Questions - Edexcel C4 - A-Level Maths

20. \begin{figure}[h]

\captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{615ec68b-3a32-4309-bb54-acf39ed09f96-14_637_749_322_657}

\end{figure} Figure 1 shows part of the curve with equation $y = \mathrm { f } ( x )$, where $$\mathrm { f } ( x ) = \frac { x ^ { 2 } + 1 } { ( 1 + x ) ( 3 - x ) } , 0 \leq x < 3$$

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 $x = 2$.
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.
21. (a) Prove that, when $x = \frac { 1 } { 15 }$, the value of $( 1 + 5 x ) ^ { - \frac { 1 } { 2 } }$ is exactly equal to $\sin 60 ^ { \circ }$.
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.
Use your answer to part (b) to find an approximation for $\sin 60 ^ { \circ }$.
Find the difference between the exact value of $\sin 60 ^ { \circ }$ and the approximation in part (c).
22. A curve is given parametrically by the equations $$x = 5 \cos t , \quad y = - 2 + 4 \sin t , \quad 0 \leq t < 2 \pi$$
Find the coordinates of all the points at which $C$ intersects the coordinate axes, giving your answers in surd form where appropriate.
Sketch the graph at $C$.
$P$ is the point on $C$ where $t = \frac { 1 } { 6 } \pi$.
Show that the normal to $C$ at $P$ has equation $$8 \sqrt { 3 } y = 10 x - 25 \sqrt { 3 }$$ [P3 June 2002 Question 6]
23. A Pancho car has value $\pounds V$ at time $t$ years. A model for $V$ assumes that the rate of decrease of $V$ at time $t$ is proportional to $V$.
By forming and solving an appropriate differential equation, show that $V = A \mathrm { e } ^ { - k t }$, where $A$ and $k$ are positive constants. The value of a new Pancho car is $\pounds 20000$, and when it is 3 years old its value is $\pounds 11000$.
Find, to the nearest $\pounds 100$, an estimate for the value of the Pancho when it is 10 years old. A Pancho car is regarded as 'scrap' when its value falls below $\pounds 500$.
Find the approximate age of the Pancho when it becomes 'scrap'.
[0pt] [P3 June 2002 Question 7]
24. 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 $D$.
25. \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 2} \includegraphics[alt={},max width=\textwidth]{615ec68b-3a32-4309-bb54-acf39ed09f96-16_504_940_1213_584}
\end{figure} Figure 2 shows part of the curve with equation $y = x ^ { 2 } + 2$.
The finite region $R$ is bounded by the curve, the $x$-axis and the lines $x = 0$ and $x = 2$.
Use the trapezium rule with 4 strips of equal width to estimate the area of $R$.
State, with a reason, whether your answer in part (a) is an under-estimate or over-estimate of the area of $R$.
Using integration, find the volume of the solid generated when $R$ is rotated through $360 ^ { \circ }$ about the $x$-axis, giving your answer in terms of $\pi$.
26. \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{615ec68b-3a32-4309-bb54-acf39ed09f96-17_723_987_287_655}
\end{figure} Figure 1 shows part of the curve with equation $y = 1 + \frac { 1 } { 2 \sqrt { x } }$. The shaded region $R$, bounded by the curve, that $x$-axis and the lines $x = 1$ and $x = 4$, is rotated through $360 ^ { \circ }$ about the $x$-axis. Using integration, show that the volume of the solid generated is $\pi \left( 5 + \frac { 1 } { 2 } \ln 2 \right)$.
(8)
[0pt] [P2 January 2003 Question 4]
27. \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 2} \includegraphics[alt={},max width=\textwidth]{615ec68b-3a32-4309-bb54-acf39ed09f96-18_757_1150_285_494}
\end{figure} Figure 2 shows the cross-section of a road tunnel and its concrete surround. The curved section of the tunnel is modelled by the curve with equation $y = 8 \sqrt { \left( \sin \frac { \pi x } { 10 } \right) }$, in the interval $0 \leq x \leq 10$. The concrete surround is represented by the shaded area bounded by the curve, the $x$-axis and the lines $x = - 2 , x = 12$ and $y = 10$. The units on both axes are metres.
Using this model, copy and complete the table below, giving the values of $y$ to 2 decimal places.
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.
47. 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.
48. The curve $C$ has equation $5 x ^ { 2 } + 2 x y - 3 y ^ { 2 } + 3 = 0$. The point $P$ on the curve $C$ has coordinates $( 1,2 )$.
Find the gradient of the curve at $P$.
Find the equation of the normal to the curve $C$ at $P$, in the form $y = a x + b$, where $a$ and $b$ are constants.
49. \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{615ec68b-3a32-4309-bb54-acf39ed09f96-32_545_824_850_837}
\end{figure} 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. 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 $R$.
Calculate the percentage error in the estimate of the area of the cross-section $R$ that you found in part (a).
50. (a) Express $\frac { 13 - 2 x } { ( 2 x - 3 ) ( x + 1 ) }$ in partial fractions.
Given that $y = 4$ at $x = 2$, use your answer to part (a) to find the solution of the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { y ( 13 - 2 x ) } { ( 2 x - 3 ) ( x + 1 ) } , \quad x > 1.5$$ Express your answer in the form $y = \mathrm { f } ( x )$.
[0pt] [P3 January 2004 Question 6]
51. The curve $C$ has equation $y = \frac { x } { 4 + x ^ { 2 } }$.
Use calculus to find the coordinates of the turning points of $C$. Using the result $\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } = \frac { 2 x \left( x ^ { 2 } - 12 \right) } { \left( 4 + x ^ { 2 } \right) ^ { 3 } }$, or otherwise,
determine the nature of each of the turning points.
Sketch the curve $C$.
52. The equations of the lines $l _ { 1 }$ and $l _ { 2 }$ are given by $$\begin{array} { l l } l _ { 1 } : & \mathbf { r } = \mathbf { i } + 3 \mathbf { j } + 5 \mathbf { k } + \lambda ( \mathbf { i } + 2 \mathbf { j } - \mathbf { k } ) ,
l _ { 2 } : & \mathbf { r } = - 2 \mathbf { i } + 3 \mathbf { j } - 4 \mathbf { k } + \mu ( 2 \mathbf { i } + \mathbf { j } + 4 \mathbf { k } ) , \end{array}$$ where $\lambda$ and $\mu$ are parameters.
Show that $l _ { 1 }$ and $l _ { 2 }$ intersect and find the coordinates of $Q$, their point of intersection.
Show that $l _ { 1 }$ is perpendicular to $l _ { 2 }$. The point $P$ with $x$-coordinate 3 lies on the line $l _ { 1 }$ and the point $R$ with $x$-coordinate 4 lies on the line $l _ { 2 }$.
Find, in its simplest form, the exact area of the triangle $P Q R$.
[0pt] [P3 January 2004 Question 8]
53. Figure 1
\includegraphics[max width=\textwidth, alt={}, center]{615ec68b-3a32-4309-bb54-acf39ed09f96-35_568_886_335_640} Figure 1 shows parts of the curve $C$ with equation $$y = \frac { x + 2 } { \sqrt { } x } .$$ The shaded region $R$ is bounded by $C$, the $x$-axis and the lines $x = 1$ and $x = 4$. This region is rotated through $360 ^ { \circ }$ about the $x$-axis to form a solid $S$.
Find, by integration, the exact volume of $S$. The solid $S$ is used to model a wooden support with a circular base and a circular top.
Show that the base and the top have the same radius. Given that the actual radius of the base is 6 cm ,
show that the volume of the wooden support is approximately $630 \mathrm {~cm} ^ { 3 }$.
54. $\mathrm { f } ( x ) = x + \frac { \mathrm { e } ^ { x } } { 5 } , \quad x \in \mathbb { R }$.
Find $\mathrm { f } ^ { \prime } ( x )$. The curve $C$, with equation $y = \mathrm { f } ( x )$, crosses the $y$-axis at the point $A$.
Find an equation for the tangent to $C$ at $A$.
Complete the table, giving the values of $\sqrt { \left( x + \frac { \mathrm { e } ^ { x } } { 5 } \right) }$ to 2 decimal places.
$x$ 0 0.5 1 1.5 2
$\sqrt { \left( x + \frac { \mathrm { e } ^ { x } } { 5 } \right) }$ 0.45 0.91
Use the trapezium rule, with all the values from your table, to find an approximation for the value of $$\int _ { 0 } ^ { 2 } \sqrt { \left( x + \frac { \mathrm { e } ^ { x } } { 5 } \right) } \mathrm { d } x$$
1. The circle $C$ has centre $( 5,13 )$ and touches the $x$-axis.
2. Find an equation of $C$ in terms of $x$ and $y$.
3. Find an equation of the tangent to $C$ at the point (10, 1), giving your answer in the form $a y + b x + c = 0$, where $a , b$ and $c$ are integers.
4. Use the substitution $u = 1 + \sin x$ and integration to show that
$$\int \sin x \cos x ( 1 + \sin x ) ^ { 5 } \mathrm {~d} x = \frac { 1 } { 42 } ( 1 + \sin x ) ^ { 6 } [ 6 \sin x - 1 ] + \text { constant. }$$
1. Given that
$$\frac { 3 + 5 x } { ( 1 + 3 x ) ( 1 - x ) } \equiv \frac { A } { 1 + 3 x } + \frac { B } { 1 - x } ,$$
find the values of the constants $A$ and $B$.
Hence, or otherwise, find the series expansion in ascending powers of $x$, up to and including the term in $x ^ { 2 }$, of $$\frac { 3 + 5 x } { ( 1 + 3 x ) ( 1 - x ) } .$$
State, with a reason, whether your series expansion in part (b) is valid for $x = \frac { 1 } { 2 }$.
[0pt] [P3 June 2004 Question 5]
58. \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{615ec68b-3a32-4309-bb54-acf39ed09f96-38_528_1028_300_311}
\end{figure} Figure 1 shows a sketch of the curve $C$ with parametric equations $$x = 3 t \sin t , y = 2 \sec t , \quad 0 \leq t < \frac { \pi } { 2 } .$$ The point $P ( a , 4 )$ lies on $C$.
Find the exact value of $a$. The region $R$ is enclosed by $C$, the axes and the line $x = a$ as shown in Fig. 1 .
Show that the area of $R$ is given by $$6 \int _ { 0 } ^ { \frac { \pi } { 3 } } ( \tan t + t ) \mathrm { d } t .$$
Find the exact value of the area of $R$.
59. A drop of oil is modelled as a circle of radius $r$. At time $t$ $$r = 4 \left( 1 - \mathrm { e } ^ { - \lambda t } \right) , \quad t > 0$$ where $\lambda$ is a positive constant.
Show that the area $A$ of the circle satisfies $$\frac { \mathrm { d } A } { \mathrm {~d} t } = 32 \pi \lambda \left( \mathrm { e } ^ { - \lambda t } - \mathrm { e } ^ { - 2 \lambda t } \right)$$ In an alternative model of the drop of oil its area $A$ at time $t$ satisfies $$\frac { \mathrm { d } A } { \mathrm {~d} t } = \frac { A ^ { \frac { 3 } { 2 } } } { t ^ { 2 } } , \quad \quad t > 0$$ Given that the area of the drop is 1 at $t = 1$,
find an expression for $A$ in terms of t for this alternative model.
Show that, in the alternative model, the value of $A$ cannot exceed 4 .
60. Relative to a fixed origin $O$, the vector equations of the two lines $l _ { 1 }$ and $l _ { 2 }$ are $$l _ { 1 } : \mathbf { r } = 9 \mathbf { i } + 2 \mathbf { j } + 4 \mathbf { k } + t ( - 8 \mathbf { i } - 3 \mathbf { j } + 5 \mathbf { k } ) ,$$ and $$l _ { 2 } : \mathbf { r } = - 16 \mathbf { i } + \alpha \mathbf { j } + 10 \mathbf { k } + s ( \mathbf { i } - 4 \mathbf { j } + 9 \mathbf { k } ) ,$$ where $\alpha$ is a constant.
The two lines intersect at the point $A$.
Find the value of $\alpha$.
Find the position vector of the point $A$.
Prove that the acute angle between $l _ { 1 }$ and $l _ { 2 }$ is $60 ^ { \circ }$. Point $B$ lies on $l _ { 1 }$ and point $C$ lies on $l _ { 2 }$. The triangle $A B C$ is equilateral with sides of length $14 \sqrt { } 2$.
Find one of the possible position vectors for the point $B$ and the corresponding position vector for the point $C$.

\(t\)	0	5	10	15	20	25
\(\boldsymbol { v }\)		1.56	7.23	17.36

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\(x\)	0	0.5	1	1.5	2
\(\sqrt { \left( x + \frac { \mathrm { e } ^ { x } } { 5 } \right) }\)	0.45	0.91