OCR MEI C3 (Core Mathematics 3) 2016 June

Find the exact value of \(\int_0^{\frac{1}{4}\pi} (1 + \cos \frac{1}{2}x) dx\). [3]

The functions \(f(x)\) and \(g(x)\) are defined by \(f(x) = \ln x\) and \(g(x) = 2 + e^x\), for \(x > 0\). Find the exact value of \(x\), given that \(fg(x) = 2x\). [5]

Find \(\int_{-1}^4 x^{-\frac{1}{2}} \ln x dx\), giving your answer in an exact form. [5]

By sketching the graphs of \(y = |2x + 1|\) and \(y = -x\) on the same axes, show that the equation \(|2x + 1| = -x\) has two roots. Find these roots. [4]

The volume \(V\) m³ of a pile of grain of height \(h\) metres is modelled by the equation $$V = 4\sqrt{h^3 + 1} - 4.$$

1. Find \(\frac{dV}{dh}\) when \(h = 2\). [4]

At a certain time, the height of the pile is 2 metres, and grain is being added so that the volume is increasing at a rate of 0.4 m³ per minute.

1. Find the rate at which the height is increasing at this time. [3]

Fig. 6 shows part of the curve \(\sin 2y = x - 1\). P is the point with coordinates \((1.5, \frac{1}{12}\pi)\) on the curve. \includegraphics{figure_6}

1. Find \(\frac{dy}{dx}\) in terms of \(y\). Hence find the exact gradient of the curve \(\sin 2y = x - 1\) at the point P. [4]

The part of the curve shown is the image of the curve \(y = \arcsin x\) under a sequence of two geometrical transformations.

1. Find \(y\) in terms of \(x\) for the curve \(\sin 2y = x - 1\). Hence describe fully the sequence of transformations. [4]

You are given that \(n\) is a positive integer. By expressing \(x^{2n} - 1\) as a product of two factors, prove that \(2^{2n} - 1\) is divisible by 3. [4]

Fig. 8 shows the curve \(y = \frac{x}{\sqrt{x+4}}\) and the line \(x = 5\). The curve has an asymptote \(l\). The tangent to the curve at the origin O crosses the line \(l\) at P and the line \(x = 5\) at Q. \includegraphics{figure_8}

1. Show that for this curve \(\frac{dy}{dx} = \frac{x + 8}{2(x + 4)^{\frac{3}{2}}}\). [5]
2. Find the coordinates of the point P. [4]
3. Using integration by substitution, find the exact area of the region enclosed by the curve, the tangent OQ and the line \(x = 5\). [9]

Fig. 9 shows the curve \(y = f(x)\), where \(f(x) = e^{2x} + k e^{-2x}\) and \(k\) is a constant greater than 1. The curve crosses the \(y\)-axis at P and has a turning point Q. \includegraphics{figure_9}

1. Find the \(y\)-coordinate of P in terms of \(k\). [1]
2. Show that the \(x\)-coordinate of Q is \(\frac{1}{4}\ln k\), and find the \(y\)-coordinate in its simplest form. [5]
3. Find, in terms of \(k\), the area of the region enclosed by the curve, the \(x\)-axis, the \(y\)-axis and the line \(x = \frac{1}{4}\ln k\). Give your answer in the form \(ak + b\). [4]

The function \(g(x)\) is defined by \(g(x) = f(x + \frac{1}{4}\ln k)\).

1. 1. Show that \(g(x) = \sqrt{k}(e^{2x} + e^{-2x})\). [3]
   2. Hence show that \(g(x)\) is an even function. [2]
   3. Deduce, with reasons, a geometrical property of the curve \(y = f(x)\). [3]