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ARAB OPEN UNIVERSITY
FACULTY OF COMPUTER STUDIES
INFORMATION TECHNOLOGY AND COMPUTING
MST 121: USING MATHEMATICS
FALL 2004 -PART-1I- EXAM
EXAM PERIOD: 150 MINUTES
FINAL EXAM ( FORM A )
Student ID:
Student Name:
Group Number:
Tutor Name:
THIS EXAM IS IN TWO PARTS:
GRADE FOR PART I (OUT OF 36 POINTS)
GRADE FOR PART II (OUT OF 64 POINTS)
EXAM GRADE (OUT OF 100 POINTS)
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I. MULTIPLE CHOICE PART (WORTH 36 POINTS).
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Please answer exactly twelve (12) questions from the available sixteen
(16). Each question is worth 3 points. From the given choices, a, b, c &
d, please do box the choice that solves the question best, and write down
the letter corresponding to your choice in the space provided. Should
you think that the correct answer is not among the four choices, please
write the word: NONE, for none of the above, in the space provided.
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Question - 1:The acceleration of a moving particle traveling according to
the trajectory y (t) = - 5 t 2 + 5t + 7, is:
(a) -10
(b) -10t +5
(c) 7
(d) infinite
The answer is:
……………………………………………………………………………………..
Question - 2: For positive t, the solution of the differential equation,
dy/dt = 1/ t , with y (1) = 0, is:
(a) 1+ ln(t)
(b) ln (t)
(c) ln(t)-1
(d) exp(t-1)
The answer is:
………………………………………………………………………………..
Question - 3: The second derivative f ″ (x), for f (x) = e 2x - sin (x+1), is:
(a) 4ex - cos (x+1)
(b) e2x - cos (x)
(c) 4e2x + sin (x +1)
(d) ln (2x ) + cos (x) +1
The answer is:
………………………………………………………………………………...
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Question - 4: The area represented by the definite integral:
(a) 28
(b) 56
(c) 3
2
 3( x
2
 x / 2)dx
1
(d) 0
The answer is:
………………………………………………………………………………..
Question - 5: The integral  exp (0.5 x)dx , is equal to:
(a) exp( x)  C
(b) C  2 exp( x / 2)
(c) 1/2
(d) C  2 exp( x)
The answer is:
……………………………………………………………………………………..
Question - 6: If y (x) = ln (3x2 ) + x, then the derivative of y is:
(a) 2/ x
(b) 6x + 1
(c) x3 + 1
1
(d) 1+ 2/ x
The answer is:
……………………………………………………………………………………..
Question - 7: The function, f (x) = x2 - 6x, has a local minimum at the point:
(a)(3, -9)
(b) (3, 9)
(c) (2, -8)
(d) (0, 0)
The answer is:
………………………………………………………………………………..
Question - 8: The function f (x) = 1/3 x3 +5/2 x2 + 6x, is stationary at:
(a) only x = -2
(b) only x = 3
(c) x = -2, -3
(d) x = 2, 3
The answer is:
……………………………………………………………………………………..
Question - 9: The number of 4 digit passwords, not starting with zero, that
can be made from the numbers, 0,1,2…9, is:
(a)10000
(b) 1000
(c) 64
(d) 9000
The answer is:
……………………………………………………………………………………..
Question - 10: The average mark on an MST final in a particular section of
25 students is 73 with a standard deviation of 12. A 95% confidence interval
for the MST student population mean mark is:
(a) (51.5, 97.5)
(b) (68.3, 77.7)
(c) (65, 85)
(d) (12/73, 73/12)
The answer is:
3
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Question - 11: The median of the score list: 50, 90, 100, 90, 80, 70, 70, is:
(a)100
(b) 70
(c) 80
(d) 90
The answer is:
………………………………………………………………………………………
Question -12: The score list: 50, 90, 100, 90, 80, 70, 70:
(a)is unimodal
(c) has a mean= 70
(b)has a mode =100
(d)is bimodal
The answer is:
………………………………………………………………………………………
Question - 13: The upper quartile of the list: 50, 90,100, 90, 80, 70, 70, is:
(a)100
(b) 70
(c) 80
(d) 90
The answer is:
………………………………………………………………………………………
Question - 14: In a game, two fair dice are tossed once, and the sum is
recorded. The probability P of the event {the sum is not equal to 7} is:
(a)5/6
(b) 5/36
(c) 7/36
(d)1/6
The answer is:
………………………………………………………………..……..
Question - 15: For a Japanese island population, the mean life expectancy of
an individual is 82 years, with a population standard deviation of 9 years.
The life expectancy of family made of 36 individuals is considered for study.
Considered a sample, the family mean life expectancy is then expected to be
82 years, with sample standard deviation equal to:
(a) 3/2
(b) 9/36
(c) 82/9
(d) 9/82
The answer is:
……………………………………………………………………………………..
Question - 16: In a Tutor assessment survey, a section of 17 students, 4
students give the Tutor the score of 1 for poor, 5 rate him 2 for average, 5
rate him 3 for very good, and the remaining 3 rate him 4for excellent. The
Tutor’s assessment survey approximate mean is:
(a)40/16
(b) 1.06
(c) 2.41
(d)10 / 17
The answer is:
4
II. ESSAY QUESTIONS PART:(WORTH 64 POINTS).
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Please answer exactly four (4) questions from the available six (6). Each
question has 4 sub-parts. Each sub- part is worth 4 points, for a total of
16 points for each question. Please show your work, and write down
your answers neatly in the space provided directly below each sub- part.
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Question – 1: Consider the function f(x) = 1 – e - x in [0,  ).
a)Find the area A under the function in the given interval [0, 1].
(b) Show that f (x) satisfies the (differential) equation: f(x) + df (x)/dx = 1.
(c) Find the second derivative, f " (x), of the function f(x), when x = 1.
What happens to f (x), f ' (x) and f " (x), as x goes to  (infinity).
x
(d) Consider the function G (x) =
 f (t )dt . Find G(0), G(1) and G' (x).
0
5
...……………………………………………………………………………..
Question – 2: Consider the functions:
f ( x)  x  4 x , ( x  0) , g ( x)  3x 2  3 / x, ( x  0) , and h( x)  2 f ( x)  g ( x), ( x  0) .
(a) Find the stationary points of f , if any, and classify them.
Find the equation of the tangent line at x =1.
(b) Find the first and the second derivatives, g ' and g ' ' , of the function g .
Show that g has no stationary points. Find the intervals over which g is
decreasing, if any. Find the intervals over which g ' is decreasing, if any.
(c) Find the gradient of the graph of h(x), at x = 1.
4
(d) Compute the value of the definite integral W =  h( x)dx .
1
6
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Question – 3: At t = 0, a soccer player, hits an incoming ball with his head
in the air, 2 m above the ground, with an initial head velocity v0, and an
angle  . The ball takes the x and y system of trajectories given by:
{x (t) = v0 cos (  ) t, and y(t) = - 5 t2 + v0 sin (  )t + 2}, (*) .
(a) Rewrite the equations in (*) if the ball is initially coming horizontally
(  =0) and v0 = 20 m/s. Find the time at which the ball hits the ground. Find
the x range the ball can reach. In the absence of obstacles ( such as a goal
keeper), can the ball penetrate the goal net which is 4 m away.?
(b)Rewrite the equations in (*) if the ball is vertical (  =90) and v0 = 10 m/s.
Find the maximum y height the ball can reach.
(c) Find the equations for the velocities in the x and y directions, for  = 45,
and v0 = 10 2 m/s. Find the maximum height, and range the ball can attain.
(d) Find the x and y accelerations of the ball at any incoming angle  , and
any head velocity v0. Can you justify the distinction between them, if any.
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…………………………………………………………………………….....
Question -4: In an experiment a pair of fair dice are tossed once, the SUM
of the numbers on the pair of dice is recorded. .
(a) Write down the list of outcomes (numbers recorded) and the probability
of each outcome. Find the expected value E (mean) for this experiment.
(b) Find the probabilities P(A), P(B), and P (C), of the events A = {the
number recorded is greater than 20 or equal to 1}, B= { the number
recorded is a positive integer}, and C={the number recorded is prime}.
(c) In a particular toss, consider the event given that number recorded is 8,
what is the probability P (D) that the tossed dice form a pair of 4’s.
d)Find the probability P (E), where E = {the number recorded is in the
interval [4.5, 9.5]}.
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Question - 5: Two AOU Branches A and B have each a large MST course
student population. The student population mean grade in branch A is 72
with a population standard deviation of 12, while the population mean
grade in Branch B is 68 with a population standard deviation of 16.
(a) Find how the average grade of a sample MST section of 25 students in
branch A is distributed (give the sample mean and standard deviation).
(b) Find a 95% confidence interval for the grade of an MST student picked
at random from a sample section of 25 students in Branch B.
(c ) If a sample of 36 MST students are picked from Branch A, and 64
students are picked in Branch B, Find the ESE (Estimated Standard Error)
two sample statistic.
(d) Use a 5% level of significance to test for the null hypothesis:
H0 = "The mean grades in Branch A & B are equal".
.
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Question – 6: Let a and b be positive numbers, and consider the functions:
f (x) = a exp (-b x), and, P (x) = ( ½)exp (- x /2) in [0,  ).
(a)Find f (0 ). Show that f (x) is decreasing in [0,  ). Find the number f (x )
converges to, as x goes to infinity, if any. What relation must be satisfied by
the constants a and b, so that the area under f (x) in [0,  ) is 1.

(  f ( x)dx  1. ).
0
(b)Verify that the function, P (x), is a probability density in [0,  ).
(c) Compute P (x is in the interval [0,  ) )
(d) Write down the integrals expressing the mean,  , and the variance,  2 ,
of the probability density function f (x). (Do not compute the integrals).
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