Questions C4 - A-Level Maths

Find a vector equation for $l _ { 1 }$. The line $l _ { 2 }$ has the equation $$\mathbf { r } = ( 5 \mathbf { j } - 7 \mathbf { k } ) + \mu ( \mathbf { i } - 2 \mathbf { j } + 7 \mathbf { k } )$$ The point $C$ lies on $l _ { 2 }$ and is such that $A C$ is perpendicular to $B C$.
Show that one possible position vector for $C$ is $( \mathbf { i } + 3 \mathbf { j } )$ and find the other. Assuming that $C$ has position vector $( \mathbf { i } + 3 \mathbf { j } )$,
find the area of triangle $A B C$, giving your answer in the form $k \sqrt { 5 }$.
8. continued
8. continued

Edexcel C4 Q1

Use integration by parts to show that

$$\int _ { 1 } ^ { 2 } x \ln x \mathrm {~d} x = 2 \ln 2 - \frac { 3 } { 4 }$$

Edexcel C4 Q2

(a) Use the trapezium rule with two intervals of equal width to find an approximate value for the integral

$$\int _ { 0 } ^ { 2 } \arctan x \mathrm {~d} x$$ (b) Use the trapezium rule with four intervals of equal width to find an improved approximation for the value of the integral.

Edexcel C4 Q3

3. A curve has the equation $$3 x ^ { 2 } - 2 x + x y + y ^ { 2 } - 11 = 0$$ The point $P$ on the curve has coordinates $( - 1,3 )$.

Show that the normal to the curve at $P$ has the equation $y = 2 - x$.
Find the coordinates of the point where the normal to the curve at $P$ meets the curve again.
3. continued

Edexcel C4 Q4

4. The points $A$ and $B$ have coordinates $( 3,9 , - 7 )$ and $( 13 , - 6 , - 2 )$ respectively.

Find, in vector form, an equation for the line $l$ which passes through $A$ and $B$.
Show that the point $C$ with coordinates $( 9,0 , - 4 )$ lies on $l$. The point $D$ is the point on $l$ closest to the origin, $O$.
Find the coordinates of $D$.
Find the area of triangle $O A B$ to 3 significant figures.
4. continued

Edexcel C4 Q5

5. A bath is filled with hot water which is allowed to cool. The temperature of the water is $\theta ^ { \circ } \mathrm { C }$ after cooling for $t$ minutes and the temperature of the room is assumed to remain constant at $20 ^ { \circ } \mathrm { C }$. Given that the rate at which the temperature of the water decreases is proportional to the difference in temperature between the water and the room,

write down a differential equation connecting $\theta$ and $t$. Given also that the temperature of the water is initially $37 ^ { \circ } \mathrm { C }$ and that it is $36 ^ { \circ } \mathrm { C }$ after cooling for four minutes,
find, to 3 significant figures, the temperature of the water after ten minutes. Advice suggests that the temperature of the water should be allowed to cool to $33 ^ { \circ } \mathrm { C }$ before a child gets in.
Find, to the nearest second, how long a child should wait before getting into the bath.
5. continued

Edexcel C4 Q6

6. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{0f2d48ab-1f61-4fb9-b35a-25d684dbd50f-10_454_602_255_479} \captionsetup{labelformat=empty} \caption{Figure 1}

\end{figure} Figure 1 shows the curve with parametric equations $$x = 3 \sin t , \quad y = 2 \sin 2 t , \quad 0 \leq t < \pi .$$ The curve meets the $x$-axis at the origin, $O$, and at the point $A$.

Find the value of $t$ at $O$ and the value of $t$ at $A$. The region enclosed by the curve is rotated through $\pi$ radians about the $x$-axis.
Show that the volume of the solid formed is given by $$\int _ { 0 } ^ { \frac { \pi } { 2 } } 12 \pi \sin ^ { 2 } 2 t \cos t \mathrm {~d} t$$
Using the substitution $u = \sin t$, or otherwise, evaluate this integral, giving your answer as an exact multiple of $\pi$.
6. continued

Edexcel C4 Q7

7. $$f ( x ) = \frac { 8 - x } { ( 1 + x ) ( 2 - x ) } , \quad | x | < 1$$

Express $\mathrm { f } ( x )$ in partial fractions.
Show that $$\int _ { 0 } ^ { \frac { 1 } { 2 } } \mathrm { f } ( x ) \mathrm { d } x = \ln k$$ where $k$ is an integer to be found.
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.
7. continued
7. continued

Edexcel C4 Q1

(a) Find the binomial expansion of $( 2 - 3 x ) ^ { - 3 }$ 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.
A curve has the equation

$$x ^ { 2 } + 3 x y - 2 y ^ { 2 } + 17 = 0$$ (a) Find an expression for $\frac { \mathrm { d } y } { \mathrm {~d} x }$ in terms of $x$ and $y$.
(b) Find an equation for the normal to the curve at the point $( 3 , - 2 )$.

Edexcel C4 Q3

3. (a) Find the values of the constants $A , B , C$ and $D$ such that $$\frac { 2 x ^ { 3 } - 5 x ^ { 2 } + 6 } { x ^ { 2 } - 3 x } \equiv A x + B + \frac { C } { x } + \frac { D } { x - 3 } .$$ (b) Evaluate $$\int _ { 1 } ^ { 2 } \frac { 2 x ^ { 3 } - 5 x ^ { 2 } + 6 } { x ^ { 2 } - 3 x } \mathrm {~d} x$$ giving your answer in the form $p + q \ln 2$, where $p$ and $q$ are integers.
3. continued

Edexcel C4 Q4

4. A mathematician is selling goods at a car boot sale. She believes that the rate at which she makes sales depends on the length of time since the start of the sale, $t$ hours, and the total value of sales she has made up to that time, $\pounds x$. She uses the model $$\frac { \mathrm { d } x } { \mathrm {~d} t } = \frac { k ( 5 - t ) } { x }$$ where $k$ is a constant.
Given that after two hours she has made sales of $\pounds 96$ in total,

solve the differential equation and show that she made $\pounds 72$ in the first hour of the sale. The mathematician believes that is it not worth staying at the sale once she is making sales at a rate of less than $\pounds 10$ per hour.
Verify that at 3 hours and 5 minutes after the start of the sale, she should have already left.
4. continued

Edexcel C4 Q5

5. Relative to a fixed origin, two lines have the equations $$\begin{aligned} & \mathbf { r } = \left( \begin{array} { l } 4
1
1 \end{array} \right) + s \left( \begin{array} { l } 1
4
5 \end{array} \right)
& \mathbf { r } = \left( \begin{array} { c } - 3
1
- 6 \end{array} \right) + t \left( \begin{array} { l } 3
a
b \end{array} \right) \end{aligned}$$ and
where $a$ and $b$ are constants and $s$ and $t$ are scalar parameters.
Given that the two lines are perpendicular,

find a linear relationship between $a$ and $b$. Given also that the two lines intersect,
find the values of $a$ and $b$,
find the coordinates of the point where they intersect.
5. continued

Edexcel C4 Q6

6. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{a4ac7e65-267e-45c0-bbf2-2c38608eacc3-10_581_823_146_477} \captionsetup{labelformat=empty} \caption{Figure 1}

\end{figure} Figure 1 shows the curve with equation $y = x \sqrt { 1 - x } , 0 \leq x \leq 1$.

Use the substitution $u ^ { 2 } = 1 - x$ to show that the area of the region bounded by the curve and the $x$-axis is $\frac { 4 } { 15 }$.
Find, in terms of $\pi$, the volume of the solid formed when the region bounded by the curve and the $x$-axis is rotated through $360 ^ { \circ }$ about the $x$-axis.
6. continued

Edexcel C4 Q7

7. A curve has parametric equations $$x = 3 \cos ^ { 2 } t , \quad y = \sin 2 t , \quad 0 \leq t < \pi$$

Show that $\frac { \mathrm { d } y } { \mathrm {~d} x } = - \frac { 2 } { 3 } \cot 2 t$.
Find the coordinates of the points where the tangent to the curve is parallel to the $x$-axis.
Show that the tangent to the curve at the point where $t = \frac { \pi } { 6 }$ has the equation $$2 x + 3 \sqrt { 3 } y = 9$$
Find a cartesian equation for the curve in the form $y ^ { 2 } = \mathrm { f } ( x )$.
7. continued
7. continued

Edexcel C4 Q1

Find

$$\int \cot ^ { 2 } 2 x \mathrm {~d} x$$

Edexcel C4 Q2

A curve has the equation

$$4 \cos x + 2 \sin y = 3$$

Show that $\frac { \mathrm { d } y } { \mathrm {~d} x } = 2 \sin x \sec y$.
Find an equation for the tangent to the curve at the point $\left( \frac { \pi } { 3 } , \frac { \pi } { 6 } \right)$, giving your answer in the form $a x + b y = c$, where $a$ and $b$ are integers.

Edexcel C4 Q3

3. (a) Express $\frac { 2 + 20 x } { 1 + 2 x - 8 x ^ { 2 } }$ as a sum of partial fractions.
(b) Hence find the series expansion of $\frac { 2 + 20 x } { 1 + 2 x - 8 x ^ { 2 } } , | x | < \frac { 1 } { 4 }$, in ascending powers of $x$ up to and including the term in $x ^ { 3 }$, simplifying each coefficient.
3. continued

Edexcel C4 Q4

4. The line $l _ { 1 }$ passes through the points $P$ and $Q$ with position vectors ( $- \mathbf { i } - 8 \mathbf { j } + 3 \mathbf { k }$ ) and ( $2 \mathbf { i } - 9 \mathbf { j } + \mathbf { k }$ ) respectively, relative to a fixed origin.

Find a vector equation for $l _ { 1 }$. The line $l _ { 2 }$ has the equation $$\mathbf { r } = ( 6 \mathbf { i } + a \mathbf { j } + b \mathbf { k } ) + \mu ( \mathbf { i } + 4 \mathbf { j } - \mathbf { k } )$$ and also passes through the point $Q$.
Find the values of the constants $a$ and $b$.
Find, in degrees to 1 decimal place, the acute angle between lines $l _ { 1 }$ and $l _ { 2 }$.
4. continued

Edexcel C4 Q5

5. At time $t = 0$, a tank of height 2 metres is completely filled with water. Water then leaks from a hole in the side of the tank such that the depth of water in the tank, $y$ metres, after $t$ hours satisfies the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} t } = - k \mathrm { e } ^ { - 0.2 t }$$ where $k$ is a positive constant,

Find an expression for $y$ in terms of $k$ and $t$. Given that two hours after being filled the depth of water in the tank is 1.6 metres,
find the value of $k$ to 4 significant figures. Given also that the hole in the tank is $h \mathrm {~cm}$ above the base of the tank,
show that $h = 79$ to 2 significant figures.
5. continued

Edexcel C4 Q6

6. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{922f404e-12d5-490b-9c8d-509f3a304c1e-10_438_700_255_518} \captionsetup{labelformat=empty} \caption{Figure 1}

\end{figure} Figure 1 shows the curve with parametric equations $$x = 2 - t ^ { 2 } , \quad y = t ( t + 1 ) , \quad t \geq 0$$

Find the coordinates of the points where the curve meets the coordinate axes.
Find the exact area of the region bounded by the curve and the coordinate axes.
6. continued

Edexcel C4 Q7

7. (a) Prove that $$\frac { \mathrm { d } } { \mathrm {~d} x } \left( a ^ { x } \right) = a ^ { x } \ln a .$$ A curve has the equation $y = 4 ^ { x } - 2 ^ { x - 1 } + 1$.
(b) Show that the tangent to the curve at the point where it crosses the $y$-axis has the equation $$3 x \ln 2 - 2 y + 3 = 0 .$$ (c) Find the exact coordinates of the stationary point of the curve.
7. continued

Edexcel C4 Q8

8. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{922f404e-12d5-490b-9c8d-509f3a304c1e-14_656_999_146_429} \captionsetup{labelformat=empty} \caption{Figure 2}

\end{figure} Figure 2 shows the curve with equation $y = \sqrt { \frac { x } { x + 1 } }$.
The shaded region is bounded by the curve, the $x$-axis and the line $x = 3$.

1. Use the trapezium rule with three strips to find an estimate for the area of the shaded region.
2. Use the trapezium rule with six strips to find an improved estimate for the area of the shaded region. The shaded region is rotated through $2 \pi$ radians about the $x$-axis.
Show that the volume of the solid formed is $\pi ( 3 - \ln 4 )$.
8. continued
8. continued

Edexcel C4 Q1

A curve has the equation

$$2 x ^ { 2 } + x y - y ^ { 2 } + 18 = 0$$ Find the coordinates of the points where the tangent to the curve is parallel to the $x$-axis.

Edexcel C4 Q2

2. Use the substitution $x = 2 \tan u$ to show that $$\int _ { 0 } ^ { 2 } \frac { x ^ { 2 } } { x ^ { 2 } + 4 } \mathrm {~d} x = \frac { 1 } { 2 } ( 4 - \pi )$$

Edexcel C4 Q3

(a) Show that $\left( 1 \frac { 1 } { 24 } \right) ^ { - \frac { 1 } { 2 } } = k \sqrt { 6 }$, where $k$ is rational.
(b) Expand $\left( 1 + \frac { 1 } { 2 } x \right) ^ { - \frac { 1 } { 2 } } , | x | < 2$, in ascending powers of $x$ up to and including the term in $x ^ { 3 }$, simplifying each coefficient.
(c) Use your answer to part (b) with $x = \frac { 1 } { 12 }$ to find an approximate value for $\sqrt { 6 }$, giving your answer to 5 decimal places.
continued
Relative to a fixed origin, two lines have the equations

$$\mathbf { r } = ( 7 \mathbf { j } - 4 \mathbf { k } ) + s ( 4 \mathbf { i } - 3 \mathbf { j } + \mathbf { k } )$$ and $$\mathbf { r } = ( - 7 \mathbf { i } + \mathbf { j } + 8 \mathbf { k } ) + t ( - 3 \mathbf { i } + 2 \mathbf { k } )$$ where $s$ and $t$ are scalar parameters.
(a) Show that the two lines intersect and find the position vector of the point where they meet.
(b) Find, in degrees to 1 decimal place, the acute angle between the lines.

Questions C4 (1162 questions)

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