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

1 Find the set of all real values of \(x\) which satisfy the equation $$| 2 x + 5 | < 7$$

2 The function f is defined by \(\mathrm { f } ( x ) = \frac { 2 x + 1 } { x - 3 }\) for all real \(x , x \neq 3\). Find an expression for \(\mathrm { f } ^ { - 1 } ( x )\) and state the domain of \(\mathrm { f } ^ { - 1 }\).

3 Show that the equation of the tangent to the curve \(y = \ln \left( x ^ { 2 } + 3 \right)\) at the point \(( 1 , \ln 4 )\) is $$2 y - x = \ln ( 16 ) - 1$$

4 The diagram shows triangle \(A B C\), in which \(A B = 1\) unit , \(A C = k\) units and \(B C = 2\) units .

1. Express \(\cos C\) in terms of \(k\).
2. Given that \(\cos C < \frac { 7 } { 8 }\), show that \(2 k ^ { 2 } - 7 k + 6 < 0\) and find the set of possible values of \(k\).

5

1. Show that the equation \(4 - \frac { 4 } { \operatorname { cosec } x } = 3 \cos ^ { 2 } x\) can be expressed in the form $$3 \sin ^ { 2 } x - 4 \sin x + 1 = 0$$
2. Hence find all values of \(x\) for which \(0 < x < \pi\) that satisfy the equation $$4 - \frac { 4 } { \operatorname { cosec } x } = 3 \cos ^ { 2 } x$$

6 The equation \(x ^ { 3 } - x - 1 = 0\) has exactly one real root in the interval \(0 \leq x \leq 3\).

1. Denoting this root by \(\alpha\), find the integer \(n\) such that \(n < \alpha < n + 1\).
2. Taking \(n\) as a first approximation, use the Newton-Raphson method to find \(\alpha\), correct to 2 decimal places. You must show the result of each iteration correct to an appropriate degree of accuracy.

7 Express \(\frac { 1 - 6 x - 2 x ^ { 2 } } { ( x + 2 ) \left( x ^ { 2 } + 1 \right) }\) in the form \(\frac { A } { x + 2 } + \frac { B x + C } { x ^ { 2 } + 1 }\) where the numerical values of \(A , B\) and \(C\) are to be found. Hence show that \(\int _ { 0 } ^ { 1 } \frac { 1 - 6 x - 2 x ^ { 2 } } { ( x + 2 ) \left( x ^ { 2 } + 1 \right) } \mathrm { d } x = \ln 3 - \frac { 5 } { 2 } \ln 2\).

8

1. Show that the lines $$\mathbf { r } = - 3 \mathbf { i } + \mathbf { j } - 5 \mathbf { k } + \lambda ( \mathbf { i } + 2 \mathbf { j } + \mathbf { 6 } \mathbf { k } ) \quad \text { and } \quad \mathbf { r } = 4 \mathbf { i } + \mathbf { j } + 5 \mathbf { k } + \mu ( - 3 \mathbf { i } + \mathbf { j } - 2 \mathbf { k } )$$ intersect and write down the coordinates of their point of intersection.
2. Find in degrees the obtuse angle between the two lines.

9

1. Show that \(z = ( 1 + \mathrm { i } )\) is a root of the cubic equation \(3 z ^ { 3 } - 8 z ^ { 2 } + 10 z - 4 = 0\).
2. Show that the equation \(3 z ^ { 3 } - 8 z ^ { 2 } + 10 z - 4 = 0\) has a quadratic factor with real coefficients and hence solve this equation completely.

10

1. Show that \(\frac { \mathrm { d } } { \mathrm { d } x } ( \sin 3 x - 3 x \cos 3 x ) = 9 x \sin 3 x\). The curve shown in the figure below is part of the graph of the function \(y = x \sin 3 x\). \includegraphics[max width=\textwidth, alt={}, center]{3e4281d1-dbad-46a2-bbb7-97706bda2dfa-3_508_1136_1939_466}
2. Show that \(\int _ { 0 } ^ { \frac { 2 \pi } { 3 } } | x \sin 3 x | \mathrm { d } x = \frac { 4 \pi } { 9 }\).

11 A sequence of terms \(x _ { n }\) generated by a recurrence relation is said to be strictly increasing if, for each \(x _ { n } , x _ { n + 1 } > x _ { n }\).

1. Let a recurrence relation be defined by $$x _ { n + 1 } = \frac { x _ { n } ^ { 2 } + 2 } { 3 } \quad \text { and } \quad x _ { 0 } = \frac { 1 } { 2 } \quad \text { for } n \geq 0$$ Calculate \(x _ { 1 } , x _ { 2 } , x _ { 3 }\) and \(x _ { 4 }\) correct to 3 significant figures where appropriate.
2. Given the recurrence relation $$x _ { n + 1 } = \frac { x _ { n } ^ { 2 } + 2 } { 3 }$$ show that the sequence is strictly increasing when \(x _ { n } > 2\) or \(x _ { n } < 1\).
3. If \(- 1 < x _ { 0 } < 1\), then the sequence \(x _ { n } ( n \geq 0 )\) converges to a limit. Explain briefly why this limit is 1 .
4. Given the recurrence relation $$x _ { n + 1 } = \frac { x _ { n } ^ { 2 } + k } { m } \text { with } m > 0$$ prove that \(x _ { n }\) is a strictly increasing sequence for all \(x _ { n }\) if \(m ^ { 2 } < 4 k\).

12 A set of data is shown in the table below.

1. Calculate the mean and standard deviation of the data. The value 8 may be regarded as an outlier.
2. Explain how you would treat this outlier if the data represents
   1. the difference of the scores obtained when throwing a pair of ordinary dice,
   2. the number of thunderstorms per year in Cambridgeshire over a 23-year period.
   3. Without doing any further calculations state what effect, if any, removing the outlier would have on the mean and standard deviation.

13 A seed company investigated how well African Marigold seeds germinated when the seeds were past their sell-by date. The table shows the average number of seeds which germinated per packet, \(y\), and the number of months past their sell-by date, \(t\). The summary data for the investigation were as follows. $$\Sigma t = 150 \quad \Sigma t ^ { 2 } = 5500 \quad \Sigma y = 101.2 \quad \Sigma y ^ { 2 } = 2146.86 \quad \Sigma t y = 2740$$

1. Calculate the equation of the regression line of \(y\) on \(t\).
2. Use your regression line to calculate \(y\) when \(t = 10\). Compare your answer with the value of \(y\) when \(t = 10\) in the table and comment on the result.
3. Use your regression line to calculate \(y\) when \(t = 100\). Comment on the validity of this result.
4. Suggest with reasons whether the regression line provides a good model for predicting the germination of seeds past their sell-by date.

14 The maximum pressure exerted by the blood on the arteries in a population of elderly male patients may be modelled by a random variable having a normal distribution with a mean of 150 and standard deviation 15, measured in suitable units.

1. Find the probability that the maximum pressure for a randomly chosen patient is more than 160.
2. If the maximum pressure is found to be \(t\) or more, the patient must be referred to a consultant. If \(5 \%\) of the patients are referred to a consultant, find the value of \(t\).
3. Find the percentage of patients whose maximum pressure is between 130 and 160 . The probability that a randomly chosen patient attending a doctor's surgery has their blood pressure measured is 0.4 .
4. Find the probability that of 18 people attending a doctor's surgery more than 8 have their blood pressure measured, assuming that each measurement is random and independent of any other.
5. If 450 patients visited the surgery in a week, find the expected number of patients whose blood pressure would be measured.

15 In order to be accepted on a university course, a student needs to pass three exams.
The probability that the student passes the first exam is \(\frac { 3 } { 4 }\).
For each of the second and third exams, the probability of passing the exam is

• the same as the probability of passing the preceding exam if the student passed the preceding exam,
• half of the probability of passing the preceding exam if the student failed the preceding exam.
  1. Draw a tree diagram to represent the above information.
  2. Find the probability that the student passes all three exams.
  3. Find the probability that the student passes at least two of the exams.
  4. Find the probability that the student passes the third exam given that exactly two of the three exams are passed.