AQA AS Paper 1 (AS Paper 1) 2020 June

At the point \((1, 0)\) on the curve \(y = \ln x\), which statement below is correct? Tick (\(\checkmark\)) one box. [1 mark] The gradient is negative and decreasing The gradient is negative and increasing The gradient is positive and decreasing The gradient is positive and increasing

Given that \(f(x) = 10\) when \(x = 4\), which statement below must be correct? Tick (\(\checkmark\)) one box. [1 mark] \(f(2x) = 5\) when \(x = 4\) \(f(2x) = 10\) when \(x = 2\) \(f(2x) = 10\) when \(x = 8\) \(f(2x) = 20\) when \(x = 4\)

Jia has to solve the equation $$2 - 2\sin^2 \theta = \cos \theta$$ where \(-180° \leq \theta \leq 180°\) Jia's working is as follows: $$2 - 2(1 - \cos^2 \theta) = \cos \theta$$ $$2 - 2 + 2\cos^2 \theta = \cos \theta$$ $$2\cos^2 \theta = \cos \theta$$ $$2\cos \theta = 1$$ $$\cos \theta = 0.5$$ $$\theta = 60°$$ Jia's teacher tells her that her solution is incomplete.

1. Explain the two errors that Jia has made. [2 marks]
2. Write down all the values of \(\theta\) that satisfy the equation $$2 - 2\sin^2 \theta = \cos \theta$$ where \(-180° \leq \theta \leq 180°\) [2 marks]

In the binomial expansion of \((\sqrt{3} + \sqrt{2})^4\) there are two irrational terms. Find the difference between these two terms. [3 marks]

Differentiate from first principles $$y = 4x^2 + x$$ [4 marks]

1. It is given that $$f(x) = x^3 - x^2 + x - 6$$ Use the factor theorem to show that \((x - 2)\) is a factor of \(f(x)\). [2 marks]
2. Find the quadratic factor of \(f(x)\). [1 mark]
3. Hence, show that there is only one real solution to \(f(x) = 0\) [3 marks]
4. Find the exact value of \(x\) that solves $$e^{3x} - e^{2x} + e^x - 6 = 0$$ [3 marks]

Curve C has equation \(y = x^2\) C is translated by vector \(\begin{pmatrix} 3 \\ 0 \end{pmatrix}\) to give curve \(C_1\) Line L has equation \(y = x\) L is stretched by scale factor 2 parallel to the \(x\)-axis to give line \(L_1\) Find the exact distance between the two intersection points of \(C_1\) and \(L_1\) [6 marks]

1. Find the equation of the tangent to the curve \(y = e^{4x}\) at the point \((a, e^{4a})\). [3 marks]
2. Find the value of \(a\) for which this tangent passes through the origin. [2 marks]
3. Hence, find the set of values of \(m\) for which the equation $$e^{4x} = mx$$ has no real solutions. [3 marks]

The diagram below shows a circle and four triangles. \(AB\) is a diameter of the circle. \(C\) is a point on the circumference of the circle. Triangles \(ABK\), \(BCL\) and \(CAM\) are equilateral. Prove that the area of triangle \(ABK\) is equal to the sum of the areas of triangle \(BCL\) and triangle \(CAM\). [5 marks]

Raj is investigating how the price, \(P\) pounds, of a brilliant-cut diamond ring is related to the weight, \(C\) carats, of the diamond. He believes that they are connected by a formula $$P = aC^n$$ where \(a\) and \(n\) are constants.

1. Express \(\ln P\) in terms of \(\ln C\). [2 marks]
2. Raj researches the price of three brilliant-cut diamond rings on a website with the following results.

   \(C\)	0.60	1.15	1.50
   \(P\)	495	1200	1720

   1. Plot \(\ln P\) against \(\ln C\) for the three rings on the grid below. [2 marks] \includegraphics{figure_10b}
   2. Explain which feature of the plot suggests that Raj's belief may be correct. [1 mark]
   3. Using the graph on page 15, estimate the value of \(a\) and the value of \(n\). [4 marks]
3. Explain the significance of \(a\) in this context. [1 mark]
4. Raj wants to buy a ring with a brilliant-cut diamond of weight 2 carats. Estimate the price of such a ring. [2 marks]

A go-kart and driver, of combined mass 55 kg, move forward in a straight line with a constant acceleration of \(0.2\text{ m s}^{-2}\) The total driving force is 14 N Find the total resistance force acting on the go-kart and driver. Circle your answer. [1 mark] 0N 3N 11N 14N

One of the following is an expression for the distance between the points represented by position vectors \(5\mathbf{i} - 3\mathbf{j}\) and \(18\mathbf{i} + 7\mathbf{j}\) Identify the correct expression. Tick (\(\checkmark\)) one box. [1 mark] \(\sqrt{13^2 + 4^2}\) \(\sqrt{13^2 + 10^2}\) \(\sqrt{23^2 + 4^2}\) \(\sqrt{23^2 + 10^2}\)

An object is moving in a straight line, with constant acceleration \(a\text{ m s}^{-2}\), over a time period of \(t\) seconds. It has an initial velocity \(u\) and final velocity \(v\) as shown in the graph below. \includegraphics{figure_13} Use the graph to show that $$v = u + at$$ [3 marks]

A particle of mass 0.1 kg is initially stationary. A single force \(\mathbf{F}\) acts on this particle in a direction parallel to the vector \(7\mathbf{i} + 24\mathbf{j}\) As a result, the particle accelerates in a straight line, reaching a speed of \(4\text{ m s}^{-1}\) after travelling a distance of 3.2 m Find \(\mathbf{F}\). [5 marks]

A particle, \(P\), is moving in a straight line with acceleration \(a\text{ m s}^{-2}\) at time \(t\) seconds, where $$a = 4 - 3t^2$$

1. Initially \(P\) is stationary. Find an expression for the velocity of \(P\) in terms of \(t\). [2 marks]
2. When \(t = 2\), the displacement of \(P\) from a fixed point, O, is 39 metres. Find the time at which \(P\) passes through O, giving your answer to three significant figures. Fully justify your answer. [5 marks]

A simple lifting mechanism comprises a light inextensible wire which is passed over a smooth fixed pulley. One end of the wire is attached to a rigid triangular container of mass 2 kg, which rests on horizontal ground. A load of \(m\) kg is placed in the container. The other end of the wire is attached to a particle of mass 5 kg, which hangs vertically downwards. The mechanism is initially held at rest as shown in the diagram below. \includegraphics{figure_16} The mechanism is released from rest, and the container begins to move upwards with acceleration \(a\text{ m s}^{-2}\) The wire remains taut throughout the motion.

1. Show that $$a = \left(\frac{3 - m}{m + 7}\right)g$$ [4 marks]
2. State the range of possible values of \(m\). [1 mark]
3. In this question use \(g = 9.8\text{ m s}^{-2}\) The load reaches a height of 2 metres above the ground 1 second after it is released. Find the mass of the load. [4 marks]
4. Ignoring air resistance, describe one assumption you have made in your model. [1 mark]