Questions C4 (1219 questions)

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AQA C4 2010 June Q3
8 marks Moderate -0.3
    1. Express \(\frac{7x - 3}{(x + 1)(3x - 2)}\) in the form \(\frac{A}{x + 1} + \frac{B}{3x - 2}\). [3 marks]
    2. Hence find \(\int \frac{7x - 3}{(x + 1)(3x - 2)} dx\). [2 marks]
  2. Express \(\frac{6x^2 + x + 2}{2x^2 - x + 1}\) in the form \(P + \frac{Qx + R}{2x^2 - x + 1}\). [3 marks]
AQA C4 2010 June Q4
7 marks Moderate -0.3
    1. Find the binomial expansion of \((1 + x)^{\frac{3}{2}}\) up to and including the term in \(x^2\). [2 marks]
    2. Find the binomial expansion of \((16 + 9x)^{\frac{3}{2}}\) up to and including the term in \(x^2\). [3 marks]
  2. Use your answer to part (a)(ii) to show that \(13^{\frac{3}{2}} \approx 46 + \frac{a}{b}\), stating the values of the integers \(a\) and \(b\). [2 marks]
AQA C4 2010 June Q5
11 marks Standard +0.3
    1. Show that the equation \(3\cos 2x + 2\sin x + 1 = 0\) can be written in the form $$3\sin^2 x - \sin x - 2 = 0$$ [3 marks]
    2. Hence, given that \(3\cos 2x + 2\sin x + 1 = 0\), find the possible values of \(\sin x\). [2 marks]
    1. Express \(3\cos 2x + 2\sin 2x\) in the form \(R\cos(2x - \alpha)\), where \(R > 0\) and \(0° < \alpha < 90°\), giving \(\alpha\) to the nearest \(0.1°\). [3 marks]
    2. Hence solve the equation $$3\cos 2x + 2\sin 2x + 1 = 0$$ for all solutions in the interval \(0° < x < 180°\), giving \(x\) to the nearest \(0.1°\). [3 marks]
AQA C4 2010 June Q6
7 marks Standard +0.3
A curve has equation \(x^3 y + \cos(\pi y) = 7\).
  1. Find the exact value of the \(x\)-coordinate at the point on the curve where \(y = 1\). [2 marks]
  2. Find the gradient of the curve at the point where \(y = 1\). [5 marks]
AQA C4 2010 June Q7
12 marks Standard +0.3
The point \(A\) has coordinates \((4, -3, 2)\). The line \(l_1\) passes through \(A\) and has equation \(\mathbf{r} = \begin{bmatrix} 4 \\ -3 \\ 2 \end{bmatrix} + \lambda \begin{bmatrix} 2 \\ 0 \\ 1 \end{bmatrix}\). The line \(l_2\) has equation \(\mathbf{r} = \begin{bmatrix} -1 \\ 3 \\ 4 \end{bmatrix} + \mu \begin{bmatrix} 1 \\ -2 \\ -1 \end{bmatrix}\). The point \(B\) lies on \(l_2\) where \(\mu = 2\).
  1. Find the vector \(\overrightarrow{AB}\). [3 marks]
    1. Show that the lines \(l_1\) and \(l_2\) intersect. [4 marks]
    2. The lines \(l_1\) and \(l_2\) intersect at the point \(P\). Find the coordinates of \(P\). [1 mark]
  3. The point \(C\) lies on a line which is parallel to \(l_1\) and which passes through the point \(B\). The points \(A\), \(B\), \(C\) and \(P\) are the vertices of a parallelogram. Find the coordinates of the two possible positions of the point \(C\). [4 marks]
AQA C4 2010 June Q8
14 marks Standard +0.3
  1. Solve the differential equation $$\frac{dx}{dt} = -\frac{1}{5}(x + 1)^{\frac{1}{2}}$$ given that \(x = 80\) when \(t = 0\). Give your answer in the form \(x = f(t)\). [6 marks]
  2. A fungus is spreading on the surface of a wall. The proportion of the wall that is unaffected after time \(t\) hours is \(x\%\). The rate of change of \(x\) is modelled by the differential equation $$\frac{dx}{dt} = -\frac{1}{5}(x + 1)^{\frac{1}{2}}$$ At \(t = 0\), the proportion of the wall that is unaffected is 80%. Find the proportion of the wall that will still be unaffected after 60 hours. [2 marks]
  3. A biologist proposes an alternative model for the rate at which the fungus is spreading on the wall. The total surface area of the wall is \(9\text{ m}^2\). The surface area that is affected at time \(t\) hours is \(A\text{ m}^2\). The biologist proposes that the rate of change of \(A\) is proportional to the product of the surface area that is affected and the surface area that is unaffected.
    1. Write down a differential equation for this model. (You are not required to solve your differential equation.) [2 marks]
    2. A solution of the differential equation for this model is given by $$A = \frac{9}{1 + 4e^{-0.09t}}$$ Find the time taken for 50% of the area of the wall to be affected. Give your answer in hours to three significant figures. [4 marks]
AQA C4 2016 June Q1
11 marks Moderate -0.3
  1. Express \(\frac{19x - 3}{(1 + 2x)(3 - 4x)}\) in the form \(\frac{A}{1 + 2x} + \frac{B}{3 - 4x}\). [3 marks]
    1. Find the binomial expansion of \(\frac{19x - 3}{(1 + 2x)(3 - 4x)}\) up to and including the term in \(x^2\). [7 marks]
    2. State the range of values of \(x\) for which this expansion is valid. [1 mark]
AQA C4 2016 June Q2
5 marks Standard +0.3
By forming and solving a suitable quadratic equation, find the solutions of the equation $$3 \cos 2\theta - 5 \cos \theta + 2 = 0$$ in the interval \(0° < \theta < 360°\), giving your answers to the nearest \(0.1°\). [5 marks]
AQA C4 2016 June Q3
8 marks Standard +0.3
  1. Express \(\frac{3 + 13x - 6x^2}{2x - 3}\) in the form \(Ax + B + \frac{C}{2x - 3}\). [4 marks]
  2. Show that \(\int_3^6 \frac{3 + 13x - 6x^2}{2x - 3} \, dx = p + q \ln 3\), where \(p\) and \(q\) are rational numbers. [4 marks]
AQA C4 2016 June Q4
7 marks Moderate -0.3
The mass of radioactive atoms in a substance can be modelled by the equation $$m = m_0 k^t$$ where \(m_0\) grams is the initial mass, \(m\) grams is the mass after \(t\) days and \(k\) is a constant. The value of \(k\) differs from one substance to another.
    1. A sample of radioactive iodine reduced in mass from 24 grams to 12 grams in 8 days. Show that the value of the constant \(k\) for this substance is 0.917004, correct to six decimal places. [1 mark]
    2. A similar sample of radioactive iodine reduced in mass to 1 gram after 60 days. Calculate the initial mass of this sample, giving your answer to the nearest gram. [2 marks]
  2. The half-life of a radioactive substance is the time it takes for a mass of \(m_0\) to reduce to a mass of \(\frac{1}{2}m_0\). A sample of radioactive vanadium reduced in mass from exactly 10 grams to 8.106 grams in 100 days. Find the half-life of radioactive vanadium, giving your answer to the nearest day. [4 marks]
AQA C4 2016 June Q5
10 marks Standard +0.3
It is given that \(\sin A = \frac{\sqrt{5}}{3}\) and \(\sin B = \frac{1}{\sqrt{5}}\), where the angles \(A\) and \(B\) are both acute.
    1. Show that the exact value of \(\cos B = \frac{2}{\sqrt{5}}\). [1 mark]
    2. Hence show that the exact value of \(\sin 2B\) is \(\frac{4}{5}\). [2 marks]
    1. Show that the exact value of \(\sin(A - B)\) can be written as \(p(5 - \sqrt{5})\), where \(p\) is a rational number. [4 marks]
    2. Find the exact value of \(\cos(A - B)\) in the form \(r + s\sqrt{5}\), where \(r\) and \(s\) are rational numbers. [3 marks]
AQA C4 2016 June Q6
15 marks Standard +0.3
The line \(l_1\) passes through the point \(A(0, 6, 9)\) and the point \(B(4, -6, -11)\). The line \(l_2\) has equation \(\mathbf{r} = \begin{bmatrix} -1 \\ 5 \\ -2 \end{bmatrix} + \lambda \begin{bmatrix} 3 \\ -5 \\ 1 \end{bmatrix}\).
  1. The acute angle between the lines \(l_1\) and \(l_2\) is \(\theta\). Find the value of \(\cos \theta\) as a fraction in its lowest terms. [5 marks]
  2. Show that the lines \(l_1\) and \(l_2\) intersect and find the coordinates of the point of intersection. [5 marks]
  3. The points \(C\) and \(D\) lie on line \(l_2\) such that \(ACBD\) is a parallelogram. \includegraphics{figure_6} The length of \(AB\) is three times the length of \(CD\). Find the coordinates of the points \(C\) and \(D\). [5 marks]
AQA C4 2016 June Q7
9 marks Standard +0.8
A curve \(C\) is defined by the parametric equations $$x = \frac{4 - e^{-6t}}{4}, \quad y = \frac{e^{3t}}{3t}, \quad t \neq 0$$
  1. Find the exact value of \(\frac{dy}{dx}\) at the point on \(C\) where \(t = \frac{2}{3}\). [5 marks]
  2. Show that \(x = \frac{4 - e^{-6t}}{4}\) can be rearranged into the form \(e^{3t} = \frac{e}{2\sqrt{(1-x)}}\). [2 marks]
  3. Hence find the Cartesian equation of \(C\), giving your answer in the form $$y = \frac{e}{f(x)[1 - \ln(f(x))]}$$ [2 marks]
AQA C4 2016 June Q8
10 marks Standard +0.8
It is given that \(\theta = \tan^{-1}\left(\frac{3x}{2}\right)\).
  1. By writing \(\theta = \tan^{-1}\left(\frac{3x}{2}\right)\) as \(2\tan\theta = 3x\), use implicit differentiation to show that $$\frac{d\theta}{dx} = \frac{k}{4 + 9x^2}$$, where \(k\) is an integer. [3 marks]
  2. Hence solve the differential equation $$9y(4 + 9x^2)\frac{dy}{dx} = \cosec 3y$$ given that \(x = 0\) when \(y = \frac{\pi}{3}\). Give your answer in the form \(\mathbf{g}(y) = \mathbf{h}(x)\). [7 marks]
Edexcel C4 Q1
6 marks Standard +0.3
Use integration by parts to find the exact value of \(\int_1^3 x^2 \ln x \, dx\). [6]
Edexcel C4 Q2
12 marks Moderate -0.3
Fluid flows out of a cylindrical tank with constant cross section. At time \(t\) minutes, \(t \geq 0\), the volume of fluid remaining in the tank is \(V\) m\(^3\). The rate at which the fluid flows, in m\(^3\) min\(^{-1}\), is proportional to the square root of \(V\).
  1. Show that the depth \(h\) metres of fluid in the tank satisfies the differential equation $$\frac{dh}{dt} = -k\sqrt{h}, \quad \text{where } k \text{ is a positive constant.}$$ [3]
  2. Show that the general solution of the differential equation may be written as $$h = (A - Bt)^2, \quad \text{where } A \text{ and } B \text{ are constants.}$$ [4] Given that at time \(t = 0\) the depth of fluid in the tank is 1 m, and that 5 minutes later the depth of fluid has reduced to 0.5 m,
  3. find the time, \(T\) minutes, which it takes for the tank to empty. [3]
  4. Find the depth of water in the tank at time \(0.5T\) minutes. [2]
Edexcel C4 Q3
14 marks Standard +0.3
  1. Use the identity for \(\cos(A + B)\) to prove that \(\cos 2A = 2\cos^2 A - 1\). [2]
  2. Use the substitution \(x = 2\sqrt{2} \sin \theta\) to prove that $$\int_2^{\sqrt{6}} \sqrt{(8 - x^2)} \, dx = \frac{1}{3}(\pi + 3\sqrt{3} - 6).$$ [7]
A curve is given by the parametric equations $$x = \sec \theta, \quad y = \ln(1 + \cos 2\theta), \quad 0 \leq \theta < \frac{\pi}{2}.$$
  1. Find an equation of the tangent to the curve at the point where \(\theta = \frac{\pi}{3}\). [5]
Edexcel C4 Q4
11 marks Standard +0.3
\includegraphics{figure_2} Figure 2 shows a sketch of the curve \(C\) with equation \(y = \frac{4}{x - 3}\), \(x \neq 3\). The points \(A\) and \(B\) on the curve have \(x\)-coordinates 3.25 and 5 respectively.
  1. Write down the \(y\)-coordinates of \(A\) and \(B\). [1]
  2. Show that an equation of \(C\) is \(\frac{3y + 4}{y} = 0\), \(y \neq 0\). [1]
The shaded region \(R\) is bounded by \(C\), the \(y\)-axis and the lines through \(A\) and \(B\) parallel to the \(x\)-axis. The region \(R\) is rotated through 360° about the \(y\)-axis to form a solid shape \(S\).
  1. Find the volume of \(S\), giving your answer in the form \(\pi(a + b \ln c)\), where \(a\), \(b\) and \(c\) are integers. [7]
The solid shape \(S\) is used to model a cooling tower. Given that 1 unit on each axis represents 3 metres,
  1. show that the volume of the tower is approximately 15500 m\(^3\). [2]
Edexcel C4 Q5
11 marks Standard +0.3
Relative to a fixed origin \(O\), the point \(A\) has position vector \(3\mathbf{i} + 2\mathbf{j} - \mathbf{k}\), the point \(B\) has position vector \(5\mathbf{i} + \mathbf{j} + \mathbf{k}\), and the point \(C\) has position vector \(7\mathbf{i} - \mathbf{j}\).
  1. Find the cosine of angle \(ABC\). [4]
  2. Find the exact value of the area of triangle \(ABC\). [3]
The point \(D\) has position vector \(7\mathbf{i} + 3\mathbf{k}\).
  1. Show that \(AC\) is perpendicular to \(CD\). [2]
  2. Find the ratio \(AD:DB\). [2]
Edexcel C4 Q6
9 marks Standard +0.3
\includegraphics{figure_2} 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 \(y = 8\sqrt{\sin \frac{\pi x}{10}}\), 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.
  1. Using this model, copy and complete the table below, giving the values of \(y\) to 2 decimal places.
    \(x\)0246810
    \(y\)06.130
    [2]
The area of the cross-section of the tunnel is given by \(\int_0^{10} y \, dx\).
  1. Estimate this area, using the trapezium rule with all the values from your table. [4]
  2. Deduce an estimate of the cross-sectional area of the concrete surround. [1]
  3. State, with a reason, whether your answer in part (c) over-estimates or under-estimates the true value. [2]
Edexcel C4 Q7
16 marks Standard +0.8
$$\text{f}(x) = \frac{25}{(3 + 2x)^2(1 - x)}, \quad |x| < 1.$$
  1. Express f(x) as a sum of partial fractions. [4]
  2. Hence find \(\int \text{f}(x) \, dx\). [5]
  3. Find the series expansion of f(x) in ascending powers of \(x\) up to and including the term in \(x^2\). Give each coefficient as a simplified fraction. [7]
Edexcel C4 Q1
8 marks Standard +0.3
The curve \(C\) has equation \(5x^2 + 2xy - 3y^2 + 3 = 0\). The point \(P\) on the curve \(C\) has coordinates \((1, 2)\).
  1. Find the gradient of the curve at \(P\). [5]
  2. Find the equation of the normal to the curve \(C\) at \(P\), in the form \(y = ax + b\), where \(a\) and \(b\) are constants. [3]
Edexcel C4 Q2
8 marks Standard +0.3
\includegraphics{figure_1} In Fig. 1, the curve \(C\) has equation \(y = f(x)\), where $$f(x) = x + \frac{2}{x^2}, \quad x > 0.$$ The shaded region is bounded by \(C\), the \(x\)-axis and the lines with equations \(x = 1\) and \(x = 2\). The shaded region is rotated through \(2\pi\) radians about the \(x\)-axis. Using calculus, calculate the volume of the solid generated. Give your answer in the form \(\pi(a + \ln b)\), where \(a\) and \(b\) are constants. [8]
Edexcel C4 Q3
9 marks Moderate -0.3
\includegraphics{figure_2} Figure 2 shows part of the curve with equation $$y = e^x \cos x, \quad 0 \leq x \leq \frac{\pi}{2}.$$ The finite region \(R\) is bounded by the curve and the coordinate axes.
  1. Calculate, to 2 decimal places, the \(y\)-coordinates of the points on the curve where \(x = 0\), \(\frac{\pi}{6}\), \(\frac{\pi}{3}\) and \(\frac{\pi}{2}\). [3]
  2. Using the trapezium rule and all the values calculated in part (a), find an approximation for the area of \(R\). [4]
  3. State, with a reason, whether your approximation underestimates or overestimates the area of \(R\). [2]
Edexcel C4 Q4
10 marks Moderate -0.3
A curve is given parametrically by the equations $$x = 5 \cos t, \quad y = -2 + 4 \sin t, \quad 0 \leq t < 2\pi.$$
  1. Find the coordinates of all the points at which \(C\) intersects the coordinate axes, giving your answers in surd form where appropriate. [4]
  2. Sketch the graph at \(C\). [2]
\(P\) is the point on \(C\) where \(t = \frac{1}{6}\pi\).
  1. Show that the normal to \(C\) at \(P\) has equation $$8\sqrt{3}y = 10x - 25\sqrt{3}.$$ [4]