GATE 2016 CS Set 2

9) In a 2 × 4 rectangle grid shown below, each cell is a rectangle. How many rectangles can be observed in the grid?

Answer: C

Explanation:

In a given rectangle, there are five vertical lines and three horizontal lines. To form a rectangle, we need to choose 2 horizontal lines and 2 vertical lines. Hence, The number of rectangles we can build is

= ⁵C₂ * ³C₂
= {(5*4)/2}*{(3*2)/2}
= 10*3
= 30
Therefore option (C) is the right answer.

10)

Choose the correct expression for f(x) given in the graph.

f(x) = 1 - |x - 1|
f(x) = 1 + |x - 1|
f(x) = 2 - |x - 1|
f(x) = 2 + |x - 1|

Answer: C

Explanation:

In the graph at x=0, then f(x) = 1. Now evaluate each option to find the correct expression. Thus,

f(x) = 1 - |x - 1| <=> f(o) = 1 - |0 - 1| = 1 - |- 1| = 1-1 = 0
f(x) = 1 + |x - 1| <=> f(o) = 1 + |0 - 1| = 1 + |- 1| = 1+1 = 2
f(x) = 2 - |x - 1| <=> f(o) = 2 - |0 - 1| = 2 - |- 1| = 2-1 = 1
f(x) = 2 + |x - 1| <=> f(o) = 2 + |0 - 1| = 2 + |- 1| = 2+1 = 3

Therefore option (C) will be the right answer.

11) Consider the following expressions:

(i) false
(ii) Q
(iii) true
(iv) P ∨ Q
(v) ¬Q ∨ P

The number of expressions given above that are logically implied by P ∧ (P ⇒ Q) is ______________.

Answer: B

Explanation:

Suppose P ∧ (P ⇒ Q) = X

Now, construct a truth table for the notation X:

P	Q	P ⇒ Q	P ∧ (P ⇒ Q)
F	F	T	F
F	T	T	F
T	F	F	F
T	T	T	T

Now we need to check for each option wheather it is tautology or not for X.

Thus option (ii), (iii), (iv), (v) are tautology. Hence option (B) is the right answer.

12) Let f(x) be a polynomial and g(x) = f'(x) be its derivative. If the degree of (f (x) + f(-x)) is 10, then the degree of (g(x) - g(-x)) is ______________.

Answer: C

Explanation:

Given,
f(x) be a polynomial and g(x) = f'(x)
As f(x) + f'(x) be a polynomial of degree 10. Then.
d/dx [f(x) + f'(x)] be a polynomial of degree 9.
or, f'(x) + f'(-x) (-1) be a polynomial of degree 9.
=> g(x) - g(-x) is a polynomial of degree 9. ( ∵ g(x) = f'(x))
Therefore option (C) is the right answer.

13) The minimum number of colours that is sufficient to vertex-colour any planar graph is ______________.

Answer: D

Explanation:

A planar graph is a graph on a plane where no two edges are crossing each other. According to Four color theorem, it says that any planar graphs can be colored with at most 4 colors. Hence option (D) is the right answer.

14) Consider the systems, each consisting of m linear equations in n variables.

I. If m < n, then all such systems have a solution
II. If m > n, then none of these systems has a solution
III. If m = n, then there exists a system which has a solution

Which one of the following is CORRECT?

I, II and III are true
Only II and III are true
Only III is true
None of them is true

Answer: C

Explanation:

I) This is false since a system with m < n has no solution because of inconsistency.

II) This is false. A system with m > n has no solution for none of the system. Suppose if this system of equations has two equations which are dependent on each other. i.e.,
a+b = 2, and 2a+2b = 4, then a-b=0
Thus a=1, b=1 is solutions. So, (II) Is false.

III) This is true, M=2, N=2
x+y=2, and x-y=0, Then m=1, n=1
Now, we can say that there exists a system which has a solution.
Therefore option (C) is the right answer.

15) Suppose that a shop has an equal number of LED bulbs of two different types. The probability of an LED bulb lasting more than 100 hours given that it is of Type 1 is 0.7, and given that it is of Type 2 is 0.4. The probability that an LED bulb chosen uniformly at random lasts more than 100 hours is ______________.

0.55
0.57
0.88
0.75

Answer: A

Explanation:

Suppose Type-1 bulbs = 10 and Type-2 bulbs = 10
According to the question, 7 Bulbs of Type-1 are going for 100 hours, and 4 Bulbs of Type-2 are going for 100 hours.
Then total bulbs going for 100 hours = 7+4 = 11
So, probability = 11/20 = 0.55
Therefore option (A) will be the right answer.

16) Suppose that the eigen values of matrix A are 1, 2, 4. The determinant of (A^-1)^T is ______________.

Answer: B

Explanation:

Determinant of matrix A = Product of eigen values = 1*2*4 = 8
We know Determinant of Inverse of Matrix is always inverse of determinant.
So, det(A^-1) = 1/det(A) = 1/8
Also the determinant of matrix is always remains same after the transpose. Hence,
determinant of (A^-1)^T = 1/8
Therefore option (B) is the right answer.