Questions C4 - A-Level Maths

show that $1.48 < k < 1.49$. Given also that the point $( 0,5 )$ lies on $C$,
find $\mathrm { f } ( x )$. The finite region $R$ is bounded by $C$, the coordinate axes and the line $x = 2$.
Use integration to find the exact area of $R$.
(4)

Edexcel C4 Q6

6. When $( 1 + a x ) ^ { n }$ is expanded as a series in ascending powers of $x$, the coefficients of $x$ and $x ^ { 2 }$ are - 6 and 27 respectively.

Find the value of $a$ and the value of $n$.
Find the coefficient of $x ^ { 3 }$.
State the set of values of $x$ for which the expansion is valid.

Edexcel C4 Q7

7. Two submarines are travelling in straight lines through the ocean. Relative to a fixed origin, the vector equations of the two lines, $l _ { 1 }$ and $l _ { 2 }$, along which they travel are $$\text { and } \quad \begin{aligned} & \mathbf { r } = 3 \mathbf { i } + 4 \mathbf { j } - 5 \mathbf { k } + \lambda ( \mathbf { i } - 2 \mathbf { j } + 2 \mathbf { k } )
& \mathbf { r } = 9 \mathbf { i } + \mathbf { j } - 2 \mathbf { k } + \mu ( 4 \mathbf { i } + \mathbf { j } - \mathbf { k } ) , \end{aligned}$$ where $\lambda$ and $\mu$ are scalars.

Show that the submarines are moving in perpendicular directions.
Given that $l _ { 1 }$ and $l _ { 2 }$ intersect at the point $A$, find the position vector of $A$. The point $b$ has position vector $10 \mathbf { j } - 11 \mathbf { k }$.
Show that only one of the submarines passes through the point $B$.
Given that 1 unit on each coordinate axis represents 100 m , find, in km , the distance $A B$.

Edexcel C4 Q8

8. In a chemical reaction two substances combine to form a third substance. At time $t , t \geq 0$, the concentration of this third substance is $x$ and the reaction is modelled by the differential equation $$\frac { \mathrm { d } x } { \mathrm {~d} t } = k ( 1 - 2 x ) ( 1 - 4 x ) \text {, where } k \text { is a positive constant. }$$

Solve this differential equation and hence show that $$\ln \left| \frac { 1 - 2 x } { 1 - 4 x } \right| = 2 k t + c \text {, where } c \text { is an arbitrary constant. }$$
Given that $x = 0$ when $t = 0$, find an expression for $x$ in terms of $k$ and $t$.
Find the limiting value of the concentration $x$ as $t$ becomes very large. END

Edexcel C4 Q1

(a) Express $1.5 \sin 2 x + 2 \cos 2 x$ in the form $R \sin ( 2 x + \alpha )$, where $R > 0$ and $0 < \alpha < \frac { 1 } { 2 } \pi$, giving your values of $R$ and $\alpha$ to 3 decimal places where appropriate.
(b) Express $3 \sin x \cos x + 4 \cos ^ { 2 } x$ in the form $a \cos 2 x + b \sin 2 x + c$, where $a , b$ and $c$ are constants to be found.
(c) Hence, using your answer to part (a), deduce the maximum value of $3 \sin x \cos x + 4 \cos ^ { 2 } x$.

\begin{figure}[h]

\captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{a1b078fe-96e3-4d62-bf0d-415294ba022f-2_668_796_863_507}

\end{figure} Figure 1 shows part of the curve with equation $y = \mathrm { f } ( x )$, where $$f ( x ) = \frac { x ^ { 2 } + 1 } { ( 1 + x ) ( 3 - x ) } , 0 \leq x < 3$$ (a) Given that $\mathrm { f } ( x ) = A + \frac { B } { 1 + x } + \frac { C } { 3 - x }$, find the values of the constants $A , B$ and $C$. The finite region $R$, shown in Fig. 1, is bounded by the curve with equation $y = \mathrm { f } ( x )$, the $x$-axis, the $y$-axis and the line $x = 2$.
(b) Find the area of $R$, giving your answer in the form $p + q \ln r$, where $p , q$ and $r$ are rational constants to be found.

Edexcel C4 Q3

3. A student tests the accuracy of the trapezium rule by evaluating $I$, where $$I = \int _ { 0.5 } ^ { 1.5 } \left( \frac { 3 } { x } + x ^ { 4 } \right) \mathrm { d } x$$

Complete the student's table, giving values to 2 decimal places where appropriate.
$x$ 0.5 0.75 1 1.25 1.5
$\frac { 3 } { x } + x ^ { 4 }$ 6.06 4.32
Use the trapezium rule, with all the values from your table, to calculate an estimate for the value of $I$.
Use integration to calculate the exact value of $I$.
Verify that the answer obtained by the trapezium rule is within $3 \%$ of the exact value.

Edexcel C4 Q4

4. Figure 1
\includegraphics[max width=\textwidth, alt={}, center]{a1b078fe-96e3-4d62-bf0d-415294ba022f-4_588_1008_242_566} Figure 1 shows a cross-section $R$ of a dam. The line $A C$ is the vertical face of the dam, $A B$ is the horizontal base and the curve $B C$ is the profile. Taking $x$ and $y$ to be the horizontal and vertical axes, then $A , B$ and $C$ have coordinates $( 0,0 ) , \left( 3 \pi ^ { 2 } , 0 \right)$ and $( 0,30 )$ respectively. The area of the cross-section is to be calculated. Initially the profile $B C$ is approximated by a straight line.

Find an estimate for the area of the cross-section $R$ using this approximation.
(1) The profile $B C$ is actually described by the parametric equations. $$x = 16 t ^ { 2 } - \pi ^ { 2 } , \quad y = 30 \sin 2 t , \quad \frac { \pi } { 4 } \leq t \leq \frac { \pi } { 2 }$$
Find the exact area of the cross-section $R$.
(7)
Calculate the percentage error in the estimate of the area of the cross-section $R$ that you found in part (a).
(2)

Edexcel C4 Q5

5. (a) Prove that, when $x = \frac { 1 } { 15 }$, the value of $( 1 + 5 x ) ^ { - \frac { 1 } { 2 } }$ is exactly equal to $\sin 60 ^ { \circ }$.
(3)
(b) Expand $( 1 + 5 x ) ^ { - \frac { 1 } { 2 } } , | x | < 0.2$, in ascending powers of $x$ up to and including the term in $x ^ { 3 }$, simplifying each term.
(c) Use your answer to part (b) to find an approximation for $\sin 60 ^ { \circ }$.
(d) Find the difference between the exact value of $\sin 60 ^ { \circ }$ and the approximation in part (c).

Edexcel C4 Q6

6. (a) Use integration by parts to show that $$\int _ { 0 } ^ { \frac { \pi } { 4 } } x \sec ^ { 2 } x \mathrm {~d} x = \frac { 1 } { 4 } \pi - \frac { 1 } { 2 } \ln 2 .$$ \begin{figure}[h]

\captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{a1b078fe-96e3-4d62-bf0d-415294ba022f-5_841_1202_459_434}

\end{figure} The finite region $R$, bounded by the equation $y = x ^ { \frac { 1 } { 2 } } \sec x$, the line $x = \frac { \pi } { 4 }$ and the $x$-axis is shown in Fig. 1. The region $R$ is rotated through $2 \pi$ radians about the $x$-axis.
(b) Find the volume of the solid of revolution generated.
(c) Find the gradient of the curve with equation $y = x ^ { \frac { 1 } { 2 } } \sec x$ at the point where $x = \frac { \pi } { 4 }$.

Edexcel C4 Q7

7. \begin{figure}[h]

\captionsetup{labelformat=empty} \caption{Figure 3} \includegraphics[alt={},max width=\textwidth]{a1b078fe-96e3-4d62-bf0d-415294ba022f-6_805_1445_269_230}

\end{figure} The curve $C$ with equation $y = 2 \mathrm { e } ^ { x } + 5$ meets the $y$-axis at the point $M$, as shown in Fig. 3 .

Find the equation of the normal to $C$ at $M$ in the form $a x + b y = c$, where $a , b$ and $c$ are integers. This normal to $C$ at $M$ crosses the $x$-axis at the point $N ( n , 0 )$.
Show that $n = 14$. The point $P ( \ln 4,13 )$ lies on $C$. The finite region $R$ is bounded by $C$, the axes and the line $P N$, as shown in Fig. 3.
Find the area of $R$, giving your answers in the form $p + q \ln 2$, where $p$ and $q$ are integers to be found.

Edexcel C4 Q8

8. Referred to an origin $O$, the points $A , B$ and $C$ have position vectors ( $9 \mathbf { i } - 2 \mathbf { j } + \mathbf { k }$ ), $( 6 \mathbf { i } + 2 \mathbf { j } + 6 \mathbf { k } )$ and $( 3 \mathbf { i } + p \mathbf { j } + q \mathbf { k } )$ respectively, where $p$ and $q$ are constants.

Find, in vector form, an equation of the line $l$ which passes through $A$ and $B$. Given that $C$ lies on $l$,
find the value of $p$ and the value of $q$,
calculate, in degrees, the acute angle between $O C$ and $A B$. The point $D$ lies on $A B$ and is such that $O D$ is perpendicular to $A B$.
Find the position vector of $D$.

Edexcel C4 Q1

A curve has the equation

$$x ^ { 2 } ( 2 + y ) - y ^ { 2 } = 0 .$$ Find an expression for $\frac { \mathrm { d } y } { \mathrm {~d} x }$ in terms of $x$ and $y$.

Edexcel C4 Q2

2. $$f ( x ) = \frac { 3 } { \sqrt { 1 - x } } , | x | < 1$$

Show that $\mathrm { f } \left( \frac { 1 } { 10 } \right) = \sqrt { 10 }$.
Expand $\mathrm { f } ( x )$ in ascending powers of $x$ up to and including the term in $x ^ { 3 }$, simplifying each coefficient.
Use your expansion to find an approximate value for $\sqrt { 10 }$, giving your answer to 8 significant figures.
Find, to 1 significant figure, the percentage error in your answer to part (c).

Edexcel C4 Q3

3. Relative to a fixed origin, $O$, the line $l$ has the equation $$\mathbf { r } = ( \mathbf { i } + p \mathbf { j } - 5 \mathbf { k } ) + \lambda ( 3 \mathbf { i } - \mathbf { j } + q \mathbf { k } ) ,$$ where $p$ and $q$ are constants and $\lambda$ is a scalar parameter.
Given that the point $A$ with coordinates $( - 5,9 , - 9 )$ lies on $l$,

find the values of $p$ and $q$,
show that the point $B$ with coordinates $( 25 , - 1,11 )$ also lies on $l$. The point $C$ lies on $l$ and is such that $O C$ is perpendicular to $l$.
Find the coordinates of $C$.
Find the ratio $A C : C B$
3. continued

Edexcel C4 Q4

4. During a chemical reaction, a compound is being made from two other substances. At time $t$ hours after the start of the reaction, $x \mathrm {~g}$ of the compound has been produced. Assuming that $x = 0$ initially, and that $$\frac { \mathrm { d } x } { \mathrm {~d} t } = 2 ( x - 6 ) ( x - 3 )$$

show that it takes approximately 7 minutes to produce 2 g of the compound.
Explain why it is not possible to produce 3 g of the compound.
4. continued

Edexcel C4 Q5

5. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{e877dc80-4cfc-4c8b-9640-9b186cd7ab13-08_617_917_146_475} \captionsetup{labelformat=empty} \caption{Figure 1}

\end{figure} Figure 1 shows the curve with equation $y = 4 x ^ { \frac { 1 } { 2 } } \mathrm { e } ^ { - x }$.
The shaded region is bounded by the curve, the $x$-axis and the line $x = 2$.

Use the trapezium rule with four intervals of equal width to estimate the area of the shaded region. The shaded region is rotated through $2 \pi$ radians about the $x$-axis.
Find, in terms of $\pi$ and e, the exact volume of the solid formed.
5. continued

Edexcel C4 Q6

6. (a) Find $$\int 2 \sin 3 x \sin 2 x d x$$ (b) Use the substitution $u ^ { 2 } = x + 1$ to evaluate $$\int _ { 0 } ^ { 3 } \frac { x ^ { 2 } } { \sqrt { x + 1 } } \mathrm {~d} x$$ 6. continued

Edexcel C4 Q7

7. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{e877dc80-4cfc-4c8b-9640-9b186cd7ab13-12_556_860_246_452} \captionsetup{labelformat=empty} \caption{Figure 2}

\end{figure} Figure 2 shows the curve with parametric equations $$x = \cos 2 t , \quad y = \operatorname { cosec } t , \quad 0 < t < \frac { \pi } { 2 } .$$ The point $P$ on the curve has $x$-coordinate $\frac { 1 } { 2 }$.

Find the value of the parameter $t$ at $P$.
Show that the tangent to the curve at $P$ has the equation $$y = 2 x + 1$$ The shaded region is bounded by the curve, the coordinate axes and the line $x = \frac { 1 } { 2 }$.
Show that the area of the shaded region is given by $$\int _ { \frac { \pi } { 6 } } ^ { \frac { \pi } { 4 } } k \cos t \mathrm {~d} t$$ where $k$ is a positive integer to be found.
Hence find the exact area of the shaded region.
7. continued
7. continued

Edexcel C4 Q1

Use integration by parts to find

$$\int x ^ { 2 } \sin x d x$$

Edexcel C4 Q2

Given that $y = - 2$ when $x = 1$, solve the differential equation

$$\frac { \mathrm { d } y } { \mathrm {~d} x } = y ^ { 2 } \sqrt { x }$$ giving your answer in the form $y = \mathrm { f } ( x )$.

Edexcel C4 Q3

3. A curve has the equation $$4 x ^ { 2 } - 2 x y - y ^ { 2 } + 11 = 0$$ Find an equation for the normal to the curve at the point with coordinates $( - 1 , - 3 )$. (8)
3. continued

Edexcel C4 Q4

4. (a) Expand $( 1 + a x ) ^ { - 3 } , | a x | < 1$, in ascending powers of $x$ up to and including the term in $x ^ { 3 }$. Give each coefficient as simply as possible in terms of the constant $a$. Given that the coefficient of $x ^ { 2 }$ in the expansion of $\frac { 6 - x } { ( 1 + a x ) ^ { 3 } } , | a x | < 1$, is 3 ,
(b) find the two possible values of $a$. Given also that $a < 0$,
(c) show that the coefficient of $x ^ { 3 }$ in the expansion of $\frac { 6 - x } { ( 1 + a x ) ^ { 3 } }$ is $\frac { 14 } { 9 }$.
4. continued

Edexcel C4 Q5

5. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{b71c9832-e502-4a25-85fb-a49c03ea9209-08_663_899_146_495} \captionsetup{labelformat=empty} \caption{Figure 1}

\end{figure} Figure 1 shows the curve with equation $y = \frac { 1 } { \sqrt { 3 x + 1 } }$.
The shaded region is bounded by the curve, the $x$-axis and the lines $x = 1$ and $x = 5$.

Find the area of the shaded region. The shaded region is rotated completely about the $x$-axis.
Find the volume of the solid formed, giving your answer in the form $k \pi \ln 2$, where $k$ is a simplified fraction.
5. continued

Edexcel C4 Q6

6. $$f ( x ) = \frac { 15 - 17 x } { ( 2 + x ) ( 1 - 3 x ) ^ { 2 } } , \quad x \neq - 2 , \quad x \neq \frac { 1 } { 3 }$$

Find the values of the constants $A , B$ and $C$ such that $$\mathrm { f } ( x ) = \frac { A } { 2 + x } + \frac { B } { 1 - 3 x } + \frac { C } { ( 1 - 3 x ) ^ { 2 } }$$
Find the value of $$\int _ { - 1 } ^ { 0 } f ( x ) d x$$ giving your answer in the form $p + \ln q$, where $p$ and $q$ are integers.
6. continued

Edexcel C4 Q7

7. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{b71c9832-e502-4a25-85fb-a49c03ea9209-12_495_784_246_461} \captionsetup{labelformat=empty} \caption{Figure 2}

\end{figure} Figure 2 shows the curve with parametric equations $$x = - 1 + 4 \cos \theta , \quad y = 2 \sqrt { 2 } \sin \theta , \quad 0 \leq \theta < 2 \pi$$ The point $P$ on the curve has coordinates $( 1 , \sqrt { 6 } )$.

Find the value of $\theta$ at $P$.
Show that the normal to the curve at $P$ passes through the origin.
Find a cartesian equation for the curve.
7. continued

\(x\)	0.5	0.75	1	1.25	1.5
\(\frac { 3 } { x } + x ^ { 4 }\)	6.06	4.32

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