Pre-U Pre-U 9794/1 (Pre-U Mathematics Paper 1) Specimen

1 It is given that \(8 ^ { 4 x } = 4 ^ { 3 x - 6 }\).

1. By expressing each side as a power of 2 , find the value of \(x\).
2. Write down the value of \(\log _ { 4 } | x |\).

2 Find the value of the positive constant \(k\) for which \(\int _ { 1 } ^ { k } ( 2 x - 1 ) \mathrm { d } x = 6\).

3

1. Find the value of \(a\) for which ( \(x - 2\) ) is a factor of \(5 x ^ { 3 } + a x ^ { 2 } + 6 a x - 8\).
2. Show that, for this value of \(a\), the cubic equation \(5 x ^ { 3 } + a x ^ { 2 } + 6 a x - 8 = 0\) has only one real root.

4

1. Sketch the graph of \(y = \sqrt { 2 } \sin x\) for \(0 \leqslant x \leqslant 2 \pi\). The points \(P\) and \(Q\) on the graph have \(x\)-coordinates \(\frac { 1 } { 4 } \pi\) and \(\frac { 3 } { 4 } \pi\), respectively.
2. Determine the equation of the tangent to the curve at \(P\). The normals to the curve at \(P\) and \(Q\) intersect at the point \(R\).
3. Determine the exact coordinates of \(R\).

5 The complex number \(z\) satisfies the equation \(2 z - \mathrm { i } = \mathrm { i } z + 2\).

1. Express \(z\) in the form \(a + \mathrm { i } b\) where \(a\) and \(b\) are rational numbers.
2. Find the exact value of \(| z |\) and of \(\arg ( z )\).
3. Express \(z ^ { 2 }\) in the form \(c + \mathrm { i } d\) where \(c\) and \(d\) are rational numbers.
4. Verify that \(\tan ( 2 \arg ( z ) ) = \tan \left( \arg \left( z ^ { 2 } \right) \right)\) using an appropriate trigonometrical identity.

6

1. (a) Using the substitution \(u = \frac { 1 } { 2 } \pi - x\), show that $$\int _ { 0 } ^ { \frac { 1 } { 2 } \pi } \cos ^ { 2 } x \mathrm {~d} x = \int _ { 0 } ^ { \frac { 1 } { 2 } \pi } \sin ^ { 2 } u \mathrm {~d} u$$ (b) Hence find the common value of these definite integrals.
2. Find the exact value of $$\int _ { 0 } ^ { \frac { 1 } { 3 } \pi } \cos ^ { 2 } x \mathrm {~d} x$$

7 With respect to the origin \(O\), the points \(A\) and \(B\) have position vectors \(\mathbf { a }\) and \(\mathbf { b }\) respectively, where \(\mathbf { a } = 2 \mathbf { i } - 3 \mathbf { j } + 6 \mathbf { k }\) and \(\mathbf { b } = - \mathbf { i } - \mathbf { j } + 2 \mathbf { k }\). The lines \(L _ { 1 }\) and \(L _ { 2 }\) have the vector equations $$\mathbf { r } = \mathbf { a } + \lambda \mathbf { b } , \quad \mathbf { r } = 2 \mathbf { b } + \mu \mathbf { a }$$ respectively.

1. Determine whether or not \(L _ { 1 }\) and \(L _ { 2 }\) intersect.
2. Find the acute angle between the directions of \(L _ { 1 }\) and \(L _ { 2 }\). The point \(C\) has position vector \(\mathbf { c } = p \mathbf { i } + \mathbf { j } + r \mathbf { k }\).
3. Given that \(O C\) is perpendicular to the triangle \(O A B\), determine \(p\) and \(r\).
4. Determine the volume of the tetrahedron \(O A B C\).

8

1. The sum of the first \(n\) terms of the arithmetic series \(1 + 3 + 5 + \ldots\) exceeds the sum of the first \(n\) terms of the arithmetic series \(100 + 97 + 94 + \ldots\). Find the least possible value of \(n\).
2. \(3 \sqrt { 2 }\) and \(2 - \sqrt { 2 }\) are the first two terms of a geometric progression.
   1. Show that the third term is \(\sqrt { 2 } - \frac { 4 } { 3 }\).
   2. Find the index \(n\) of the first term that is less than 0.01.
   3. Find the exact value of the sum to infinity of this progression.
   4. Which of the terms 'alternating', 'periodic', 'convergent' apply to the sequences generated by the following \(n\)th terms, where \(n\) is a positive integer?
      (a) \(1 - \left( \frac { 3 } { 4 } \right) ^ { n }\) (b) \(\frac { 1 } { n } \cos n \pi\) (c) \(\sec n \pi\)

9 The cubic polynomial \(x ^ { 3 } + a x ^ { 2 } + b x + c\), where \(a , b\) and \(c\) are real, is denoted by \(\mathrm { p } ( x )\).

1. Give a reason why the equation \(\mathrm { p } ( x ) = 0\) has at least one real root.
2. Suppose that the curve with equation \(y = \mathrm { p } ( x )\) has a local minimum point and a local maximum point with \(y\)-coordinates \(y _ { \text {min } }\) and \(y _ { \text {max } }\) respectively.
   1. Prove that if \(y _ { \text {min } } y _ { \text {max } } < 0\), then the equation \(\mathrm { p } ( x ) = 0\) has three real roots.
   2. Comment on the number of distinct real roots of the equation \(\mathrm { p } ( x ) = 0\) in the case \(y _ { \text {min } } y _ { \text {max } } = 0\).
   3. Suppose instead that the equation \(\mathrm { p } ( x ) = 0\) has only one real root for all values of \(c\). Prove that \(a ^ { 2 } \leqslant 3 b\).
   4. The iterative scheme $$x _ { n + 1 } = \frac { 2 x _ { n } ^ { 3 } + 1 } { 3 x _ { n } ^ { 2 } + 1 } , \quad x _ { 0 } = 0$$ converges to a root of the cubic equation \(\mathrm { p } ( x ) = 0\).
      (a) Find \(\mathrm { p } ( x )\).
      (b) Find the limit of the iteration, correct to 4 decimal places.
   5. Determine the rate of convergence of the iterative scheme.

10 Cheeky Cola is sold in bottles of two sizes, small and large. For each size, the content of a randomly chosen bottle is normally distributed with mean and standard deviation, in litres, as given in the table.

1. Find the probability that a randomly chosen small bottle contains more than 0.51 litres.
2. Find \(x\) if the probability that a randomly chosen large bottle contains less than 1.45 litres is 0.1 . The manufacturer introduces a new size of bottle of Cheeky Cola, called the mega bottle. It is found that the probabilities that a randomly chosen mega bottle contains less than 2.97 litres or more than 3.05 litres are both 0.05 .
3. Assuming that the contents of the mega bottle are normally distributed, find the mean and variance of the distribution.

11 During the 30-day month of April, the probability that it will rain on any given day is 0.25 .

1. Find the probability that the first rainy day in April is the 7th of April, explaining any modelling assumptions you have made.
2. Given that it does not rain on the first 6 days of April, find the probability that it first rains on the 10th of April.
3. The probability that it first rains on the \(n\)th of April and next on the ( \(n + 3\) )th of April is 0.02 , correct to 1 significant figure. Determine \(n\).
4. Determine the expected number of dry days in April, given that it first rains on the 8th of April.

12 A faulty random number generator generates odd digits according to the probability distribution for the random variable \(X\) given in the following table.

1. Find \(p\).
2. Find \(\mathrm { E } ( X )\) and \(\mathrm { E } \left( X ^ { 2 } \right)\).
3. Deduce the value of \(\operatorname { Var } ( X )\). A second random number generator generates odd digits each with equal probability. Both random generators are operated once.
4. Find the probability that both generate a prime number.
5. Given that the first generates 1, 3 or 5, find the probability that both generate a power of 3 . 1315 pupils, including two sisters, are placed in random order in a line.
6. What is the probability that the sisters are not next to each other?
7. How many arrangements are there with 9 pupils between the sisters? A team of 5 is chosen from the 15 pupils.
8. How many ways are there of choosing the team if no more than one of the sisters can be in the team? Having chosen the first team, a second team of 5 pupils is chosen from the remaining 10 pupils.
9. How many ways are there of choosing the teams if each sister is in one or other of the teams?