { 1 } { 2 }$$

Express $\mathrm { f } ( x )$ in partial fractions.
Show that $$\int _ { 1 } ^ { 2 } \mathrm { f } ( x ) \mathrm { d } x = p - \ln q$$ where $p$ is rational and $q$ is an integer.
3. continued

Edexcel C4 Q4

4. Relative to a fixed origin, two lines have the equations $$\begin{aligned} & \mathbf { r } = \left( \begin{array} { c } 7
0
- 3 \end{array} \right) + \lambda \left( \begin{array} { c }

\(x\)	0	\(\frac { \pi } { 4 }\)	\(\frac { \pi } { 2 }\)	\(\frac { 3 \pi } { 4 }\)	\(\pi\)
\(y\)	1.0986				0

Questions — Edexcel C4 (360 questions)

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