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Mathematics C Final Term 2 Examination Paper SAMPLE C
发布时间：2024-07-18
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SAMPLE C
Mathematics C
Final Term 2 Examination Paper
Question 1 (12 marks) Use a SEPARATE book clearly marked Question 1
(i) Write down the coefficient of x3 in the expansion of (1+ x)6.
(ii) Find 
.
(iii) Find all asymptotes for the rational function 
.
(iv) A regular 6-sided die is rolled once and the number on the uppermost face is recorded as a score.
(a) Find the probability of scoring a number greater than 4 or an even number.
(b) Find the probability of scoring a number greater than 4 given that an even number has been scored.
(v) For the scores 8, 3, 6, 8, 7, 4, find:
(a) the mode.
(b) the median.
(c) the mean.
(d) the standard deviation. [Give answer correct to 2 significant figures.]
(e) the number of scores within one standard deviation of the mean.
Question 2 (12 marks) Use a SEPARATE book clearly marked Question 2
(i) Differentiate each of the following with respect to x :
(a) ln(3x - 1).
(b) e(x2 -1).
(ii) Find the number of different arrangements that can be made from all of the letters
of the word EXERCISE if:
(a) there are no restrictions.
(b) the letters E are together.
(c) there is a vowel at each end of the arrangement.
(iii) The continuous random variable X has cumulative distribution function given by
(a) Find the probability density function f(x).
(b) Find P(X < 1.5).
Question 3 (12 marks) Use a SEPARATE book clearly marked Question 3
(i) Find the following indefinite integrals:
(ii) In this question you may use the table of areas under the standard normal curve which can be found at the end of this examination paper.
A random variable X is normally distributed with a mean of 70 and a standard deviation of 18.
(a) Find P(79 ≤ X ≤ 101. 5) .
(b) Find the value of k such that P(X ≥ k) = 0 . 006 .
(iii) Consider the following table:
|
x |
14 |
21 |
19 |
32 |
11 |
|
y |
0 .5 |
1.1 |
3.7 |
17 . 2 |
0.6 |
(a) Calculate Pearson’s correlation coefficient for the above table. Give your answer correct to 3 decimal places.
(b) Describe the strength of the linear association you found in part (a) .
Question 4 (12 marks) Use a SEPARATE book clearly marked Question 4
(i) (a) Sketch the graph of the hyperbola 
.
(b) Use the graph to solve the inequality 
.
(ii) A group of 16 disk drives coming off an assembly line includes 4 which are
defective. The group of 16 disk drives is passed on to a quality control engineer who selects a random sample of 3 of the disk drives and tests them for defects.
Let X be the random variable associated with the number of defective disk drives in the sample.
(a) Show that the probability that there are no defective disk drives in the sample is 28/11.
(b) Find the probability that exactly 2 disk drives in the sample are defective.
(c) Copy and complete the probability distribution table for the random variable X.
|
x |
0 |
|
|
|
|
P(X = x) |
11 28 |
|
|
|
(d) Find the expected number of defective disk drives in the sample.
Question 5 (12 marks) Use a SEPARATE book clearly marked Question 5
(i) The temperature TO Celsius of a cake t minutes after it is taken out of the oven is
given by the equation T = 30 +180e—0.08t .
(a) Find the temperature of the cake as it is being taken out of the oven.
(b) Find the temperature of the cake 10 minutes after it is taken out of the oven.
(c) Find the time taken for the cake to cool to a temperature of 35OC. Give the answer correct to the nearest minute.
(d) Show that dt/dT = -0.08(T - 30)
(e) Find the rate at which the cake is cooling at the time when its temperature is 35OC .
(ii)
Find 
dx by using the substitution u = ex + 3 .
(iii) (a) Find dx/d(xln x - x)
(b) Hence, or otherwise, find the exact value of 
ln x dx .
Question 6 (12 marks) Use a SEPARATE book clearly marked Question 6
(i) A bag contains 3 silver and 1 gold coin.
(a) If two coins are drawn at random in succession, without replacement, find the probability that both coins are silver.
(b) If coins are drawn at random one by one, without replacement,
until one coin is left in the bag, find the probability that this coin is gold.
(ii) Find the term independent of x in the expansion of 
(iii) In a class of 50 students 12 students study Science and 13 students are from
Indonesia. Three of the students who study Science are from Indonesia.
(a) Illustrate this information on a Venn diagram.
(b) One student in the class is chosen at random.
(α) Find the probability that the student is from Indonesia.
(β) Find the probability the student is from Indonesia but does not study Science.
(y) Find the probability that the student is from Indonesia given that the student does not study Science.
