1
GATE CSE 2014 Set 3
Numerical
+1
-0
If $$\int_0^{2\pi } {\left| {x\sin x} \right|dx = k\pi ,}$$ then the values of $$k$$ is equal to _________ .
2
GATE CSE 2014 Set 3
+2
-0.6
The value of the integral given below is $$\int_0^\pi {{x^2}\,\cos \,x\,dx}$$\$
A
$$- 2\pi$$
B
$$\pi$$
C
$$-\pi$$
D
$$2\pi$$
3
GATE CSE 2014 Set 3
Numerical
+2
-0
Suppose you want to move from 0 to 100 on the number line. In each step, you either move right by a unit distance or you take a shortcut. A shortcut is simply a pre specified pair of integers i, j with i < j. Given a shortcut i, j if you are at position i on the number line, you may directly move to j. suppose T(k) denotes the smallest number of steps needed to move from k to 100. Suppose further that there is at most 1 shortcut involving any number, and in particular from 9 there is a shortcut to 15. Let y and z be such that T(9) = 1+ min(T(y),T(z)). Then the value of the product yz is _______.
4
GATE CSE 2014 Set 3
+2
-0.6
Consider the set of all functions $$f:\left\{ {0,\,1,.....,2014} \right\} \to \left\{ {0,\,1,.....,2014} \right\}$$ such that $$f\left( {f\left( i \right)} \right) = i,\,\,\,$$ for all $$0 \le i \le 2014.$$ Consider the following statements:
$$P$$. For each such function it must be the case that for every $$i$$, $$f\left( i \right) = i$$
$$Q$$. For each such function it must be the case that for some $$i$$, $$f\left( i \right) = 1$$
$$R$$. Each such function must be onto.

Which one of the following id CORRECT?

A
$$P, Q$$ and $$R$$ are true
B
Only $$Q$$ and $$R$$ are true
C
Only $$P$$ and $$Q$$ are true
D
Only $$R$$ is true
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