1
GATE CSE 2015 Set 3
+1
-0.3
The value of $$\mathop {\lim }\limits_{x \to \alpha } {\left( {1 + {x^2}} \right)^{{e^{ - x}}}}\,\,$$ is
A
$$0$$
B
$${{1 \over 2}}$$
C
$$1$$
D
$$\infty$$
2
GATE CSE 2015 Set 3
+1
-0.3
Choose the most appropriate equation for the function drawn as a thick line, in the plot below.
A
$$x = y - \left| y \right|$$
B
$$x = - \left( {y - \left| y \right|} \right)$$
C
$$x = y + \left| y \right|$$
D
$$x = - \left( {y + \left| y \right|} \right)$$
3
GATE CSE 2014 Set 1
+1
-0.3
Let the function
$$f\left( \theta \right) = \left| {\matrix{ {\sin \,\theta } & {\cos \,\theta } & {\tan \,\theta } \cr {\sin \left( {{\pi \over 6}} \right)} & {\cos \left( {{\pi \over 6}} \right)} & {\tan \left( {{\pi \over 6}} \right)} \cr {\sin \left( {{\pi \over 3}} \right)} & {\cos \left( {{\pi \over 3}} \right)} & {\tan \left( {{\pi \over 3}} \right)} \cr } } \right|$$

Where $$\theta \in \left[ {{\pi \over 6},{\pi \over 3}} \right]$$ and $$f\left( \theta \right)$$ denote the derivative of $$f$$ with repect to $$\theta$$. Which of the following statements is/are TRUE?

$${\rm I})$$ There exists $$\theta \in \left[ {{\pi \over 6},{\pi \over 3}} \right]$$ such that $$f\left( \theta \right)$$ $$= 0$$.
$${\rm I}{\rm I})$$ There exists $$\theta \in \left[ {{\pi \over 6},{\pi \over 3}} \right]$$ such that $$f\left( \theta \right)$$ $$\ne 0$$.

A
$${\rm I}$$ only
B
$${\rm I}$$$${\rm I}$$ only
C
Both $${\rm I}$$ and $${\rm I}$$$${\rm I}$$
D
neither $${\rm I}$$ nor $${\rm I}$$$${\rm I}$$
4
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 _________ .