AQA AS Paper 1 (AS Paper 1) Specimen

The curve \(y = \sqrt{x}\) is translated onto the curve \(y = \sqrt{x + 4}\) The translation is described by a vector. Find this vector. Circle your answer. [1 mark] \(\begin{bmatrix} 4 \\ 0 \end{bmatrix}\) \(\begin{bmatrix} -4 \\ 0 \end{bmatrix}\) \(\begin{bmatrix} 0 \\ 4 \end{bmatrix}\) \(\begin{bmatrix} 0 \\ -4 \end{bmatrix}\)

Consider the two statements, A and B, below. A: \(x^2 - 6x + 8 > 0\) B: \(x > 4\) Choose the most appropriate option below. Circle your answer. [1 mark] \(A \Rightarrow B\) \(A \Leftarrow B\) \(A \Leftrightarrow B\) There is no connection between A and B

1. Write down the value of \(p\) and the value of \(q\) given that:
   1. \(\sqrt{3} = 3^p\) [1 mark]
   2. \(\frac{1}{9} = 3^q\) [1 mark]
2. Find the value of \(x\) for which \(\sqrt{3} \times 3^x = \frac{1}{9}\) [2 marks]

Show that \(\frac{5\sqrt{2} + 2}{3\sqrt{2} + 4}\) can be expressed in the form \(m + n\sqrt{2}\), where \(m\) and \(n\) are integers. [3 marks]

Jessica, a maths student, is asked by her teacher to solve the equation \(\tan x = \sin x\), giving all solutions in the range \(0° \leq x \leq 360°\) The steps of Jessica's working are shown below. \(\tan x = \sin x\) Step 1 \(\Rightarrow\) \(\frac{\sin x}{\cos x} = \sin x\) Write \(\tan x\) as \(\frac{\sin x}{\cos x}\) Step 2 \(\Rightarrow\) \(\sin x = \sin x \cos x\) Multiply by \(\cos x\) Step 3 \(\Rightarrow\) \(1 = \cos x\) Cancel \(\sin x\) \(\Rightarrow\) \(x = 0°\) or \(360°\) The teacher tells Jessica that she has not found all the solutions because of a mistake. Explain why Jessica's method is not correct. [2 marks]

A parallelogram has sides of length 6 cm and 4.5 cm. The larger interior angles of the parallelogram have size \(\alpha\) Given that the area of the parallelogram is 24 cm², find the exact value of \(\tan \alpha\) [4 marks]

Determine whether the line with equation \(2x + 3y + 4 = 0\) is parallel to the line through the points with coordinates \((9, 4)\) and \((3, 8)\). [4 marks]

1. Find the first three terms, in ascending powers of \(x\), of the expansion of \((1 - 2x)^{10}\) [3 marks]
2. Carly has lost her calculator. She uses the first three terms, in ascending powers of \(x\), of the expansion of \((1 - 2x)^{10}\) to evaluate \(0.998^{10}\) Find Carly's value for \(0.998^{10}\) and show that it is correct to five decimal places. [3 marks]

1. Given that \(f(x) = x^2 - 4x + 2\), find \(f(3 + h)\) Express your answer in the form \(h^2 + bh + c\), where \(b\) and \(c \in \mathbb{Z}\). [2 marks]
2. The curve with equation \(y = x^2 - 4x + 2\) passes through the point \(P(3, -1)\) and the point \(Q\) where \(x = 3 + h\). Using differentiation from first principles, find the gradient of the tangent to the curve at the point \(P\). [3 marks]

A student conducts an experiment and records the following data for two variables, \(x\) and \(y\).

\(x\)	1	2	3	4	5	6
\(y\)	14	45	130	1100	1300	3400
\(\log_{10} y\)

The student is told that the relationship between \(x\) and \(y\) can be modelled by an equation of the form \(y = kb^x\)

1. Plot values of \(\log_{10} y\) against \(x\) on the grid below. [2 marks] \includegraphics{figure_10}
2. State, with a reason, which value of \(y\) is likely to have been recorded incorrectly. [1 mark]
3. By drawing an appropriate straight line, find the values of \(k\) and \(b\). [4 marks]

Chris claims that, "for any given value of \(x\), the gradient of the curve \(y = 2x^3 + 6x^2 - 12x + 3\) is always greater than the gradient of the curve \(y = 1 + 60x - 6x^2\)". Show that Chris is wrong by finding all the values of \(x\) for which his claim is not true. [7 marks]

A curve has equation \(y = 6x\sqrt{x} + \frac{32}{x}\) for \(x > 0\)

1. Find \(\frac{dy}{dx}\) [4 marks]
2. The point \(A\) lies on the curve and has \(x\)-coordinate 4 Find the coordinates of the point where the tangent to the curve at \(A\) crosses the \(x\)-axis. [5 marks]

1. The unit vectors \(\mathbf{i}\) and \(\mathbf{j}\) are perpendicular. Find the magnitude of the vector \(-20\mathbf{i} + 21\mathbf{j}\) Circle your answer. [1 mark] \(-1\) \(1\) \(\sqrt{41}\) \(29\)
2. The angle between the vector \(\mathbf{i}\) and the vector \(-20\mathbf{i} + 21\mathbf{j}\) is \(\theta\) Which statement about \(\theta\) is true? Circle your answer. [1 mark] \(0° < \theta < 45°\) \(45° < \theta < 90°\) \(90° < \theta < 135°\) \(135° < \theta < 180°\)

In this question use \(g = 10\) m s⁻². A man of mass 80 kg is travelling in a lift. The lift is rising vertically. \includegraphics{figure_14} The lift decelerates at a rate of 1.5 m s⁻² Find the magnitude of the force exerted on the man by the lift. [3 marks]

The graph shows how the speed of a cyclist varies during a timed section of length 120 metres along a straight track. \includegraphics{figure_15}

1. Find the acceleration of the cyclist during the first 10 seconds. [1 mark]
2. After the first 15 seconds, the cyclist travels at a constant speed of 5 m s⁻¹ for a further \(T\) seconds to complete the 120-metre section. Calculate the value of \(T\). [4 marks]

A particle, of mass 400 grams, is initially at rest at the point \(O\). The particle starts to move in a straight line so that its velocity, \(v\) m s⁻¹, at time \(t\) seconds is given by \(v = 6t^2 - 12t^3\) for \(t > 0\)

1. Find an expression, in terms of \(t\), for the force acting on the particle. [3 marks]
2. Find the time when the particle next passes through \(O\). [5 marks]

In this question use \(g = 9.8\) m s⁻². A van of mass 1300 kg and a crate of mass 300 kg are connected by a light inextensible rope. The rope passes over a light smooth pulley, as shown in the diagram. The rope between the pulley and the van is horizontal. \includegraphics{figure_17} Initially, the van is at rest and the crate rests on the lower level. The rope is taut. The van moves away from the pulley to lift the crate from the lower level. The van's engine produces a constant driving force of 5000 N. A constant resistance force of magnitude 780 N acts on the van. Assume there is no resistance force acting on the crate.

1. 1. Draw a diagram to show the forces acting on the crate while it is being lifted. [1 mark]
   2. Draw a diagram to show the forces acting on the van while the crate is being lifted. [1 mark]
2. Show that the acceleration of the van is 0.80 m s⁻² [4 marks]
3. Find the tension in the rope. [2 marks]
4. Suggest how the assumption of a constant resistance force could be refined to produce a better model. [1 mark]