Questions — AQA (3620 questions)

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AQA C3 2011 June Q6
6 marks Standard +0.3
  1. Given that \(3\ln x = 4\), find the exact value of \(x\). [1]
  2. By forming a quadratic equation in \(\ln x\), solve \(3\ln x + \frac{20}{\ln x} = 19\), giving your answers for \(x\) in an exact form. [5]
AQA C3 2011 June Q7
12 marks Moderate -0.3
  1. On separate diagrams:
    1. sketch the curve with equation \(y = |3x + 3|\); [2]
    2. sketch the curve with equation \(y = |x^2 - 1|\). [3]
    1. Solve the equation \(|3x + 3| = |x^2 - 1|\). [5]
    2. Hence solve the inequality \(|3x + 3| < |x^2 - 1|\). [2]
AQA C3 2011 June Q8
5 marks Standard +0.8
Use the substitution \(u = 1 + 2\tan x\) to find $$\int \frac{1}{(1 + 2\tan x)^2 \cos^2 x} \, dx$$ [5]
AQA C3 2011 June Q9
11 marks Standard +0.3
  1. Use integration by parts to find \(\int x\ln x \, dx\). [3]
  2. Given that \(y = (\ln x)^2\), find \(\frac{dy}{dx}\). [2]
  3. The diagram shows part of the curve with equation \(y = \sqrt{x\ln x}\). \includegraphics{figure_9} The shaded region \(R\) is bounded by the curve \(y = \sqrt{x\ln x}\), the line \(x = e\) and the \(x\)-axis from \(x = 1\) to \(x = e\). Find the volume of the solid generated when the region \(R\) is rotated through 360° about the \(x\)-axis, giving your answer in an exact form. [6]
AQA C4 2010 June Q1
7 marks Easy -1.2
  1. The polynomial \(f(x)\) is defined by \(f(x) = 8x^3 + 6x^2 - 14x - 1\). Find the remainder when \(f(x)\) is divided by \((4x - 1)\). [2 marks]
  2. The polynomial \(g(x)\) is defined by \(g(x) = 8x^3 + 6x^2 - 14x + d\).
    1. Given that \((4x - 1)\) is a factor of \(g(x)\), find the value of the constant \(d\). [2 marks]
    2. Given that \(g(x) = (4x - 1)(ax^2 + bx + c)\), find the values of the integers \(a\), \(b\) and \(c\). [3 marks]
AQA C4 2010 June Q2
9 marks Moderate -0.3
A curve is defined by the parametric equations $$x = 1 - 3t, \quad y = 1 + 2t^3$$
  1. Find \(\frac{dy}{dx}\) in terms of \(t\). [3 marks]
  2. Find an equation of the normal to the curve at the point where \(t = 1\). [4 marks]
  3. Find a cartesian equation of the curve. [2 marks]
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]
AQA S2 2010 June Q1
9 marks Moderate -0.3
Judith, the village postmistress, believes that, since moving the post office counter into the local pharmacy, the mean daily number of customers that she serves has increased from \(79\). In order to investigate her belief, she counts the number of customers that she serves on \(12\) randomly selected days, with the following results. \(88 \quad 81 \quad 84 \quad 89 \quad 90 \quad 77 \quad 72 \quad 80 \quad 82 \quad 81 \quad 75 \quad 85\) Stating a necessary distributional assumption, test Judith's belief at the \(5\%\) level of significance. [9 marks]
AQA S2 2010 June Q2
8 marks Standard +0.3
It is claimed that a new drug is effective in the prevention of sickness in holiday-makers. A sample of \(100\) holiday-makers was surveyed, with the following results.
SicknessNo sicknessTotal
Drug taken245680
No drug taken11920
Total3565100
Assuming that the \(100\) holiday-makers are a random sample, use a \(\chi^2\) test, at the \(5\%\) level of significance, to investigate the claim. [8 marks]
AQA S2 2010 June Q3
10 marks Moderate -0.8
The continuous random variable \(X\) has a rectangular distribution defined by $$f(x) = \begin{cases} k & -3k \leqslant x \leqslant k \\ 0 & \text{otherwise} \end{cases}$$
    1. Sketch the graph of f. [2 marks]
    2. Hence show that \(k = \frac{1}{2}\). [2 marks]
  2. Find the exact numerical values for the mean and the standard deviation of \(X\). [3 marks]
    1. Find \(\mathrm{P}\left(X \geqslant -\frac{1}{4}\right)\). [2 marks]
    2. Write down the value of \(\mathrm{P}\left(X \neq -\frac{1}{4}\right)\). [1 mark]
AQA S2 2010 June Q4
5 marks Standard +0.3
The error, \(X\) °C, made in measuring a patient's temperature at a local doctors' surgery may be modelled by a normal distribution with mean \(\mu\) and standard deviation \(\sigma\). The errors, \(x\) °C, made in measuring the temperature of each of a random sample of \(10\) patients are summarised below. $$\sum x = 0.35 \quad \text{and} \quad \sum(x - \bar{x})^2 = 0.12705$$ Construct a \(99\%\) confidence interval for \(\mu\), giving the limits to three decimal places. [5 marks]
AQA S2 2010 June Q5
13 marks Standard +0.3
The number of telephone calls received, during an \(8\)-hour period, by an IT company that request an urgent visit by an engineer may be modelled by a Poisson distribution with a mean of \(7\).
  1. Determine the probability that, during a given \(8\)-hour period, the number of telephone calls received that request an urgent visit by an engineer is:
    1. at most \(5\); [1 mark]
    2. exactly \(7\); [2 marks]
    3. at least \(5\) but fewer than \(10\). [3 marks]
  2. Write down the distribution for the number of telephone calls received each hour that request an urgent visit by an engineer. [1 mark]
  3. The IT company has \(4\) engineers available for urgent visits and it may be assumed that each of these engineers takes exactly \(1\) hour for each such visit. At \(10\)am on a particular day, all \(4\) engineers are available for urgent visits.
    1. State the maximum possible number of telephone calls received between \(10\)am and \(11\)am that request an urgent visit and for which an engineer is immediately available. [1 mark]
    2. Calculate the probability that at \(11\)am an engineer will not be immediately available to make an urgent visit. [4 marks]
  4. Give a reason why a Poisson distribution may not be a suitable model for the number of telephone calls per hour received by the IT company that request an urgent visit by an engineer. [1 mark]