AQA AS Paper 1 (AS Paper 1) 2023 June

At a point \(P\) on a curve, the gradient of the tangent to the curve is 10 State the gradient of the normal to the curve at \(P\) Circle your answer. [1 mark] \(-10\) \quad \(-0.1\) \quad \(0.1\) \quad \(10\)

Identify the expression below which is equivalent to \(\left(\frac{2x}{5}\right)^{-3}\) Circle your answer. [1 mark] \(\frac{8x^3}{125}\) \quad \(\frac{125x^3}{8}\) \quad \(\frac{125}{8x^3}\) \quad \(\frac{8}{125x^3}\)

The coefficient of \(x^2\) in the binomial expansion of \((1 + ax)^6\) is \(\frac{20}{3}\) Find the two possible values of \(a\) [3 marks]

It is given that \(5\cos^2 \theta - 4\sin^2 \theta = 0\)

1. Find the possible values of \(\tan \theta\), giving your answers in exact form. [3 marks]
2. Hence, or otherwise, solve the equation $$5\cos^2 \theta - 4\sin^2 \theta = 0$$ giving all solutions of \(\theta\) to the nearest \(0.1°\) in the interval \(0° \leq \theta \leq 360°\) [2 marks]

1. Given that \(y = x\sqrt{x}\), find \(\frac{dy}{dx}\) [2 marks]
2. The line, \(L\), has equation \(6x - 2y + 5 = 0\) \(L\) is a tangent to the curve with equation \(y = x\sqrt{x} + k\) Find the value of \(k\) [5 marks]

1. The curve \(C_1\) has equation \(y = 2x^2 - 20x + 42\) Express the equation of \(C_1\) in the form $$y = a(x - h)^2 + c$$ where \(a\), \(b\) and \(c\) are integers. [3 marks]
2. Write down the coordinates of the minimum point of \(C_1\) [1 mark]
3. The curve \(C_1\) is mapped onto the curve \(C_2\) by a stretch in the \(y\)-direction. The minimum point of \(C_2\) is at \((5, -4)\) Find the equation of \(C_2\) [2 marks]

Points \(P\) and \(Q\) lie on the curve with equation \(y = x^4\) The \(x\)-coordinate of \(P\) is \(x\) The \(x\)-coordinate of \(Q\) is \(x + h\)

1. Expand \((x + h)^4\) [2 marks]
2. Hence, find an expression, in terms of \(x\) and \(h\), for the gradient of the line \(PQ\) [1 mark]
3. Explain how to use the answer from part (b) to obtain the gradient function of \(y = x^4\) [2 marks]

1. Show that $$\int_1^a \left(6 - \frac{12}{\sqrt{x}}\right) dx = 6a - 24\sqrt{a} + 18$$ [3 marks]
2. The curve \(y = 6 - \frac{12}{\sqrt{x}}\), the line \(x = 1\) and the line \(x = a\) are shown in the diagram below. The shaded region \(R_1\) is bounded by the curve, the line \(x = 1\) and the \(x\)-axis. The shaded region \(R_2\) is bounded by the curve, the line \(x = a\) and the \(x\)-axis. \includegraphics{figure_8} It is given that the areas of \(R_1\) and \(R_2\) are equal. Find the value of \(a\) Fully justify your answer. [4 marks]

A continuous curve has equation \(y = f(x)\) The curve passes through the points \(A(2, 1)\), \(B(4, 5)\) and \(C(6, 1)\) It is given that \(f'(4) = 0\) Jasmin made two statements about the nature of the curve \(y = f(x)\) at the point \(B\): Statement 1: There is a turning point at \(B\) Statement 2: There is a maximum point at \(B\)

1. Draw a sketch of the curve \(y = f(x)\) such that Statement 1 is correct and Statement 2 is correct. [1 mark]
2. Draw a sketch of the curve \(y = f(x)\) such that Statement 1 is correct and Statement 2 is not correct. [1 mark]
3. Draw a sketch of the curve \(y = f(x)\) such that Statement 1 is not correct and Statement 2 is not correct. [1 mark]

Charlie buys a car for £18000 on 1 January 2016. The value of the car decreases exponentially. The car has a value of £12000 on 1 January 2018.

1. Charlie says: • because the car has lost £6000 after two years, after another two years it will be worth £6000. Charlie's friend Kaya says: • because the car has lost one third of its value after two years, after another two years it will be worth £8000. Explain whose statement is correct, justifying the value they have stated. [2 marks]
2. The value of Charlie's car, £\(V\), \(t\) years after 1 January 2016 may be modelled by the equation $$V = Ae^{-kt}$$ where \(A\) and \(k\) are positive constants. Find the value of \(t\) when the car has a value of £10000, giving your answer to two significant figures. [5 marks]
3. Give a reason why the model, in this context, will not be suitable to calculate the value of the car when \(t = 30\) [1 mark]

1. A circle has equation $$x^2 + y^2 - 10x - 6 = 0$$ Find the centre and the radius of the circle. [2 marks]
2. An equilateral triangle has one vertex at the origin, and one side along the line \(x = 8\), as shown in the diagram below. \includegraphics{figure_11}
   1. Show that the vertex at the origin lies inside the circle \(x^2 + y^2 - 10x - 6 = 0\) [1 mark]
   2. Prove that the triangle lies completely within the circle \(x^2 + y^2 - 10x - 6 = 0\) [4 marks]

A particle, initially at rest, starts to move forward in a straight line with constant acceleration, \(a \text{ m s}^{-2}\) After 6 seconds the particle has a velocity of \(3 \text{ m s}^{-1}\) Find the value of \(a\) Circle your answer. [1 mark] \(-2\) \quad \(-0.5\) \quad \(0.5\) \quad \(2\)

A resultant force of \(\begin{bmatrix} -2 \\ 6 \end{bmatrix}\) N acts on a particle. The acceleration of the particle is \(\begin{bmatrix} -6 \\ y \end{bmatrix} \text{ m s}^{-2}\) Find the value of \(y\) Circle your answer. [1 mark] \(2\) \quad \(3\) \quad \(10\) \quad \(18\)

A ball, initially at rest, is dropped from a vertical height of \(h\) metres above the Earth's surface. After 4 seconds the ball's height above the Earth's surface is \(0.2h\) metres.

1. Assuming air resistance can be ignored, show that $$h = 10g$$ [3 marks]
2. Assuming air resistance cannot be ignored, explain the effect that this would have on the value of \(h\) in part (a). [1 mark]

A particle is moving in a straight line such that its velocity, \(v \text{ m s}^{-1}\), changes with respect to time, \(t\) seconds, as shown in the graph below. \includegraphics{figure_15}

1. Show that the acceleration of the particle over the first 4 seconds is \(3.5 \text{ m s}^{-2}\) [1 mark]
2. The particle is initially at a fixed point \(P\) Show that the displacement of the particle from \(P\), when \(t = 9\), is 62 metres. [3 marks]

A toy remote control speed boat is launched from one edge of a small pond and moves in a straight line across the pond's surface. The boat's velocity, \(v \text{ m s}^{-1}\), is modelled in terms of time, \(t\) seconds after the boat is launched, by the expression $$v = 0.9 + 0.16t - 0.06t^2$$

1. Find the acceleration of the boat when \(t = 2\) [3 marks]
2. Find the displacement of the boat, from the point where it was launched, when \(t = 2\) [4 marks]

A particle, \(P\), is initially at rest on a smooth horizontal surface. A resultant force of \(\begin{bmatrix} 12 \\ 9 \end{bmatrix}\) N is then applied to \(P\), so that it moves in a straight line.

1. Find the magnitude of the resultant force. [1 mark]
2. Two fixed points \(A\) and \(B\) have position vectors $$\overrightarrow{OA} = \begin{bmatrix} 3 \\ 7 \end{bmatrix} \text{ metres} \quad \text{and} \quad \overrightarrow{OB} = \begin{bmatrix} k \\ k-1 \end{bmatrix} \text{ metres}$$ with respect to a fixed origin, \(O\) \(P\) moves in a straight line parallel to \(\overrightarrow{AB}\)
   1. Find \(\overrightarrow{AB}\) in terms of \(k\) [1 mark]
   2. Find the value of \(k\) [2 marks]

A rescue van is towing a broken-down car by using a tow bar. The van and the car are moving with a constant acceleration of \(0.6 \text{ m s}^{-2}\) along a straight horizontal road as shown in the diagram below. \includegraphics{figure_18} The van has a total mass of 2780 kg The car has a total mass of 1620 kg The van experiences a driving force of \(D\) newtons. The van experiences a total resistance force of \(R\) newtons. The car experiences a total resistance force of \(0.6R\) newtons.

1. The tension in the tow bar, \(T\) newtons, may be modelled by $$T = kD - 18$$ where \(k\) is a constant. Find \(k\) [5 marks]
2. State one assumption that must be made in answering part (a). [1 mark]