1
GATE CSE 2018
Numerical
+1
-0
Let $$G$$ be a finite group on $$84$$ elements. The size of a largest possible proper subgroup of $$G$$ is ________.
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2
GATE CSE 2018
Numerical
+1
-0
The chromatic number of the following graph is _______. GATE CSE 2018 Discrete Mathematics - Graph Theory Question 23 English
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3
GATE CSE 2018
MCQ (Single Correct Answer)
+2
-0.6
Consider a matrix P whose only eigenvectors are the multiples of $$\left[ {\matrix{ 1 \cr 4 \cr } } \right].$$

Consider the following statements.

$$\left( {\rm I} \right)$$ $$\,\,\,\,\,\,\,\,\,\,\,\,\,\,$$ $$P$$ does not have an inverse
$$\left( {\rm II} \right)$$ $$\,\,\,\,\,\,\,\,\,\,\,$$ $$P$$ has a repeated eigenvalue
$$\left( {\rm III} \right)$$ $$\,\,\,\,\,\,\,\,\,$$ $$P$$ cannot be diagonalized

Which one of the following options is correct?

A
Only $${\rm I}$$ and $${\rm III}$$ are necessarily true
B
Only $${\rm II}$$ is necessarily true
C
Only $${\rm I}$$ and $${\rm II}$$ are necessarily true
D
Only $${\rm II}$$ and $${\rm III}$$ are necessarily true
4
GATE CSE 2018
MCQ (Single Correct Answer)
+2
-0.6
Let N be the set of natural numbers. Consider the following sets.

$$\,\,\,\,\,\,\,\,$$ $$P:$$ Set of Rational numbers (positive and negative)
$$\,\,\,\,\,\,\,\,$$ $$Q:$$ Set of functions from $$\left\{ {0,1} \right\}$$ to $$N$$
$$\,\,\,\,\,\,\,\,$$ $$R:$$ Set of functions from $$N$$ to $$\left\{ {0,1} \right\}$$
$$\,\,\,\,\,\,\,\,$$ $$S:$$ Set of finite subsets of $$N.$$

Which of the sets above are countable?

A
$$Q$$ and $$S$$ only
B
$$P$$ and $$S$$ only
C
$$P$$ and $$R$$ only
D
$$P, Q$$ and $$S$$ only
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