Veer Narmad South Gujarat University 2011-1st Sem Certification MS-CIT SC-3801 - 101 : mathematics - 1 M.sc ( IT ) ( ) - Question Paper
Time : 3 Hours] [Total Marks : 70
M. Sc. (I.T.) (Sem. I) Examination
April/May - 2011
101 - Mathematics - I
Instruction :
""'N Seat No.:
6silq<3i Puunkiufl [qaim SuwiA u* qsq d-onql. Fillup strictly the details of signs on your answer book.
Name of the Examination :
M. Sc. (I.T.) (SEM. 1)
Name of the Subject:
101 - MATHEMATICS -1
|
-Subject Code No. |
|
-Section No. (1,2,.....): NIL |
(4)
(9)
(5)
(ii)
(iiO
(iv)
(b)
(c)
(i)
(ii)
-2 -3 L 2
5
0
7J
4
1
6
-2
1
-1
If A =
and B =
then find
Answer any two of the following. (6)
Let f, g : R -> R be two functions. Let f(x) = x+3 and (gof)(x) = 3x
5.Find the formula for the function g.
Let A = {1, 2, 3,4, 5}. Let f, g, h : A-> A be functions defined by f = {(1,
2), (2,3), (3, 2), (4, 5), (5,1)}; g = {(1, 3), (2, 2), (3,1), (4,1), (5, 4)} and h = {(1, 2), (2, 4), (3, 3), (4, 5), (5,1)}. (i) Determine which of the functions f, g, h are onto, (ii) Find fo(hog).
Let f : R - R be a function defined as f(x) =(3x + 4). Prove that f is one-to-one and onto and hence find the inverse off.
Consider the real valued functions f(x) = 3x1 + 4x - 5 and g(x) = (5x-9).
Find (fog)(x) and (gof)(5).
Define any three of the following giving one illustration to each. (3)
(i) An upper triangular matrix, (ii) Transpose of 3 matrix, (iii) Non-Singular matrix, (iv)symmetric matrix.(v) Skew-Hermitian matrix.
Answer any three of the following. (9)
In the usual notations prove that (XAUB) = Max. (Xa,Xb)-
1 3 3 -2 L5 4
|
(i) a matrix C such that 12A + 3C = 6B and T3 0 (iii) Obtain the adjA of a matrix A = 2 4 .4 -1 |
(ii) a matrix BA. and calculate A(adjA). |
(iv) Find the inverse of the following matrix A using elementary row transformations.
1 2-2]
A= -3 6 2
4 5 2
(v) Show that the every square matrix can be expressed as the sum of a symmetric matrix and a skew-symmetric matrix.
3.(a) State different measures of central tendency. Which measure will you (4) consider ideal. Why?
OR
(a) Explain Dispersion. Discuss the merits and demerits of a variance as a measure of dispersion.
(b) Answer any twO of the following. (10)
(i) Find mean, median, mode and P60 for the following data.
|
Age (in years) |
30-34 |
35-39 |
40-44 |
45-49 |
50-54 |
55-59 | ||||
|
No. of persons. |
4 |
6 |
12 |
18 |
14 |
8 | ||||
|
Calculate the harmonic mean, Q3 and D6 |
for the |
Following data. | ||||||||
|
Class : |
0-10 |
10-20 |
20-30 |
30-40 |
40-50 |
50-60 | ||||
|
Frequency |
10 |
16 |
40 |
20 |
10 |
4 | ||||
|
Wages: |
>180 |
>190 |
>200 |
>210 |
>220 |
>230 |
>240 |
|
No. of workers: |
40 |
36 |
28 |
13 |
4 |
1 |
0 |
(where > means greater than)
(c) Define: (i) Random experiment, (ii) Probability function. State and (4) prove addition theorem on probability.
OR
(c) Define: (i) Sample space, (ii) Conditional probability. State and prove multiplication theorem on probability.
4.(a) Answer any two of the following. (6)
(i) A box I contains 16 non-defective and 4 defective electric bulbs and box II contains 12 non-defective and 4 defective bulbs. If integer 1 or
2 appears on the face of the die, box I is selected and a bulb is drawn at random from it. If integer other than 1 and 2 appears on the die, box II is selected and a bulb is drawn from it. A bulb is drawn at random and it was found defective. What is the probability that it was selected from the box II ?
(ii) A card is selected from a pack of well shuffled 52 cards. Let A be the event that the selected card is of ace and B is the event that the card is of heart. Find P(A/B).
(iii) If A and B are independent events then prove that A~ and B are also independent.
(b) State the probability function of a binomial distribution with (4)
parameters n and p. Show that its mean is np and variance is npq, where q=l-p.
OR
(b) State the probability function of a Poisson distribution with parameter m. Show that its variance is m.
(c) Answer any two of the following. (6)
(i) The mean and standard deviation of a random variable X are 10 and 5 respectively. Find E(4 X + 5) and V(3X + 5)
(ii) A multiple choice test consists of 8 questions with 3 answers to each question (of which only one is correct). A student answers each question by tossing a die and checking the first answer if he gets 1 or 2; the second answer if he gets 3 or 4 and third answer if he gets 5 or 6. To get a distinction, the student must secure at least 75% correct answer. If there is no negative marking, what is the probability that the student secures a distinction?
(iii) Between 10 and 11 a.m. the average number of phone calls per minute coming in to the switch board of a company is 2.5. Using Poisson distribution, find the probability that during one particular minute there will be (i) no phone call at all. (ii) exactly 3 calls.
SC-3801] 3 [ 100 ]
Attempt all questions.
|
Attachment: |
| Earning: Approval pending. |
