Questions — Edexcel C4 (386 questions)

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Edexcel C4 Q3
11 marks Standard +0.3
3. $$f ( x ) = \frac { 7 + 3 x + 2 x ^ { 2 } } { ( 1 - 2 x ) ( 1 + x ) ^ { 2 } } , \quad | x | > \frac { 1 } { 2 }$$
  1. Express \(\mathrm { f } ( x )\) in partial fractions.
  2. 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
11 marks Standard +0.3
4. Relative to a fixed origin, two lines have the equations $$\begin{aligned} & \mathbf { r } = \left( \begin{array} { c } 7 \\ 0 \\ - 3 \end{array} \right) + \lambda \left( \begin{array} { c } 5 \\ 4 \\ - 2 \end{array} \right) \end{aligned}$$
Edexcel C4 Q5
12 marks Moderate -0.8

& \mathbf { r } = \left( \begin{array} { l } a
6
3 \end{array} \right)
Edexcel C4 Q7
15 marks Moderate -0.5
7
0
- 3 \end{array} \right) + \lambda \left( \begin{array} { c } 5
4
- 2 \end{array} \right)
& \mathbf { r } = \left( \begin{array} { l } a
6
3 \end{array} \right) + \mu \left( \begin{array} { c } - 5
Edexcel C4 Q14
Standard +0.3
14
2 \end{array} \right) , $$ and\\ where \(a\) is a constant and \(\lambda\) and \(\mu\) are scalar parameters.\\ Given that the two lines intersect,
  1. find the position vector of their point of intersection,
  2. find the value of \(a\). Given also that \(\theta\) is the acute angle between the lines,
  3. find the value of \(\cos \theta\) in the form \(k \sqrt { 5 }\) where \(k\) is rational.\\ 4. continued\\ 5. A curve has the equation $$x ^ { 2 } - 4 x y + 2 y ^ { 2 } = 1$$
  4. Find an expression for \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in its simplest form in terms of \(x\) and \(y\).
  5. Show that the tangent to the curve at the point \(P ( 1,2 )\) has the equation $$3 x - 2 y + 1 = 0$$ The tangent to the curve at the point \(Q\) is parallel to the tangent at \(P\).
  6. Find the coordinates of \(Q\).\\ 5. continued\\ 6. The rate of increase in the number of bacteria in a culture, \(N\), at time \(t\) hours is proportional to \(N\).
  7. Write down a differential equation connecting \(N\) and \(t\). Given that initially there are \(N _ { 0 }\) bacteria present in a culture,
  8. Show that \(N = N _ { 0 } \mathrm { e } ^ { k t }\), where \(k\) is a positive constant. Given also that the number of bacteria present doubles every six hours,
  9. find the value of \(k\),
  10. find how long it takes for the number of bacteria to increase by a factor of ten, giving your answer to the nearest minute. of ten, giving your answer to the nearest minute.\\ 6. continued\\ 7. A curve has parametric equations $$x = \sec \theta + \tan \theta , \quad y = \operatorname { cosec } \theta + \cot \theta , \quad 0 < \theta < \frac { \pi } { 2 } .$$
  11. Show that \(x + \frac { 1 } { x } = 2 \sec \theta\). Given that \(y + \frac { 1 } { y } = 2 \operatorname { cosec } \theta\),
  12. find a cartesian equation for the curve.
  13. Show that \(\frac { \mathrm { d } x } { \mathrm {~d} \theta } = \frac { 1 } { 2 } \left( x ^ { 2 } + 1 \right)\).
  14. Find an expression for \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(x\) and \(y\).
    7. continued
    7. continued
Edexcel C4 2013 January Q5
15 marks Moderate -0.3
  1. Show that \(A\) has coordinates \(( 0,3 )\).
  2. Find the \(x\) coordinate of the point \(B\).
  3. Find an equation of the normal to \(C\) at the point \(A\). The region \(R\), as shown shaded in Figure 2, is bounded by the curve \(C\), the line \(x = - 1\) and the \(x\)-axis.
  4. Use integration to find the exact area of \(R\).
Edexcel C4 Q4
11 marks Challenging +1.2
  1. Show that the volume of the solid formed is \(\frac { 1 } { 4 } \pi ( \pi + 2 )\).
  2. Find a cartesian equation for the curve.
Edexcel C4 2014 June Q1
7 marks Standard +0.3
A curve \(C\) has the equation $$x^3 + 2xy - x - y^3 - 20 = 0$$
  1. [(a)] Find \(\frac{dy}{dx}\) in terms of \(x\) and \(y\). \hfill [5]
  2. [(b)] Find an equation of the tangent to \(C\) at the point \((3, -2)\), giving your answer in the form \(ax + by + c = 0\), where \(a\), \(b\) and \(c\) are integers. \hfill [2]
Edexcel C4 2014 June Q2
5 marks Moderate -0.3
Given that the binomial expansion of \((1 + kx)^{-4}\), \(|kx| < 1\), is $$1 - 6x + Ax^2 + \ldots$$
  1. [(a)] find the value of the constant \(k\), \hfill [2]
  2. [(b)] find the value of the constant \(A\), giving your answer in its simplest form. \hfill [3]
Edexcel C4 2014 June Q3
11 marks Standard +0.3
\includegraphics{figure_1} Figure 1 shows a sketch of part of the curve with equation \(y = \frac{10}{2x + 5\sqrt{x}}\), \(x > 0\) The finite region \(R\), shown shaded in Figure 1, is bounded by the curve, the \(x\)-axis, and the lines with equations \(x = 1\) and \(x = 4\) The table below shows corresponding values of \(x\) and \(y\) for \(y = \frac{10}{2x + 5\sqrt{x}}\)
\(x\)1234
\(y\)1.428570.903260.55556
  1. [(a)] Complete the table above by giving the missing value of \(y\) to 5 decimal places. \hfill [1]
  2. [(b)] Use the trapezium rule, with all the values of \(y\) in the completed table, to find an estimate for the area of \(R\), giving your answer to 4 decimal places. \hfill [3]
  3. [(c)] By reference to the curve in Figure 1, state, giving a reason, whether your estimate in part (b) is an overestimate or an underestimate for the area of \(R\). \hfill [1]
  4. [(d)] Use the substitution \(u = \sqrt{x}\), or otherwise, to find the exact value of $$\int_1^4 \frac{10}{2x + 5\sqrt{x}} dx$$ \hfill [6]
Edexcel C4 2014 June Q4
5 marks Moderate -0.3
\includegraphics{figure_2} A vase with a circular cross-section is shown in Figure 2. Water is flowing into the vase. When the depth of the water is \(h\) cm, the volume of water \(V\) cm\(^3\) is given by $$V = 4\pi h(h + 4), \quad 0 \leq h \leq 25$$ Water flows into the vase at a constant rate of \(80\pi\) cm\(^3\)s\(^{-1}\) Find the rate of change of the depth of the water, in cm s\(^{-1}\), when \(h = 6\) \hfill [5]
Edexcel C4 2014 June Q5
5 marks Moderate -0.3
\includegraphics{figure_3} Figure 3 shows a sketch of the curve \(C\) with parametric equations $$x = 4\cos\left(t + \frac{\pi}{6}\right), \quad y = 2\sin t, \quad 0 \leq t < 2\pi$$
  1. [(a)] Show that $$x + y = 2\sqrt{3}\cos t$$ \hfill [3]
  2. [(b)] Show that a cartesian equation of \(C\) is $$(x + y)^2 + ay^2 = b$$ where \(a\) and \(b\) are integers to be determined. \hfill [2]
Edexcel C4 2014 June Q6
12 marks Standard +0.3
  1. [(i)] Find $$\int xe^{4x} dx$$ \hfill [3]
  2. [(ii)] Find $$\int \frac{8}{(2x - 1)^3} dx, \quad x > \frac{1}{2}$$ \hfill [2]
  3. [(iii)] Given that \(y = \frac{\pi}{6}\) at \(x = 0\), solve the differential equation $$\frac{dy}{dx} = e^x \cosec 2y \cosec y$$ \hfill [7] \end{enumerate}
Edexcel C4 2014 June Q7
15 marks Challenging +1.2
\includegraphics{figure_4} Figure 4 shows a sketch of part of the curve \(C\) with parametric equations $$x = 3\tan\theta, \quad y = 4\cos^2\theta, \quad 0 \leq \theta < \frac{\pi}{2}$$ The point \(P\) lies on \(C\) and has coordinates \((3, 2)\). The line \(l\) is the normal to \(C\) at \(P\). The normal cuts the \(x\)-axis at the point \(Q\).
  1. [(a)] Find the \(x\) coordinate of the point \(Q\). \hfill [6]
The finite region \(S\), shown shaded in Figure 4, is bounded by the curve \(C\), the \(x\)-axis, the \(y\)-axis and the line \(l\). This shaded region is rotated \(2\pi\) radians about the \(x\)-axis to form a solid of revolution.
  1. [(b)] Find the exact value of the volume of the solid of revolution, giving your answer in the form \(p\pi + q\pi^2\), where \(p\) and \(q\) are rational numbers to be determined. [You may use the formula \(V = \frac{1}{3}\pi r^2 h\) for the volume of a cone.] \hfill [9] \end{enumerate} \end{enumerate}
Edexcel C4 2014 June Q8
15 marks Standard +0.3
Relative to a fixed origin \(O\), the point \(A\) has position vector \(\begin{pmatrix} -2 \\ 4 \\ 7 \end{pmatrix}\) and the point \(B\) has position vector \(\begin{pmatrix} -1 \\ 3 \\ 8 \end{pmatrix}\) The line \(l_1\) passes through the points \(A\) and \(B\).
  1. [(a)] Find the vector \(\overrightarrow{AB}\). \hfill [2]
  2. [(b)] Hence find a vector equation for the line \(l_1\) \hfill [1]
The point \(P\) has position vector \(\begin{pmatrix} 0 \\ 2 \\ 3 \end{pmatrix}\) Given that angle \(PBA\) is \(\theta\),
  1. [(c)] show that \(\cos\theta = \frac{1}{3}\) \hfill [3]
The line \(l_2\) passes through the point \(P\) and is parallel to the line \(l_1\)
  1. [(d)] Find a vector equation for the line \(l_2\) \hfill [2]
The points \(C\) and \(D\) both lie on the line \(l_2\) Given that \(AB = PC = DP\) and the \(x\) coordinate of \(C\) is positive,
  1. [(e)] find the coordinates of \(C\) and the coordinates of \(D\). \hfill [3]
  2. [(f)] find the exact area of the trapezium \(ABCD\), giving your answer as a simplified surd. \hfill [4] \end{enumerate} \end{enumerate} \end{enumerate} \end{enumerate}
Edexcel C4 Q1
5 marks Moderate -0.3
Use the binomial theorem to expand $$\sqrt{(4-9x)}, \quad |x| < \frac{4}{9},$$ in ascending powers of \(x\), up to and including the term in \(x^3\), simplifying each term. [5]
Edexcel C4 Q2
7 marks Standard +0.3
A curve has equation $$x^2 + 2xy - 3y^2 + 16 = 0.$$ Find the coordinates of the points on the curve where \(\frac{dy}{dx} = 0\). [7]
Edexcel C4 Q3
8 marks Moderate -0.3
  1. Express \(\frac{5x + 3}{(2x - 3)(x + 2)}\) in partial fractions. [3]
  2. Hence find the exact value of \(\int_0^1 \frac{5x + 3}{(2x - 3)(x + 2)} dx\), giving your answer as a single logarithm. [5]
Edexcel C4 Q4
7 marks Challenging +1.2
Use the substitution \(x = \sin \theta\) to find the exact value of $$\int_0^1 \frac{1}{(1-x^2)^{3/2}} dx.$$ [7]
Edexcel C4 Q5
10 marks Standard +0.3
\includegraphics{figure_1} Figure 1 shows the graph of the curve with equation $$y = xe^x, \quad x \geq 0.$$ The finite region \(R\) bounded by the lines \(x = 1\), the \(x\)-axis and the curve is shown shaded in Figure 1.
  1. Use integration to find the exact value of the area for \(R\). [5]
  2. Complete the table with the values of \(y\) corresponding to \(x = 0.4\) and \(0.8\).
    \(x\)00.20.40.60.8
    \(y = xe^x\)00.298361.99207
    [1]
  3. Use the trapezium rule with all the values in the table to find an approximate value for this area, giving your answer to 4 significant figures. [4]
Edexcel C4 Q6
12 marks Standard +0.3
A curve has parametric equations $$x = 2\cot t, \quad y = 2\sin^2 t, \quad 0 < t \leq \frac{\pi}{2}.$$
  1. Find an expression for \(\frac{dy}{dx}\) in terms of the parameter \(t\). [4]
  2. Find an equation of the tangent to the curve at the point where \(t = \frac{\pi}{4}\). [4]
  3. Find a cartesian equation of the curve in the form \(y = f(x)\). State the domain on which the curve is defined. [4]
Edexcel C4 Q7
13 marks Standard +0.3
The line \(l_1\) has vector equation $$\mathbf{r} = \begin{pmatrix} 3 \\ 1 \\ 2 \end{pmatrix} + \lambda \begin{pmatrix} 1 \\ -1 \\ 4 \end{pmatrix}$$ and the line \(l_2\) has vector equation $$\mathbf{r} = \begin{pmatrix} 0 \\ 4 \\ -2 \end{pmatrix} + \mu \begin{pmatrix} 1 \\ -1 \\ 0 \end{pmatrix},$$ where \(\lambda\) and \(\mu\) are parameters. The lines \(l_1\) and \(l_2\) intersect at the point \(B\) and the acute angle between \(l_1\) and \(l_2\) is \(\theta\).
  1. Find the coordinates of \(B\). [4]
  2. Find the value of \(\cos \theta\), giving your answer as a simplified fraction. [4]
The point \(A\), which lies on \(l_1\), has position vector \(\mathbf{a} = 3\mathbf{i} + \mathbf{j} + 2\mathbf{k}\). The point \(C\), which lies on \(l_2\), has position vector \(\mathbf{c} = 5\mathbf{i} - \mathbf{j} - 2\mathbf{k}\). The point \(D\) is such that \(ABCD\) is a parallelogram.
  1. Show that \(|\overrightarrow{AB}| = |\overrightarrow{BC}|\). [3]
  2. Find the position vector of the point \(D\). [2]
Edexcel C4 Q8
13 marks Standard +0.3
Liquid is pouring into a container at a constant rate of \(20\text{ cm}^3\text{s}^{-1}\) and is leaking out at a rate proportional to the volume of the liquid already in the container.
  1. Explain why, at time \(t\) seconds, the volume, \(V\text{ cm}^3\), of liquid in the container satisfies the differential equation $$\frac{dV}{dt} = 20 - kV,$$ where \(k\) is a positive constant. [2]
The container is initially empty.
  1. By solving the differential equation, show that $$V = A + Be^{-kt},$$ giving the values of \(A\) and \(B\) in terms of \(k\). [6]
Given also that \(\frac{dV}{dt} = 10\) when \(t = 5\),
  1. find the volume of liquid in the container at 10 s after the start. [5]
Edexcel C4 2013 June Q1
8 marks Moderate -0.3
  1. Find the binomial expansion of $$\sqrt{(9 + 8x)}, \quad |x| < \frac{9}{8}$$ in ascending powers of \(x\), up to and including the term in \(x^2\). Give each coefficient as a simplified fraction. [5]
  2. Use your expansion to estimate the value of \(\sqrt{11}\), giving your answer as a single fraction. [3]
Edexcel C4 2013 June Q2
11 marks Standard +0.3
\includegraphics{figure_1} Figure 1 shows a sketch of part of the curve with equation \(y = xe^{-\frac{1}{2}x}\), \(x > 0\). 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 \(y = xe^{-\frac{1}{2}x}\).
\(x\)01234
\(y\)0\(e^{-\frac{1}{2}}\)\(3e^{-\frac{3}{2}}\)\(4e^{-2}\)
  1. Complete the table with the value of \(y\) corresponding to \(x = 2\) [1]
  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. [4]
    1. Find \(\int xe^{-\frac{1}{2}x} \, dx\).
    2. Hence find the exact area of \(R\), giving your answer in the form \(a + be^{-2}\), where \(a\) and \(b\) are integers. [6]