AQA C4 (Core Mathematics 4) 2010 June

1. The polynomial \(f(x)\) is defined by \(f(x) = 8x^3 + 6x^2 - 14x - 1\). Find the remainder when \(f(x)\) is divided by \((4x - 1)\). [2 marks]
2. The polynomial \(g(x)\) is defined by \(g(x) = 8x^3 + 6x^2 - 14x + d\).
   1. Given that \((4x - 1)\) is a factor of \(g(x)\), find the value of the constant \(d\). [2 marks]
   2. Given that \(g(x) = (4x - 1)(ax^2 + bx + c)\), find the values of the integers \(a\), \(b\) and \(c\). [3 marks]

A curve is defined by the parametric equations $$x = 1 - 3t, \quad y = 1 + 2t^3$$

1. Find \(\frac{dy}{dx}\) in terms of \(t\). [3 marks]
2. Find an equation of the normal to the curve at the point where \(t = 1\). [4 marks]
3. Find a cartesian equation of the curve. [2 marks]

1. 1. Express \(\frac{7x - 3}{(x + 1)(3x - 2)}\) in the form \(\frac{A}{x + 1} + \frac{B}{3x - 2}\). [3 marks]
   2. Hence find \(\int \frac{7x - 3}{(x + 1)(3x - 2)} dx\). [2 marks]
2. Express \(\frac{6x^2 + x + 2}{2x^2 - x + 1}\) in the form \(P + \frac{Qx + R}{2x^2 - x + 1}\). [3 marks]

1. 1. Find the binomial expansion of \((1 + x)^{\frac{3}{2}}\) up to and including the term in \(x^2\). [2 marks]
   2. Find the binomial expansion of \((16 + 9x)^{\frac{3}{2}}\) up to and including the term in \(x^2\). [3 marks]
2. Use your answer to part (a)(ii) to show that \(13^{\frac{3}{2}} \approx 46 + \frac{a}{b}\), stating the values of the integers \(a\) and \(b\). [2 marks]

1. 1. Show that the equation \(3\cos 2x + 2\sin x + 1 = 0\) can be written in the form $$3\sin^2 x - \sin x - 2 = 0$$ [3 marks]
   2. Hence, given that \(3\cos 2x + 2\sin x + 1 = 0\), find the possible values of \(\sin x\). [2 marks]
2. 1. Express \(3\cos 2x + 2\sin 2x\) in the form \(R\cos(2x - \alpha)\), where \(R > 0\) and \(0° < \alpha < 90°\), giving \(\alpha\) to the nearest \(0.1°\). [3 marks]
   2. Hence solve the equation $$3\cos 2x + 2\sin 2x + 1 = 0$$ for all solutions in the interval \(0° < x < 180°\), giving \(x\) to the nearest \(0.1°\). [3 marks]

A curve has equation \(x^3 y + \cos(\pi y) = 7\).

1. Find the exact value of the \(x\)-coordinate at the point on the curve where \(y = 1\). [2 marks]
2. Find the gradient of the curve at the point where \(y = 1\). [5 marks]

The point \(A\) has coordinates \((4, -3, 2)\). The line \(l_1\) passes through \(A\) and has equation \(\mathbf{r} = \begin{bmatrix} 4 \\ -3 \\ 2 \end{bmatrix} + \lambda \begin{bmatrix} 2 \\ 0 \\ 1 \end{bmatrix}\). The line \(l_2\) has equation \(\mathbf{r} = \begin{bmatrix} -1 \\ 3 \\ 4 \end{bmatrix} + \mu \begin{bmatrix} 1 \\ -2 \\ -1 \end{bmatrix}\). The point \(B\) lies on \(l_2\) where \(\mu = 2\).

1. Find the vector \(\overrightarrow{AB}\). [3 marks]
2. 1. Show that the lines \(l_1\) and \(l_2\) intersect. [4 marks]
   2. The lines \(l_1\) and \(l_2\) intersect at the point \(P\). Find the coordinates of \(P\). [1 mark]
3. The point \(C\) lies on a line which is parallel to \(l_1\) and which passes through the point \(B\). The points \(A\), \(B\), \(C\) and \(P\) are the vertices of a parallelogram. Find the coordinates of the two possible positions of the point \(C\). [4 marks]

1. Solve the differential equation $$\frac{dx}{dt} = -\frac{1}{5}(x + 1)^{\frac{1}{2}}$$ given that \(x = 80\) when \(t = 0\). Give your answer in the form \(x = f(t)\). [6 marks]
2. A fungus is spreading on the surface of a wall. The proportion of the wall that is unaffected after time \(t\) hours is \(x\%\). The rate of change of \(x\) is modelled by the differential equation $$\frac{dx}{dt} = -\frac{1}{5}(x + 1)^{\frac{1}{2}}$$ At \(t = 0\), the proportion of the wall that is unaffected is 80%. Find the proportion of the wall that will still be unaffected after 60 hours. [2 marks]
3. A biologist proposes an alternative model for the rate at which the fungus is spreading on the wall. The total surface area of the wall is \(9\text{ m}^2\). The surface area that is affected at time \(t\) hours is \(A\text{ m}^2\). The biologist proposes that the rate of change of \(A\) is proportional to the product of the surface area that is affected and the surface area that is unaffected.
   1. Write down a differential equation for this model. (You are not required to solve your differential equation.) [2 marks]
   2. A solution of the differential equation for this model is given by $$A = \frac{9}{1 + 4e^{-0.09t}}$$ Find the time taken for 50% of the area of the wall to be affected. Give your answer in hours to three significant figures. [4 marks]