1
GATE CSE 2011
+1
-0.3
If the difference between the expectation of the square of a random variable $$\left( {E\left[ {{X^2}} \right]} \right)$$ and the square of the expectation of the random variable $${\left( {E\left[ X \right]} \right)^2}$$ is denoted by R then
A
R = 0
B
R < 0
C
$$R\, \ge \,0$$
D
R > 0
2
GATE CSE 2011
+2
-0.6
A deck of 5 cards (each carrying a distinct number from 1 to 5) is shuffled thoroughly. Two cards are then removed one at a time from the deck. What is probability that the two cards are selected with the number on the first card is one higher than the number on the second card.
A
$${{1 \over 5}}$$
B
$${{4 \over 25}}$$
C
$${{1 \over 4}}$$
D
$${{2 \over 5}}$$
3
GATE CSE 2011
+2
-0.6
Which one of the following options is correct given three positive integers $$x, y$$ and $$z$$, and a predicate
$$P\left( x \right) = \neg \left( {x = 1} \right) \wedge \forall y\left( {\exists z\left( {x = y * z} \right) \Rightarrow \left( {y = x} \right) \vee \left( {y = 1} \right)} \right)$$
A
$$P(x)$$ being true means that $$x$$ is a prime number
B
$$P(x)$$ being true means that $$x$$ is a number other than 1
C
$$P(x)$$ is always true irrespective of the value of $$x$$
D
$$P(x)$$ being true means that $$x$$ has exactly two factors other than 1 and $$x$$
4
GATE CSE 2011
+2
-0.6
Given $$i = \sqrt { - 1} ,$$ what will be the evaluation of the definite integral $$\int\limits_0^{\pi /2} {{{\cos x +i \sin x} \over {\cos x - i\,\sin x}}dx?}$$
A
$$0$$
B
$$2$$
C
$$-1$$
D
$$i$$
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