Pre-U Pre-U 9794/1 (Pre-U Mathematics Paper 1) 2010 June

Solve the equation \(2^x = 4^{2x+1}\). [3]

The equation \(x^3 - 5x + 3 = 0\) has a root between \(x = 0\) and \(x = 1\).

1. The equation can be rearranged into the form \(x = g(x)\) where \(g(x) = px^3 + q\). State the values of \(p\) and \(q\). [1]
2. By considering \(|g'(x)|\), show that the iterative form \(x_{n+1} = g(x_n)\) with a suitable starting value converges to the root between \(x = 0\) and \(x = 1\). [You are not required to find this root.] [2]

Let \(f(x) = x^2(x - 2)\) and \(g(x) = 2x - 1\) for all real \(x\).

1. Sketch the graph of \(y = f(x)\) and explain briefly why the function f has no inverse. [2]
2. Write down \(g^{-1}(x)\). [1]
3. On the same diagram, sketch the graphs of \(y = f(x - 1) - 3\) and \(y = g^{-1}(x)\) and state the number of real roots of the equation \(f(x - 1) - 3 = g^{-1}(x)\). [3]

Using the substitution \(u = 1 + \sqrt{x}\), or otherwise, find \(\int \frac{1}{1 + \sqrt{x}} dx\) giving your answer in terms of \(x\). [5]

The parametric equations of a curve are \(x = \frac{1}{1 + t^2}\) and \(y = \frac{t}{1 + t^2}\), \(t \in \mathbb{R}\).

1. Find \(\frac{dy}{dx}\) in terms of \(t\). [5]
2. Hence find the coordinates of the stationary points of the curve. [2]

A geometric progression with common ratio \(r\) consists of positive terms. The sum of the first four terms is five times the sum of the first two terms.

1. Find an equation in \(r\) and deduce that \(r = 2\). [3]
2. Given that the fifth term is 192, find the value of the first term. [1]
3. Find the smallest value of \(n\) such that the sum of the first \(n\) terms of the progression exceeds \(10^{64}\). [3]

Let \(f(x) = \frac{1 + x^2}{\sqrt{4 - 3x}}\)

1. Obtain in ascending powers of \(x\) the first three terms in the expansion of \(\frac{1}{\sqrt{4 - 3x}}\) and state the values of \(x\) for which this expansion is valid. [5]
2. Hence obtain an approximation to \(f(x)\) in the form \(a + bx + cx^2\) where \(a\), \(b\) and \(c\) are constants. [2]
3. Use your approximation to estimate \(\int_0^{0.1} f(x) dx\). [2]

The points \(A\) and \(B\) have position vectors \(\mathbf{i} - \mathbf{j} + \mathbf{k}\) and \(2\mathbf{i} + \mathbf{j} + 3\mathbf{k}\) respectively, relative to the origin \(O\). The point \(C\) is on the line \(OA\) extended so that \(\overrightarrow{AC} = 2\overrightarrow{OA}\) and the point \(D\) is on the line \(OB\) extended so that \(\overrightarrow{BD} = 3\overrightarrow{OB}\). The point \(X\) is such that \(OCXD\) is a parallelogram.

1. Show that a vector equation of the line \(AX\) is \(\mathbf{r} = \mathbf{i} - \mathbf{j} + \mathbf{k} + \lambda(5\mathbf{i} + 7\mathbf{k})\) and find an equation of the line \(CD\) in a similar form. [5]
2. Prove that the lines \(AX\) and \(CD\) intersect and find the position vector of their point of intersection. [4]

A curve has equation \(x^2 - xy + y^2 = 1\).

1. Find \(\frac{dy}{dx}\) in terms of \(x\) and \(y\). [4]
2. Find the coordinates of the points on the curve in the second and fourth quadrants where the tangent is parallel to \(y = x\). [5]

1. Solve the equation \((2 + i)z = (4 + in)\). Give your answer in the form \(a + ib\), expressing \(a\) and \(b\) in terms of the real constant \(n\). [4]
2. The roots of the equation \(z^2 + 8z + 25 = 0\) are denoted by \(z_1\) and \(z_2\).
   1. Find \(z_1\) and \(z_2\) and show these roots on an Argand diagram. [3]
   2. Find the modulus and argument in radians of each of \((z_1 + 1)\) and \((z_2 + 1)\). [3]

1. Write down an identity for \(\tan 2\theta\) in terms of \(\tan \theta\) and use this result to show that $$\tan 3\theta = \frac{3 \tan \theta - \tan^3 \theta}{1 - 3 \tan^2 \theta}.$$ [4]
2. Given that \(0 < \theta < \frac{1}{2}\pi\) and \(\theta = \sin^{-1}\left(\frac{1}{\sqrt{10}}\right)\), show that \(\tan 3\theta = \frac{13}{3}\). [3]
3. Show that the solutions of the equation $$\tan(3 \sin^{-1} x) = \frac{13}{3}$$ for \(0 < x < 2\pi\) are $$x = \frac{\sqrt{10}}{10} \quad \text{and} \quad x = \frac{\sqrt{10(1 + 3\sqrt{3})}}{20}.$$ [4]

1. Events \(A\) and \(B\) are such that \(\mathrm{P}(A' \cap B') = \frac{1}{6}\).
   1. Find \(\mathrm{P}(A \cup B)\). [2]
   2. Given that \(\mathrm{P}(A | B) = \frac{1}{4}\) and \(\mathrm{P}(B) = \frac{1}{3}\), find \(\mathrm{P}(A \cap B)\) and \(\mathrm{P}(A)\). [3]
2. In playing the UK Lottery, a set of 6 different integers is chosen irrespective of order from the integers 1 to 49 inclusive. How many different sets of 6 integers can be chosen? [2]

A survey was conducted into the annual salary offered for 19 different jobs in 2008. The results were as follows, in thousands of pounds.

15	16	18	19	21	36	36	38	41	41
43	47	51	55	56	60	62	64	110

It was decided to undertake a further study to see if self-esteem was correlated with level of annual salary. A random sample of 11 employees was taken and self-esteem was rated on a scale of 1 to 10 with the highest self-esteem being 10. The results were as follows.

Salary in £10 000's	1	2	3	4	5	6	7	8	9	10	11
Self-esteem	4	3	5	1	7	7	8	5	10	7	9

\begin{enumerate}[label=(\alph*)] \item In a game show contestants are asked up to five questions in succession to qualify for the next round. An incorrect answer eliminates a contestant from the game show. Let \(X\) denote the number of questions correctly answered by a contestant. The probability distribution of \(X\) is given below.

\(x\)	0	1	2	3	4	5
\(\mathrm{P}(X = x)\)	0.30	0.25	0.20	0.16	0.06	0.03

1. Find the expected number of correctly answered questions and the variance of the distribution. [3]
2. Find the probability that a randomly selected contestant will correctly answer 3 or more questions. [1]
3. Each show had two contestants. Find the probability that both the contestants will correctly answer at least one question. [2]

\item In a promotion, a newspaper included a token in every copy of the newspaper. A proportion, 0.002, are winning tokens and occur randomly. A reader keeps buying copies of the newspaper until he buys one with a winning token and then stops. Let \(Y\) denote the number of copies bought.

1. Explain briefly why this situation may be modelled by a geometric distribution and write down a formula for \(\mathrm{P}(Y = y)\). [2]
2. Find the probability that the reader gets a winning token with the twentieth copy bought. [2]
3. Find the probability that the reader will not have to buy more than three copies in order to get a winning token. [2] \end{enumerate]

A manufacturer produces components designed with length \(L\) mm such that \(12 < L < 15\). The Quality Control department finds that 15% of the components sampled are longer than 15 mm while 8% are shorter than 12 mm. Assume that \(L\) is normally distributed with mean \(\mu\) and standard deviation \(\sigma\).

1. Calculate \(\mu\) and \(\sigma\). [6]
2. The shortest 5% of components are rejected. Find the minimum length which a component may have before it is rejected. [3]
3. It was found in a random sample that 10% of components were longer than 16 mm. Determine whether this finding is consistent with the assumption that \(L\) is normally distributed with the \(\mu\) and \(\sigma\) found in part (i). [3]