Pre-U Pre-U 9794/1 (Pre-U Mathematics Paper 1) 2011 June

Find the equation of the line passing through the points \((-2, 5)\) and \((4, -7)\). Give your answer in the form \(y = mx + c\). [3]

\includegraphics{figure_2} The diagram shows a sector \(OAB\) of a circle with centre \(O\) and radius \(r\) cm in which angle \(AOB\) is \(\theta\) radians. The sector has a perimeter of 18 cm.

1. Show that \(\theta = \frac{18 - 2r}{r}\). [2]
2. Find the area of the sector in terms of \(r\), simplifying your answer. [2]

Solve the equation \(3 + 2x = |7 - 4x|\). [3]

1. Show that \(4 \ln x - \ln(3x - 2) - \ln x^2 = \ln\left(\frac{x^2}{3x - 2}\right)\), where \(x > \frac{2}{3}\). [3]
2. Hence solve the equation \(4 \ln x - \ln(3x - 2) - \ln x^2 = 0\). [3]

A circle has equation \(x^2 + y^2 = 16\). Find the volume generated when the region in the first quadrant which is bounded by the circle and the lines \(x = 1\) and \(x = 2\) is rotated through \(2\pi\) radians about the \(x\)-axis. [5]

1. Sketch, on a single diagram, the graphs of \(y = e^{3x}\) and \(y = x\) and state the number of roots of the equation \(e^{3x} = x\). [3]
2. Use the Newton-Raphson method with \(x_0 = 0\) to determine the value of a root of the equation \(e^{3x} = x\) correct to 3 decimal places. [4]

1. Given that the point \((-1, -2, 4)\) lies on both the lines $$\mathbf{r} = \begin{pmatrix} 2 \\ -3 \\ a \end{pmatrix} + \lambda \begin{pmatrix} -3 \\ 2 \\ 1 \end{pmatrix} \quad \text{and} \quad \mathbf{r} = \begin{pmatrix} 2 \\ 4 \\ b \end{pmatrix} + \mu \begin{pmatrix} -1 \\ -2 \\ 1 \end{pmatrix},$$ find \(a\) and \(b\). [3]
2. Find the acute angle between the lines. [4]

1. Find and simplify the first three terms in the expansion of \((1 - 4a)^{\frac{1}{2}}\) in ascending powers of \(a\), where \(|a| < \frac{1}{4}\). [4]
2. Hence show that the roots of the quadratic equation \(x^2 - x + a = 0\) are approximately \(1 - a - a^2\) and \(a + a^2\), where \(a\) is small. [4]

1. Prove that \(\sin 3\theta = 3 \sin \theta - 4 \sin^3 \theta\) and deduce that $$\sin \theta + \sin 3\theta = 4 \sin \theta \cos^2 \theta.$$ [5]
2. Hence find the values of \(\theta\) such that \(0° < \theta < 180°\) that satisfy the equation $$\cot^2 \theta = \sin \theta + \sin 3\theta.$$ [4]

1. The complex number \(z\) is such that \(|z| = 2\) and \(\arg z = -\frac{3}{4}\pi\). Find the exact value of the real part of \(z\) and of the imaginary part of \(z\). [2]
2. The complex numbers \(u\) and \(v\) are such that $$u = 1 + ia \quad \text{and} \quad v = b - i,$$ where \(a\) and \(b\) are real and \(a < b\). Given that \(uv = 7 + 9i\), find the values of \(a\) and \(b\). [7]

An arithmetic progression has first term \(a\) and common difference \(d\). The first, ninth and fourteenth terms are, respectively, the first three terms of a geometric progression with common ratio \(r\), where \(r \neq 1\).

1. Find \(d\) in terms of \(a\) and show that \(r = \frac{5}{3}\). [7]
2. Find the sum to infinity of the geometric progression in terms of \(a\). [2]

Find the general solution of the differential equation $$\frac{dy}{dx} = \frac{x}{x(1 + x^2)}$$ giving your answer in the form \(y = f(x)\). [10]

1. A random sample of young people in a certain town comprised 312 boys and 253 girls. Denoting a boy's age by \(x\) years and a girl's age by \(y\) years, the following data were obtained: $$\sum x = 4618, \quad \sum x^2 = 68812, \quad \sum y = 3719, \quad \sum y^2 = 55998.$$
   1. Calculate the mean and standard deviation of the ages of the boys in the sample and also of the girls in the sample. [3]
   2. Use these results to comment on the distribution of the ages of the boys and girls in the sample. [1]
2. How many arrangements of the letters of the word DEFEATED are there in which the Es are separated from each other? [3]

1. The table below relates the values of two variables \(x\) and \(y\).

   \(x\)	1	\(A\)	\(A + 3\)	10
   \(y\)	2	\(A - 1\)	\(A\)	5

   \(A\) is a positive integer and \(\sum xy = 92\).
   1. Calculate the value of \(A\). [3]
   2. Explain how you can tell that the product-moment correlation coefficient is 1. [1]
2. A music society has 300 members. 240 like Puccini, 100 like Wagner and 50 like neither.
   1. Calculate the probability that a member chosen at random likes Puccini but not Wagner. [3]
   2. Calculate the probability that a member chosen at random likes Puccini given that this member likes Wagner. [2]

A firm produces chocolate bars whose weights are normally distributed with mean 120 g and standard deviation 6 g.

1. Bars which weigh more than 114 g are sold at a profit of 15p per bar. The remaining bars are sold at no profit. Show that the expected profit per 100 bars is £12.62. [5]
2. It is subsequently decided that bars which weigh more than \(x\) g should be sold at a profit of 20p per bar. Those which weigh \(x\) g or less are sold to employees at a profit of 3p per bar. The expected profit per 100 bars is £19.17. Find the value of \(x\). [7]

In a factory, computer chips are produced in large batches. A quality control procedure is used for each batch which requires a random sample of 8 chips to be tested. If no faulty chip is found, the batch is accepted. If two or more are faulty, the batch is rejected. If one is faulty, a further sample of 4 is selected and the batch is accepted if none of these is faulty. The probability of any chip being faulty is \(q\).

1. Show that the probability of accepting a batch is \(p^8(1 + 8p^3 - 8p^4)\), where \(p = 1 - q\). [6]
2. Find the expected number of chips sampled per batch, giving your answer in terms of \(p\). Hence show that when \(p = 0.75\), the expected number of chips sampled per batch is approximately 9. [6]