Questions — Edexcel C4 (360 questions)

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Edexcel C4 2012 June Q2
2. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{12fbfe89-60fe-4890-9a22-2b1988d05d33-03_424_465_228_721} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 shows a metal cube which is expanding uniformly as it is heated. At time \(t\) seconds, the length of each edge of the cube is \(x \mathrm {~cm}\), and the volume of the cube is \(V \mathrm {~cm} ^ { 3 }\).
  1. Show that \(\frac { \mathrm { d } V } { \mathrm {~d} x } = 3 x ^ { 2 }\) Given that the volume, \(V \mathrm {~cm} ^ { 3 }\), increases at a constant rate of \(0.048 \mathrm {~cm} ^ { 3 } \mathrm {~s} ^ { - 1 }\),
  2. find \(\frac { \mathrm { d } x } { \mathrm {~d} t }\), when \(x = 8\)
  3. find the rate of increase of the total surface area of the cube, in \(\mathrm { cm } ^ { 2 } \mathrm {~s} ^ { - 1 }\), when \(x = 8\)
Edexcel C4 2012 June Q3
3. $$f ( x ) = \frac { 6 } { \sqrt { ( 9 - 4 x ) } } , \quad | x | < \frac { 9 } { 4 }$$
  1. Find the binomial expansion of \(\mathrm { f } ( x )\) in ascending powers of \(x\), up to and including the term in \(x ^ { 3 }\). Give each coefficient in its simplest form.
    (6) Use your answer to part (a) to find the binomial expansion in ascending powers of \(x\), up to and including the term in \(x ^ { 3 }\), of
  2. \(\quad \mathrm { g } ( x ) = \frac { 6 } { \sqrt { } ( 9 + 4 x ) } , \quad | x | < \frac { 9 } { 4 }\)
  3. \(\mathrm { h } ( x ) = \frac { 6 } { \sqrt { } ( 9 - 8 x ) } , \quad | x | < \frac { 9 } { 8 }\)
Edexcel C4 2012 June Q4
  1. Given that \(y = 2\) at \(x = \frac { \pi } { 4 }\), solve the differential equation
$$\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 3 } { y \cos ^ { 2 } x }$$
Edexcel C4 2012 June Q5
  1. The curve \(C\) has equation
$$16 y ^ { 3 } + 9 x ^ { 2 } y - 54 x = 0$$
  1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(x\) and \(y\).
  2. Find the coordinates of the points on \(C\) where \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 0\).
Edexcel C4 2012 June Q6
6. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{12fbfe89-60fe-4890-9a22-2b1988d05d33-09_831_784_127_580} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} Figure 2 shows a sketch of the curve \(C\) with parametric equations $$x = ( \sqrt { } 3 ) \sin 2 t , \quad y = 4 \cos ^ { 2 } t , \quad 0 \leqslant t \leqslant \pi$$
  1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = k ( \sqrt { } 3 ) \tan 2 t\), where \(k\) is a constant to be determined.
  2. Find an equation of the tangent to \(C\) at the point where \(t = \frac { \pi } { 3 }\). Give your answer in the form \(y = a x + b\), where \(a\) and \(b\) are constants.
  3. Find a cartesian equation of \(C\).
Edexcel C4 2012 June Q7
7. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{12fbfe89-60fe-4890-9a22-2b1988d05d33-11_754_1177_217_388} \captionsetup{labelformat=empty} \caption{Figure 3}
\end{figure} Figure 3 shows a sketch of part of the curve with equation \(y = x ^ { \frac { 1 } { 2 } } \ln 2 x\).
The finite region \(R\), shown shaded in Figure 3, is bounded by the curve, the \(x\)-axis and the lines \(x = 1\) and \(x = 4\)
  1. Use the trapezium rule, with 3 strips of equal width, to find an estimate for the area of \(R\), giving your answer to 2 decimal places.
  2. Find \(\int x ^ { \frac { 1 } { 2 } } \ln 2 x \mathrm {~d} x\).
  3. Hence find the exact area of \(R\), giving your answer in the form \(a \ln 2 + b\), where \(a\) and \(b\) are exact constants.
Edexcel C4 2012 June Q8
  1. Relative to a fixed origin \(O\), the point \(A\) has position vector \(( 10 \mathbf { i } + 2 \mathbf { j } + 3 \mathbf { k } )\), and the point \(B\) has position vector \(( 8 \mathbf { i } + 3 \mathbf { j } + 4 \mathbf { k } )\).
The line \(l\) passes through the points \(A\) and \(B\).
  1. Find the vector \(\overrightarrow { A B }\).
  2. Find a vector equation for the line \(l\). The point \(C\) has position vector \(( 3 \mathbf { i } + 12 \mathbf { j } + 3 \mathbf { k } )\).
    The point \(P\) lies on \(l\). Given that the vector \(\overrightarrow { C P }\) is perpendicular to \(l\),
  3. find the position vector of the point \(P\).
Edexcel C4 2013 June Q1
  1. Express in partial fractions
$$\frac { 5 x + 3 } { ( 2 x + 1 ) ( x + 1 ) ^ { 2 } }$$
Edexcel C4 2013 June Q2
2. The curve \(C\) has equation $$3 ^ { x - 1 } + x y - y ^ { 2 } + 5 = 0$$ Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) at the point \(( 1,3 )\) on the curve \(C\) can be written in the form \(\frac { 1 } { \lambda } \ln \left( \mu \mathrm { e } ^ { 3 } \right)\), where \(\lambda\) and \(\mu\) are integers to be found.
Edexcel C4 2013 June Q3
3. Using the substitution \(u = 2 + \sqrt { } ( 2 x + 1 )\), or other suitable substitutions, find the exact value of $$\int _ { 0 } ^ { 4 } \frac { 1 } { 2 + \sqrt { } ( 2 x + 1 ) } d x$$ giving your answer in the form \(A + 2 \ln B\), where \(A\) is an integer and \(B\) is a positive constant.
Edexcel C4 2013 June Q4
4. (a) Find the binomial expansion of $$\sqrt [ 3 ] { ( 8 - 9 x ) , \quad } \quad | x | < \frac { 8 } { 9 }$$ in ascending powers of \(x\), up to and including the term in \(x ^ { 3 }\). Give each coefficient as a simplified fraction.
(b) Use your expansion to estimate an approximate value for \(\sqrt [ 3 ] { 7100 }\), giving your answer to 4 decimal places. State the value of \(x\), which you use in your expansion, and show all your working.
Edexcel C4 2013 June Q5
5. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{08f62966-2e63-4542-a10a-c6453a3215e7-06_689_992_118_484} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 shows part of the curve with equation \(x = 4 t \mathrm { e } ^ { - \frac { 1 } { 3 } t } + 3\). The finite region \(R\) shown shaded in Figure 1 is bounded by the curve, the \(x\)-axis, the \(t\)-axis and the line \(t = 8\).
  1. Complete the table with the value of \(x\) corresponding to \(t = 6\), giving your answer to 3 decimal places.
    \(t\)02468
    \(x\)37.1077.2185.223
  2. Use the trapezium rule with all the values of \(x\) in the completed table to obtain an estimate for the area of the region \(R\), giving your answer to 2 decimal places.
  3. Use calculus to find the exact value for the area of \(R\).
  4. Find the difference between the values obtained in part (b) and part (c), giving your answer to 2 decimal places.
Edexcel C4 2013 June Q6
6. Relative to a fixed origin \(O\), the point \(A\) has position vector \(21 \mathbf { i } - 17 \mathbf { j } + 6 \mathbf { k }\) and the point \(B\) has position vector \(25 \mathbf { i } - 14 \mathbf { j } + 18 \mathbf { k }\). The line \(l\) has vector equation $$\mathbf { r } = \left( \begin{array} { r } a
b
Edexcel C4 2013 June Q7
7. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{08f62966-2e63-4542-a10a-c6453a3215e7-10_542_1164_251_477} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} Figure 2 shows a sketch of the curve \(C\) with parametric equations $$x = 27 \sec ^ { 3 } t , y = 3 \tan t , \quad 0 \leqslant t \leqslant \frac { \pi } { 3 }$$
  1. Find the gradient of the curve \(C\) at the point where \(t = \frac { \pi } { 6 }\)
  2. Show that the cartesian equation of \(C\) may be written in the form $$y = \left( x ^ { \frac { 2 } { 3 } } - 9 \right) ^ { \frac { 1 } { 2 } } , \quad a \leqslant x \leqslant b$$ stating the values of \(a\) and \(b\).
    (3) \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{08f62966-2e63-4542-a10a-c6453a3215e7-10_581_1173_1628_475} \captionsetup{labelformat=empty} \caption{Figure 3}
    \end{figure} The finite region \(R\) which is bounded by the curve \(C\), the \(x\)-axis and the line \(x = 125\) is shown shaded in Figure 3. This region is rotated through \(2 \pi\) radians about the \(x\)-axis to form a solid of revolution.
  3. Use calculus to find the exact value of the volume of the solid of revolution.
Edexcel C4 2013 June Q10
10 \end{array} \right) + \lambda \left( \begin{array} { r } 6
c
- 1 \end{array} \right)$$ where \(a , b\) and \(c\) are constants and \(\lambda\) is a parameter.
Given that the point \(A\) lies on the line \(l\),
  1. find the value of \(a\). Given also that the vector \(\overrightarrow { A B }\) is perpendicular to \(l\),
  2. find the values of \(b\) and \(c\),
  3. find the distance \(A B\). The image of the point \(B\) after reflection in the line \(l\) is the point \(B ^ { \prime }\).
  4. Find the position vector of the point \(B ^ { \prime }\).
    7. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{08f62966-2e63-4542-a10a-c6453a3215e7-10_542_1164_251_477} \captionsetup{labelformat=empty} \caption{Figure 2}
    \end{figure} Figure 2 shows a sketch of the curve \(C\) with parametric equations $$x = 27 \sec ^ { 3 } t , y = 3 \tan t , \quad 0 \leqslant t \leqslant \frac { \pi } { 3 }$$
  5. Find the gradient of the curve \(C\) at the point where \(t = \frac { \pi } { 6 }\)
  6. Show that the cartesian equation of \(C\) may be written in the form $$y = \left( x ^ { \frac { 2 } { 3 } } - 9 \right) ^ { \frac { 1 } { 2 } } , \quad a \leqslant x \leqslant b$$ stating the values of \(a\) and \(b\).
    (3) \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{08f62966-2e63-4542-a10a-c6453a3215e7-10_581_1173_1628_475} \captionsetup{labelformat=empty} \caption{Figure 3}
    \end{figure} The finite region \(R\) which is bounded by the curve \(C\), the \(x\)-axis and the line \(x = 125\) is shown shaded in Figure 3. This region is rotated through \(2 \pi\) radians about the \(x\)-axis to form a solid of revolution.
  7. Use calculus to find the exact value of the volume of the solid of revolution. 8. In an experiment testing solid rocket fuel, some fuel is burned and the waste products are collected. Throughout the experiment the sum of the masses of the unburned fuel and waste products remains constant. Let \(x\) be the mass of waste products, in kg , at time \(t\) minutes after the start of the experiment. It is known that at time \(t\) minutes, the rate of increase of the mass of waste products, in kg per minute, is \(k\) times the mass of unburned fuel remaining, where \(k\) is a positive constant. The differential equation connecting \(x\) and \(t\) may be written in the form $$\frac { \mathrm { d } x } { \mathrm {~d} t } = k ( M - x ) , \text { where } M \text { is a constant. }$$
  8. Explain, in the context of the problem, what \(\frac { \mathrm { d } x } { \mathrm {~d} t }\) and \(M\) represent. Given that initially the mass of waste products is zero,
  9. solve the differential equation, expressing \(x\) in terms of \(k , M\) and \(t\). Given also that \(x = \frac { 1 } { 2 } M\) when \(t = \ln 4\),
  10. find the value of \(x\) when \(t = \ln 9\), expressing \(x\) in terms of \(M\), in its simplest form.
Edexcel C4 2013 June Q1
  1. (a) Find the binomial expansion of
$$\sqrt { } ( 9 + 8 x ) , \quad | x | < \frac { 9 } { 8 }$$ in ascending powers of \(x\), up to and including the term in \(x ^ { 2 }\).
Give each coefficient as a simplified fraction.
(b) Use your expansion to estimate the value of \(\sqrt { } ( 11 )\), giving your answer as a single fraction.
Edexcel C4 2013 June Q2
2. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{b0d21f0f-f5f6-4ca5-8e3e-98aee0d9db7a-03_735_1171_360_490} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 shows a sketch of part of the curve with equation \(y = x \mathrm { e } ^ { - \frac { 1 } { 2 } x } , x \geqslant 0\).
The finite region \(R\), shown shaded in Figure 1, is bounded by the curve, the \(x\)-axis, and the line \(x = 4\). The table shows corresponding values of \(x\) and \(y\) for \(y = x e ^ { - \frac { 1 } { 2 } x }\).
\(x\)01234
\(y\)0\(\mathrm { e } ^ { - \frac { 1 } { 2 } }\)\(3 \mathrm { e } ^ { - \frac { 3 } { 2 } }\)\(4 \mathrm { e } ^ { - 2 }\)
  1. Complete the table with the value of \(y\) corresponding to \(x = 2\)
  2. Use the trapezium rule, with all the values of \(y\) in the completed table, to obtain an estimate for the area of \(R\), giving your answer to 2 decimal places.
    1. Find \(\int x \mathrm { e } ^ { - \frac { 1 } { 2 } x } \mathrm {~d} x\).
    2. Hence find the exact area of \(R\), giving your answer in the form \(a + b \mathrm { e } ^ { - 2 }\), where \(a\) and \(b\) are integers.
Edexcel C4 2013 June Q3
  1. A curve \(C\) has parametric equations
$$x = 2 t + 5 , \quad y = 3 + \frac { 4 } { t } , \quad t \neq 0$$
  1. Find the value of \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) at the point on \(C\) with coordinates \(( 9,5 )\).
  2. Find a cartesian equation of the curve in the form $$y = \frac { a x + b } { c x + d }$$ where \(a\), \(b\), \(c\) and \(d\) are integers.
Edexcel C4 2013 June Q4
4. With respect to a fixed origin \(O\), the line \(l _ { 1 }\) has vector equation $$\mathbf { r } = \left( \begin{array} { r } - 9
Edexcel C4 2013 June Q5
5 \end{array} \right) + \mu \left( \begin{array} { r } 5
- 4
- 3 \end{array} \right)$$ where \(\mu\) is a scalar parameter.
The point \(A\) is on \(l _ { 1 }\) where \(\mu = 2\).
  1. Write down the coordinates of \(A\). The acute angle between \(O A\) and \(l _ { 1 }\) is \(\theta\), where \(O\) is the origin.
  2. Find the value of \(\cos \theta\). The point \(B\) is such that \(\overrightarrow { O B } = 3 \overrightarrow { O A }\).
    The line \(l _ { 2 }\) passes through the point \(B\) and is parallel to the line \(l _ { 1 }\).
  3. Find a vector equation of \(l _ { 2 }\).
  4. Find the length of \(O B\), giving your answer as a simplified surd. The point \(X\) lies on \(l _ { 2 }\). Given that the vector \(\overrightarrow { O X }\) is perpendicular to \(l _ { 2 }\),
  5. find the length of \(O X\), giving your answer to 3 significant figures.
    5. The curve \(C\) has the equation $$\sin ( \pi y ) - y - x ^ { 2 } y = - 5 , \quad x > 0$$
  6. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(x\) and \(y\). The point \(P\) with coordinates \(( 2,1 )\) lies on \(C\).
    The tangent to \(C\) at \(P\) meets the \(x\)-axis at the point \(A\).
  7. Find the exact value of the \(x\)-coordinate of \(A\).
Edexcel C4 2013 June Q6
6. (i) (a) Express \(\frac { 7 x } { ( x + 3 ) ( 2 x - 1 ) }\) in partial fractions.
(b) Given that \(x > \frac { 1 } { 2 }\), find $$\int \frac { 7 x } { ( x + 3 ) ( 2 x - 1 ) } d x$$ (ii) Using the substitution \(u ^ { 3 } = x\), or otherwise, find $$\int \frac { 1 } { x + x ^ { \frac { 1 } { 3 } } } d x , \quad x > 0$$
Edexcel C4 2013 June Q8
8
5 \end{array} \right) + \mu \left( \begin{array} { r } 5
- 4
- 3 \end{array} \right)$$ where \(\mu\) is a scalar parameter.
The point \(A\) is on \(l _ { 1 }\) where \(\mu = 2\).
  1. Write down the coordinates of \(A\). The acute angle between \(O A\) and \(l _ { 1 }\) is \(\theta\), where \(O\) is the origin.
  2. Find the value of \(\cos \theta\). The point \(B\) is such that \(\overrightarrow { O B } = 3 \overrightarrow { O A }\).
    The line \(l _ { 2 }\) passes through the point \(B\) and is parallel to the line \(l _ { 1 }\).
  3. Find a vector equation of \(l _ { 2 }\).
  4. Find the length of \(O B\), giving your answer as a simplified surd. The point \(X\) lies on \(l _ { 2 }\). Given that the vector \(\overrightarrow { O X }\) is perpendicular to \(l _ { 2 }\),
  5. find the length of \(O X\), giving your answer to 3 significant figures.
    5. The curve \(C\) has the equation $$\sin ( \pi y ) - y - x ^ { 2 } y = - 5 , \quad x > 0$$
  6. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(x\) and \(y\). The point \(P\) with coordinates \(( 2,1 )\) lies on \(C\).
    The tangent to \(C\) at \(P\) meets the \(x\)-axis at the point \(A\).
  7. Find the exact value of the \(x\)-coordinate of \(A\).
    6. (i) (a) Express \(\frac { 7 x } { ( x + 3 ) ( 2 x - 1 ) }\) in partial fractions.
  8. Given that \(x > \frac { 1 } { 2 }\), find $$\int \frac { 7 x } { ( x + 3 ) ( 2 x - 1 ) } d x$$ (ii) Using the substitution \(u ^ { 3 } = x\), or otherwise, find $$\int \frac { 1 } { x + x ^ { \frac { 1 } { 3 } } } d x , \quad x > 0$$ 7. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{b0d21f0f-f5f6-4ca5-8e3e-98aee0d9db7a-11_703_1164_373_492} \captionsetup{labelformat=empty} \caption{Figure 2}
    \end{figure} Figure 2 shows a sketch of part of the curve \(C\) with parametric equations $$x = \tan \theta , \quad y = 1 + 2 \cos 2 \theta , \quad 0 \leqslant \theta < \frac { \pi } { 2 }$$ The curve \(C\) crosses the \(x\)-axis at \(( \sqrt { } 3,0 )\). The finite shaded region \(S\) shown in Figure 2 is bounded by \(C\), the line \(x = 1\) and the \(x\)-axis. This shaded region is rotated through \(2 \pi\) radians about the \(x\)-axis to form a solid of revolution.
  9. Show that the volume of the solid of revolution formed is given by the integral $$k \int _ { \frac { \pi } { 4 } } ^ { \frac { \pi } { 3 } } \left( 16 \cos ^ { 2 } \theta - 8 + \sec ^ { 2 } \theta \right) d \theta$$ where \(k\) is a constant.
  10. Hence, use integration to find the exact value for this volume.
    8. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{b0d21f0f-f5f6-4ca5-8e3e-98aee0d9db7a-13_869_545_312_811} \captionsetup{labelformat=empty} \caption{Figure 3}
    \end{figure} Figure 3 shows a large vertical cylindrical tank containing a liquid. The radius of the circular cross-section of the tank is 40 cm . At time \(t\) minutes, the depth of liquid in the tank is \(h\) centimetres. The liquid leaks from a hole \(P\) at the bottom of the tank. The liquid leaks from the tank at a rate of \(32 \pi \sqrt { } h \mathrm {~cm} ^ { 3 } \mathrm {~min} ^ { - 1 }\).
  11. Show that at time \(t\) minutes, the height \(h \mathrm {~cm}\) of liquid in the tank satisfies the differential equation $$\frac { \mathrm { d } h } { \mathrm {~d} t } = - 0.02 \sqrt { } h$$
  12. Find the time taken, to the nearest minute, for the depth of liquid in the tank to decrease from 100 cm to 50 cm .
    \includegraphics[max width=\textwidth, alt={}]{b0d21f0f-f5f6-4ca5-8e3e-98aee0d9db7a-14_2639_1834_214_217}
Edexcel C4 2013 June Q1
  1. Find \(\int x ^ { 2 } e ^ { x } d x\).
  2. Hence find the exact value of \(\int _ { 0 } ^ { 1 } x ^ { 2 } \mathrm { e } ^ { x } \mathrm {~d} x\).
Edexcel C4 2013 June Q3
3. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{5c9f77f0-9f7c-4125-9da7-20fb8d79b05e-04_814_882_258_539} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 shows the finite region \(R\) bounded by the \(x\)-axis, the \(y\)-axis, the line \(x = \frac { \pi } { 2 }\) and the curve with equation $$y = \sec \left( \frac { 1 } { 2 } x \right) , \quad 0 \leqslant x \leqslant \frac { \pi } { 2 }$$ The table shows corresponding values of \(x\) and \(y\) for \(y = \sec \left( \frac { 1 } { 2 } x \right)\).
\(x\)0\(\frac { \pi } { 6 }\)\(\frac { \pi } { 3 }\)\(\frac { \pi } { 2 }\)
\(y\)11.0352761.414214
  1. Complete the table above giving the missing value of \(y\) to 6 decimal places.
  2. Using the trapezium rule, with all of the values of \(y\) from the completed table, find an approximation for the area of \(R\), giving your answer to 4 decimal places. Region \(R\) is rotated through \(2 \pi\) radians about the \(x\)-axis.
  3. Use calculus to find the exact volume of the solid formed.
Edexcel C4 2013 June Q4
  1. A curve \(C\) has parametric equations
$$x = 2 \sin t , \quad y = 1 - \cos 2 t , \quad - \frac { \pi } { 2 } \leqslant t \leqslant \frac { \pi } { 2 }$$
  1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) at the point where \(t = \frac { \pi } { 6 }\)
  2. Find a cartesian equation for \(C\) in the form $$y = \mathrm { f } ( x ) , \quad - k \leqslant x \leqslant k$$ stating the value of the constant \(k\).
  3. Write down the range of \(\mathrm { f } ( x )\).