AQA Paper 1 (Paper 1) Specimen

Find the gradient of the line with equation \(2x + 5y = 7\) Circle your answer. [1 mark] \(\frac{2}{5}\) \quad \(\frac{5}{2}\) \quad \(-\frac{2}{5}\) \quad \(-\frac{5}{2}\)

A curve has equation \(y = \frac{2}{\sqrt{x}}\) Find \(\frac{dy}{dx}\) Circle your answer. [1 mark] \(\frac{\sqrt{x}}{3}\) \quad \(\frac{1}{x\sqrt{x}}\) \quad \(-\frac{1}{x\sqrt{x}}\) \quad \(-\frac{1}{2x\sqrt{x}}\)

When \(\theta\) is small, find an approximation for \(\cos 3\theta + \theta \sin 2\theta\), giving your answer in the form \(a + b\theta^2\) [3 marks]

\(p(x) = 2x^3 + 7x^2 + 2x - 3\)

1. Use the factor theorem to prove that \(x + 3\) is a factor of \(p(x)\) [2 marks]
2. Simplify the expression \(\frac{2x^3 + 7x^2 + 2x - 3}{4x^2 - 1}\), \(x \neq \pm \frac{1}{2}\) [4 marks]

The diagram shows a sector \(AOB\) of a circle with centre \(O\) and radius \(r\) cm. \includegraphics{figure_5} The angle \(AOB\) is \(\theta\) radians The sector has area 9 cm\(^2\) and perimeter 15 cm.

1. Show that \(r\) satisfies the equation \(2r^2 - 15r + 18 = 0\) [4 marks]
2. Find the value of \(\theta\). Explain why it is the only possible value. [4 marks]

Sam goes on a diet. He assumes that his mass, \(m\) kg after \(t\) days, decreases at a rate that is inversely proportional to the cube root of his mass.

1. Construct a differential equation involving \(m\), \(t\) and a positive constant \(k\) to model this situation. [3 marks]
2. Explain why Sam's assumption may not be appropriate. [1 mark]

Find the values of \(k\) for which the equation \((2k - 3)x^2 - kx + (k - 1) = 0\) has equal roots. [4 marks]

1. Given that \(u = 2^x\), write down an expression for \(\frac{du}{dx}\) [1 mark]
2. Find the exact value of \(\int_0^1 2^x \sqrt{3 + 2^x}\) dx Fully justify your answer. [6 marks]

A curve has equation \(y = \frac{2x + 3}{4x^2 + 7}\)

1. 1. Find \(\frac{dy}{dx}\) [2 marks]
   2. Hence show that \(y\) is increasing when \(4x^2 + 12x - 7 < 0\) [4 marks]
2. Find the values of \(x\) for which \(y\) is increasing. [2 marks]

The function f is defined by $$f(x) = 4 + 3^{-x}, \quad x \in \mathbb{R}$$

1. Using set notation, state the range of f [2 marks]
2. The inverse of f is \(f^{-1}\)
   1. Using set notation, state the domain of \(f^{-1}\) [1 mark]
   2. Find an expression for \(f^{-1}(x)\) [3 marks]
3. The function g is defined by $$g(x) = 5 - \sqrt{x}, \quad (x \in \mathbb{R} : x > 0)$$
   1. Find an expression for \(gf(x)\) [1 mark]
   2. Solve the equation \(gf(x) = 2\), giving your answer in an exact form. [3 marks]

A circle with centre \(C\) has equation \(x^2 + y^2 + 8x - 12y = 12\)

1. Find the coordinates of \(C\) and the radius of the circle. [3 marks]
2. The points \(P\) and \(Q\) lie on the circle. The origin is the midpoint of the chord \(PQ\). Show that \(PQ\) has length \(n\sqrt{3}\), where \(n\) is an integer. [5 marks]

A sculpture formed from a prism is fixed on a horizontal platform, as shown in the diagram. The shape of the cross-section of the sculpture can be modelled by the equation \(x^2 + 2xy + 2y^2 = 10\), where \(x\) and \(y\) are measured in metres. The \(x\) and \(y\) axes are horizontal and vertical respectively. \includegraphics{figure_12} Find the maximum vertical height above the platform of the sculpture. [8 marks]

Prove the identity \(\cot^2 \theta - \cos^2 \theta = \cot^2 \theta \cos^2 \theta\) [3 marks]

An open-topped fish tank is to be made for an aquarium. It will have a square horizontal base, rectangular vertical sides and a volume of 60 m\(^3\) The materials cost:

• £15 per m\(^2\) for the base
• £8 per m\(^2\) for the sides.

1. Modelling the sides and base of the fish tank as laminae, use calculus to find the height of the tank for which the overall cost of the materials has its minimum value. Fully justify your answer. [8 marks]
2. 1. In reality, the thickness of the base and sides of the tank is 2.5 cm Briefly explain how you would refine your modelling to take account of the thickness of the sides and base of the tank. [1 mark]
   2. How would your refinement affect your answer to part (a)? [1 mark]

The height \(x\) metres, of a column of water in a fountain display satisfies the differential equation \(\frac{dx}{dt} = \frac{8\sin 2t}{3\sqrt{x}}\), where \(t\) is the time in seconds after the display begins.

1. Solve the differential equation, given that initially the column of water has zero height. Express your answer in the form \(x = f(t)\) [7 marks]
2. Find the maximum height of the column of water, giving your answer to the nearest cm. [1 mark]

A student argues that when a rational number is multiplied by an irrational number the result will always be an irrational number.

1. Identify the rational number for which the student's argument is not true. [1 mark]
2. Prove that the student is right for all rational numbers other than the one you have identified in part (a). [4 marks]

\(f(x) = \sin x\) Using differentiation from first principles find the exact value of \(f'\left(\frac{\pi}{6}\right)\) Fully justify your answer. [6 marks]