GATE 2016 CS Set 1-2

9) If f(x) = 2x⁷ + 3x - 5, which of the following is a factor of f(x)?

(x³ +8)
(x-1)
(2x-5)
(x+1)

Answer: B

Explanation:

Given,
f(x) = 2x⁷ + 3x - 5
Putting x = 1, we get f(x) = 0.
So, x-1 is a factor of f(x).

Therefore option (B) will be the correct answer.

10) In a process, the number of cycles to failure decreases exponentially with an increase in load. At a load of 80 units, it takes 100 cycles for failure. When the load is halved, it takes 10000 cycles for failure. The load for which the failure will happen in 5000 cycles is ________.

40.00
46.02
60.01
92.02

Answer: B

Explanation:

According to the question, the number of cycles to failure decrease exponentially with an increase in load. So, we have general equation

y = ae^-bx

where, y = number of cycles to failure, and

x = load

GIven,

At load = 80 units , it takes 100 cycles for failure.

100 = ae^-80b ...................(1)

when load = halved, it takes 10000 cycles for failure.

10000 = ae^-40b ...............(2)

Divide (2) by (1), we get

=> e^40b = 100

=> b = log_e100 / 40 ..................(3)

Now, at 5000 cycles to failure

5000 = ae^-xb .................(4)

Again, divide (2) by (4), we get

=> eb^(x-40) = 2

=> b(x-40) = loge2

=> log_e100 / 40 * (x-40) = log_e2 ...............By using (3)

=> x = 40 * (log_e2 + log_e100) / log_e100

=> 40 * log_e200 / log_e100

=> 46.02

Therefore option (B) will be the correct answer.

11) Let p, q, r, s represent the following propositions.

p: x ∈ {8, 9, 10, 11, 12}
q: x is a composite number
r: x is a perfect square
s: x is a prime number

The integer x ≥ 2 which satisfies ¬((p ⇒ q) ∧ (¬r ∨ ¬s)) is

Answer: C

Explanation:

(p ⇒ q) will give {8, 9, 10, 12}

¬r will give {8, 10, 11, 12}

¬s will give {8, 9, 10, 12}

(¬r ∨ ¬s) will give {8, 9, 10, 11, 12}

(p ⇒ q) ∧ (¬r ∨ ¬s) will give {8, 9, 10, 12}

¬((p ⇒ q) ∧ (¬r ∨ ¬s)) will give 11.

Therefore option (C) will be the correct answer.

12) Let a_n be the number of n-bit strings that do NOT contain two consecutive 1s. Which one of the following is the recurrence relation for a_n?

a_n = a_n-1 + 2a_n-2
a_n = a_n-1 + a_n-2
a_n = 2a_n-1 + a_n-2
a_n = 2a_n-1 + 2a_n-2

Answer: B

Explanation:

Here, we assume that the least value of n for the recursion would be 3.

n	n-bit string without consecutive 1's	a_n
1	{0,1}	a₁ = 2
2	{00,01,10}	a₂ = 3
3	{000,001,010,100,101}	a₃ = 5

This seems to follow the recurrence relation: a_n = a_n-1 + a_n-2.

Therefore option (B) is the correct answer.

13) Gate 2016 CS set 1

Answer: B

Explanation:

Given,

Substitute h = x - 4
So it becomes lim (sinh) / h
h->0
Now solving this we get 1.

Therefore option (B) will be the correct answer.

14) A probability density function on the interval [a, 1] is given by 1/x₂ and outside this interval the value of the function is zero. The value of "a" is ________.

0.5
0
0.7
-1

Answer: A

Explanation:

We know that the sum of all the probabilities always equals to 1.
Therefore, on integrating 1/x₂ with limits [a to 1], the result always equals to 1.

Gate 2016 CS set 1 = 1

Hence,
Gate 2016 CS set 1 = 1

-1 + 1/a = 1
or, -a + 1 = a
2a = 1
Hence, a = 0.5

Therefore option (A) will be the correct answer.

15) Two eigenvalues of a 3 × 3 real matrix P are (2 + √-1) and 3. The determinant of P is ________.

Answer: D

Explanation:

We know, the Eigenvalues are roots of Characteristic equation |A-λI| = 0. And the determinant of a real matrix cannot be an imaginary. So, if one eigenvalue is complex roots, then this value always occur in pairs. Hence, the eigenvalues of the matrix will be:
λ1 = 2+i, λ₂ = 2-i and λ₃ = 3
And, det(A) = λ₁λ₂λ₃ = (2+i)*(2-i)*3 = (2₂-i₂)∗3 = 5∗3 = 15

Therefore option (D) will be the correct answer.

16) Consider the Boolean operator # with the following properties:

x # 0 = x, x # 1 = x, x # x = 0 and x # x = 1. Then x # y is equivalent to