AQA Paper 1 (Paper 1) 2019 June

Given that \(a > 0\), determine which of these expressions is not equivalent to the others. Circle your answer. [1 mark] $$-2\log_{10}\left(\frac{1}{a}\right) \quad 2\log_{10}(a) \quad \log_{10}(a^2) \quad -4\log_{10}(\sqrt{a})$$

Given \(y = e^{kx}\), where \(k\) is a constant, find \(\frac{dy}{dx}\) Circle your answer. [1 mark] $$\frac{dy}{dx} = e^{kx} \quad \frac{dy}{dx} = ke^{kx} \quad \frac{dy}{dx} = kxe^{x-1} \quad \frac{dy}{dx} = \frac{e^{kx}}{k}$$

The diagram below shows a sector of a circle. \includegraphics{figure_3} The radius of the circle is 4cm and \(\theta = 0.8\) radians. Find the area of the sector. Circle your answer. [1 mark] $$1.28\text{cm}^2 \quad 3.2\text{cm}^2 \quad 6.4\text{cm}^2 \quad 12.8\text{cm}^2$$

The point \(A\) has coordinates \((-1, a)\) and the point \(B\) has coordinates \((3, b)\) The line \(AB\) has equation \(5x + 4y = 17\) Find the equation of the perpendicular bisector of the points \(A\) and \(B\). [4 marks]

An arithmetic sequence has first term \(a\) and common difference \(d\). The sum of the first 16 terms of the sequence is 260

1. Show that \(4a + 30d = 65\) [2 marks]
2. Given that the sum of the first 60 terms is 315, find the sum of the first 41 terms. [3 marks]
3. \(S_n\) is the sum of the first \(n\) terms of the sequence. Explain why the value you found in part (b) is the maximum value of \(S_n\) [2 marks]

The function f is defined by $$f(x) = \frac{1}{2}(x^2 + 1), \quad x \geq 0$$

1. Find the range of f. [1 mark]
2. 1. Find \(f^{-1}(x)\) [3 marks]
   2. State the range of \(f^{-1}(x)\) [1 mark]
3. State the transformation which maps the graph of \(y = f(x)\) onto the graph of \(y = f^{-1}(x)\) [1 mark]
4. Find the coordinates of the point of intersection of the graphs of \(y = f(x)\) and \(y = f^{-1}(x)\) [2 marks]

1. By sketching the graphs of \(y = \frac{1}{x}\) and \(y = \sec 2x\) on the axes below, show that the equation $$\frac{1}{x} = \sec 2x$$ has exactly one solution for \(x > 0\) [3 marks] \includegraphics{figure_7a}
2. By considering a suitable change of sign, show that the solution to the equation lies between 0.4 and 0.6 [2 marks]
3. Show that the equation can be rearranged to give $$x = \frac{1}{2}\cos^{-1}x$$ [2 marks]
4. 1. Use the iterative formula $$x_{n+1} = \frac{1}{2}\cos^{-1}x_n$$ with \(x_1 = 0.4\), to find \(x_2\), \(x_3\) and \(x_4\), giving your answers to four decimal places. [2 marks]
   2. On the graph below, draw a cobweb or staircase diagram to show how convergence takes place, indicating the positions of \(x_2\), \(x_3\) and \(x_4\). [2 marks] \includegraphics{figure_7d}

$$P(n) = \sum_{k=0}^{n} k^3 - \sum_{k=0}^{n-1} k^3 \text{ where } n \text{ is a positive integer.}$$

1. Find P(3) and P(10) [2 marks]
2. Solve the equation \(P(n) = 1.25 \times 10^8\) [2 marks]

Prove that the sum of a rational number and an irrational number is always irrational. [5 marks]

The volume of a spherical bubble is increasing at a constant rate. Show that the rate of increase of the radius, \(r\), of the bubble is inversely proportional to \(r^2\) Volume of a sphere = \(\frac{4}{3}\pi r^3\) [4 marks]

Jodie is attempting to use differentiation from first principles to prove that the gradient of \(y = \sin x\) is zero when \(x = \frac{\pi}{2}\) Jodie's teacher tells her that she has made mistakes starting in Step 4 of her working. Her working is shown below. \includegraphics{figure_11} Complete Steps 4 and 5 of Jodie's working below, to correct her proof. [4 marks] Step 4 \quad For gradient of curve at A, Step 5 \quad Hence the gradient of the curve at A is given by

1. Show that the equation $$2\cot^2 x + 2\cosec^2 x = 1 + 4\cosec x$$ can be written in the form $$a\cosec^2 x + b\cosec x + c = 0$$ [2 marks]
2. Hence, given \(x\) is obtuse and $$2\cot^2 x + 2\cosec^2 x = 1 + 4\cosec x$$ find the exact value of \(\tan x\) Fully justify your answer. [5 marks]

A curve, C, has equation $$y = \frac{e^{3x-5}}{x^2}$$ Show that C has exactly one stationary point. Fully justify your answer. [7 marks]

The graph of \(y = \frac{2x^3}{x^2 + 1}\) is shown for \(0 \leq x \leq 4\) Caroline is attempting to approximate the shaded area, A, under the curve using the trapezium rule by splitting the area into \(n\) trapezia.

1. When \(n = 4\)
   1. State the number of ordinates that Caroline uses. [1 mark]
   2. Calculate the area that Caroline should obtain using this method. Give your answer correct to two decimal places. [3 marks]
2. Show that the exact area of \(A\) is $$16 - \ln 17$$ Fully justify your answer. [5 marks]
3. Explain what would happen to Caroline's answer to part (a)(ii) as \(n \to \infty\) [1 mark]

At time \(t\) hours after a high tide, the height, \(h\) metres, of the tide and the velocity, \(v\) knots, of the tidal flow can be modelled using the parametric equations $$v = 4 - \left(\frac{2t}{3} - 2\right)^2$$ $$h = 3 - 2\sqrt[3]{t - 3}$$ High tides and low tides occur alternately when the velocity of the tidal flow is zero. A high tide occurs at 2am.

1. 1. Use the model to find the height of this high tide. [1 mark]
   2. Find the time of the first low tide after 2am. [3 marks]
   3. Find the height of this low tide. [1 mark]
2. Use the model to find the height of the tide when it is flowing with maximum velocity. [3 marks]
3. Comment on the validity of the model. [2 marks]

1. \(y = e^{-x}(\sin x + \cos x)\) Find \(\frac{dy}{dx}\) Simplify your answer. [3 marks]
2. Hence, show that $$\int e^{-x}\sin x \, dx = ae^{-x}(\sin x + \cos x) + c$$ where \(a\) is a rational number. [2 marks]
3. A sketch of the graph of \(y = e^{-x}\sin x\) for \(x \geq 0\) is shown below. The areas of the finite regions bounded by the curve and the \(x\)-axis are denoted by \(A_1, A_2, \ldots, A_n, \ldots\) \includegraphics{figure_16c}
   1. Find the exact value of the area \(A_1\) [3 marks]
   2. Show that $$\frac{A_2}{A_1} = e^{-\pi}$$ [4 marks]
   3. Given that $$\frac{A_{n+1}}{A_n} = e^{-\pi}$$ show that the exact value of the total area enclosed between the curve and the \(x\)-axis is $$\frac{1 + e^\pi}{2(e^\pi - 1)}$$ [4 marks]