Pre-U Pre-U 9794/2 (Pre-U Mathematics Paper 2) 2016 June

1. Find the remainder when \(x^3 + 2x\) is divided by \(x + 2\). [2]
2. Write down the value of \(k\) for which \(x + 2\) is a factor of \(x^3 + 2x + k\). [1]

Solve the equation \(4 \times 3^x = 5\), giving the solution in an exact form. [4]

The graph of \(\log_{10} y\) against \(x\) is a straight line with gradient 2 and the intercept on the vertical axis at 4. Write down an equation for this straight line and show that \(y = 10000 \times 100^x\). [4]

The complex numbers \(z_1\) and \(z_2\) are given by \(z_1 = 2 + i\) and \(z_2 = 3 + 4i\).

1. Verify that \(|z_1| + |z_2| > |z_1 + z_2|\). [4]
2. Sketch on an Argand diagram the locus \(|z - z_1| = 2\). [2]

1. Show that \(\frac{3}{x+2} + \frac{1}{x+1} \equiv \frac{4x+5}{x^2+3x+2}\). [2]
2. Differentiate \(\frac{4x+5}{x^2+3x+2}\) with respect to \(x\). [3]
3. Hence show that the function given by $$f(x) = \frac{4x+5}{x^2+3x+2}, \quad x \neq -1, x \neq -2,$$ is a decreasing function. [2]

The points \(A\) and \(B\) are at \((2, 3, 5)\) and \((8, 2, 4)\) with respect to the origin \(O\).

1. Find the size of angle \(AOB\). [4]
2. Show that triangle \(AOB\) is isosceles. [3]

1. Use a change of sign to verify that the equation \(\cos x - x = 0\) has a root \(\alpha\) between \(x = 0.7\) and \(x = 0.8\). [2]
2. Sketch, on a single diagram, the curve \(y = \cos x\) and the line \(y = x\) for \(0 \leqslant x \leqslant \frac{1}{2}\pi\), giving the coordinates of all points of intersection with the coordinate axes. [2]

An iteration of the form \(x_{n+1} = \cos(x_n)\) is to be used to find \(\alpha\).

1. By considering the gradient of \(y = \cos x\), show that this iteration will converge. [3]
2. On a copy of your sketch from part (ii), illustrate how this iteration converges to \(\alpha\). [2]
3. Use a change of sign to verify that \(\alpha = 0.7391\) to 4 decimal places. [2]

\(P\) and \(Q\) are points on the circumference of a circle with centre \(O\) and radius \(r\). The angle \(POQ\) is \(\theta\) radians. Given that the chord \(PQ\) has length 4, find an expression for the length of the arc \(PQ\) in terms of \(\theta\) of only. [5]

1. Show that \(\frac{\sin x}{1 + \sin x} \equiv \sec x \tan x - \sec^2 x + 1\). [5]
2. Hence show that \(\int_0^{\frac{\pi}{4}} \frac{\sin x}{1 + \sin x} \, dx = \frac{1}{4}\pi + \sqrt{2} - 2\). [6]

1. Using the substitution \(u = \frac{1}{x}\), or otherwise, find \(\int \frac{\sin\left(\frac{1}{x}\right)}{x^2} \, dx\). [4]
2. Evaluate \(\int_{\frac{1}{2\pi}}^{\frac{1}{\pi}} \frac{\sin\left(\frac{1}{x}\right)}{x^2} \, dx\) and \(\int_{\frac{1}{2\pi}}^{\frac{1}{\pi}} \frac{\sin\left(\frac{1}{x}\right)}{x^2} \, dx\). [3]
3. Show that, when \(n\) is a positive integer, the integral \(\int_{\frac{1}{(n+1)\pi}}^{\frac{1}{n\pi}} \frac{\sin\left(\frac{1}{x}\right)}{x^2} \, dx\) takes one of the two values found in part (ii), distinguishing between the two cases. [3]

The function f is defined by \(f(x) = \sqrt{x}, x > 0\).

1. Use differentiation from first principles to find an expression for \(f'(x)\). [5]

The lines \(l_1\) and \(l_2\) are the tangents to the curve \(y = f(x)\) at the points \(A\) and \(B\) where \(x = a\) and \(x = b\) respectively, \(a \neq b\).

1. 1. Show that the tangents intersect at the point \(\left(\sqrt{ab}, \frac{1}{2}(\sqrt{a} + \sqrt{b})\right)\). [5]
   2. Given that \(l_1\) and \(l_2\) intersect at a point with integer coordinates, write down a possible pair of values for \(a\) and \(b\). [2]