Electronics & Communication Gate Yearwise
Electronics and Communication Gate 2015 Set-3 Questions with Answer
Ques 40 GATE 2015 SET-3
A fair die with faces {1, 2, 3, 4, 5, 6} is thrown repeatedly till '3' is observed for the first time. Let X denote the number of times the die is thrown. The expected value of X is __________.
Ques 41 GATE 2015 SET-3
Consider the differential equation d2x(t)/dt2 + 3dx(t)/dt + 2x(t) = 0. Given x(0) = 20 and x(1) = 10/e, where e = 2.718, the value of x(2) is __________.
Given differential equation: d2x(t)/dt2 + 3dx(t)/dt + 2x(t) = 0
Initial conditions: x(0) = 20 and x(1) = 10/e
We need to find x(2)
First, I'll write the characteristic equation:
r2 + 3r + 2 = 0
Factoring: (r + 1)(r + 2) = 0
So the roots are r1 = -1 and r2 = -2
The general solution is:
x(t) = C1e-t + C2e-2t
Now applying the first condition x(0) = 20:
C1 + C2 = 20 ... (1)
Applying the second condition x(1) = 10/e:
C1e-1 + C2e-2 = 10/e
This simplifies to: C1 + C2/e = 10 ... (2)
From equation (1): C1 = 20 - C2
Substituting in equation (2):
(20 - C2) + C2/e = 10
20 - C2 + C2/e = 10
C2(1/e - 1) = -10
C2 = 10e/(e - 1)
C1 = (10e - 20)/(e - 1)
Finally calculating x(2):
x(2) = C1e-2 + C2e-4
x(2) = [(10e - 20)/(e - 1)] × e-2 + [10e/(e - 1)] × e-4
x(2) = [1/(e - 1)] × [(10e - 20)e-2 + 10e·e-4]
Putting e = 2.718:
e - 1 = 1.718
After calculation: x(2) ≈ 0.856
Answer: x(2) = 0.856
Ques 42 GATE 2015 SET-3
A vector field D = 2ρ2 aρ + z az exists inside a cylindrical region enclosed by the surfaces ρ=1, z=0 and z=5. Let S be the surface bounding this cylindrical region. The surface integral of this field on S (∮S D · ds) is __________.
Ques 43 GATE 2015 SET-3
The variance of the random variable X with probability density function f(x) = (1/2)|x|e-|x| is __________.
Ques 44 GATE 2015 SET-3
Choose the most appropriate word from the options given below to complete the following sentence.
If the athlete had wanted to come first in the race, he __________ several hours every day.
Ques 45 GATE 2015 SET-3
Choose the most suitable one word substitute for the following expression:
Connotation of a road or way
Ques 46 GATE 2015 SET-3
Choose the correct verb to fill in the blank below:
Let us __________.
Ques 47 GATE 2015 SET-3
Find the missing sequence in the letter series below:
A, CD, GHI, ?, UVWXY
Ques 48 GATE 2015 SET-3
If x > y > 1, which of the following must be true?
i. ln x > ln y
ii. ex > ey
iii. yx > xy
iv. cos x > cos y
Ques 49 GATE 2015 SET-3
Ram and Shyam shared a secret and promised to each other that it would remain between them. Ram expressed himself in one of the following ways as given in the choices below. Identify the correct way as per standard English.
Ques 50 GATE 2015 SET-3
In the following question, the first and the last sentence of the passage are in order and numbered 1 and 6. The rest of the passage is split into 4 parts and numbered as 2, 3, 4, and 5. These 4 parts are not arranged in proper order. Read the sentences and arrange them in a logical sequence to make a passage and choose the correct sequence from the given options.
1. On Diwali, the family rises early in the morning.
2. The whole family, including the young and the old enjoy doing this.
3. Children let off fireworks later in the night with their friends.
4. At sunset, the lamps are lit and the family performs various rituals.
5. Father, mother, and children visit relatives and exchange gifts and sweets.
6. Houses look so pretty with lighted lamps all around.
Ques 51 GATE 2015 SET-3
From a circular sheet of paper of radius 30 cm, a sector of 10% area is removed. If the remaining part is used to make a conical surface, then the ratio of the radius and height of the cone is __________.
Ques 52 GATE 2015 SET-3
The value of log tan 1° + log tan 2° + ....... + log tan 89° is ________.
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