Pre-U Pre-U 9794/2 (Pre-U Mathematics Paper 2) 2012 June

1. Solve the equation \(x^2 - 8x + 4 = 0\), giving your answer in the form \(p \pm q\sqrt{3}\), where \(p\) and \(q\) are integers. [2]
2. Expand and simplify \((6 + 2\sqrt{3})(2 - \sqrt{3})\). [3]

\includegraphics{figure_2} The diagram shows a triangle \(ABC\). The vertices have coordinates \(A(3, -7)\), \(B(9, 1)\) and \(C(-1, -5)\).

1. 1. Find the length of the side \(AB\). [2]
   2. Find the coordinates of the mid-point of \(AB\). [1]
   3. A circle has diameter \(AB\). Find the equation of the circle in the form \((x - a)^2 + (y - b)^2 = r^2\), where \(a\), \(b\) and \(r\) are constants to be found. [3]
2. Find the equation of the line \(l\) passing through \(B\) parallel to \(AC\). [3]

Find the exact value of \(\int_0^1 (e^x - x) dx\). [4]

Use logarithms to solve the equation \(2^{2x-1} = 5\). [4]

Sketch, on separate diagrams, the graphs of the following functions for \(0 \leqslant x \leqslant 2\pi\) giving the coordinates of all points of intersection with the axes.

1. \(y = \sin x\). [1]
2. \(y = \sin\left(x + \frac{1}{6}\pi\right)\). [2]

1. An arithmetic sequence has first term 5 and fifth term 37.
   1. Find an expression for \(u_n\), the \(n\)th term of the sequence, in terms of \(n\). [4]
   2. Find an expression for \(S_n\), the sum of the first \(n\) terms of this sequence, in terms of \(n\). [2]
2. Hence, or otherwise, calculate \(\sum_{n=5}^{25} (8n - 3)\). [2]

Let \(y = (2x - 3)e^{-2x}\).

1. Find \(\frac{dy}{dx}\), giving your answer in the form \(e^{-2x}(ax + b)\), where \(a\) and \(b\) are integers. [3]
2. Determine the set of values of \(x\) for which \(y\) is increasing. [2]

Solve the differential equation \(\frac{dy}{dx} = -y^2 x^3\), where \(y = 2\) when \(x = 1\), expressing your solution in the form \(y = f(x)\). [6]

\includegraphics{figure_9} The diagram shows a sector of a circle, \(OMN\). The angle \(MON\) is \(2x\) radians, the radius of the circle is \(r\) and \(O\) is the centre.

1. Find expressions, in terms of \(r\) and \(x\), for the area, \(A\), and perimeter, \(P\), of the sector. [2]
2. Given that \(P = 20\), show that \(A = \frac{100x}{(1 + x)^2}\). [2]
3. Find \(\frac{dA}{dx}\), and hence find the value of \(x\) for which the area of the sector is a maximum. [5]

1. 1. Find \(\int \frac{e^x}{1 + e^x} dx\). [2]
   2. Hence evaluate \(\int_0^{\ln 3} \frac{e^x}{1 + e^x} dx\), giving your answer in the form \(\ln k\), where \(k\) is an integer. [3]
2. 1. Using the substitution \(u = 1 + e^x\), find \(\int \left(\frac{e^x}{1 + e^x}\right)^2 dx\). [5]
   2. Hence find the exact volume of the solid of revolution generated when the curve given by \(y = \frac{e^x}{1 + e^x}\), between \(x = -\ln 3\) and \(x = \ln 3\), is rotated through \(2\pi\) radians about the \(x\)-axis. [2]

The function f is defined by \(f : t \mapsto 2 \sin t + \cos 2t\) for \(0 \leqslant t < 2\pi\).

1. Show that \(\frac{df}{dt} = 2 \cos t(1 - 2 \sin t)\). [2]
2. Determine the range of f. [5]

A curve \(C\) is given parametrically by \(x = 2 \cos t + \sin 2t\), \(y = f(t)\) for \(0 \leqslant t < 2\pi\).

1. Show that \(x^2 + y^2 = 5 + 4 \sin 3t\). [3]
2. Deduce that \(C\) lies between two circles centred at the origin, and touches both. [2]
3. Find the gradient of the tangent to \(C\) at the point at which \(t = 0\). [3]