1
GATE CE 2002
+2
-0.6
The value of the following improper integral is $$\,\int\limits_0^1 {x\,\log \,x\,dx} = \_\_\_\_\_.$$
A
$${1 \over 4}$$
B
$$0$$
C
$${{ - 1} \over 4}$$
D
$$1$$
2
GATE CE 2002
+1
-0.3
The value of the following definite integral in $$\int\limits_{{\raise0.5ex\hbox{\scriptstyle { - \pi }} \kern-0.1em/\kern-0.15em \lower0.25ex\hbox{\scriptstyle 2}}}^{{\raise0.5ex\hbox{\scriptstyle \pi } \kern-0.1em/\kern-0.15em \lower0.25ex\hbox{\scriptstyle 2}}} {{{Sin2x} \over {1 + \cos x}}dx = \_\_\_\_\_\_\_\_.}$$
A
$$-2log$$ $$2$$
B
$$2$$
C
$$0$$
D
None
3
GATE CE 2002
+2
-0.6
The directional derivative of the following function at $$(1, 2)$$ in the direction of $$(4i+3j)$$ is : $$f\left( {x,y} \right) = {x^2} + {y^2}$$
A
$$4/5$$
B
$$4$$
C
$$2/5$$
D
$$1$$
4
GATE CE 2002
+2
-0.6
The Laplace transform of the following function is $$f\left( t \right) = \left\{ {\matrix{ {\sin t} & {for\,\,0 \le t \le \pi } \cr 0 & {for\,\,t > \pi } \cr } } \right.$$\$
A
$$1/\left( {1 + {s^2}} \right)\,$$ for all $$\,s > 0$$
B
$$1/\left( {1 + {s^2}} \right)\,$$ for all $$\,s < \pi$$
C
$$\left( {1 + {e^{ - \pi s}}} \right)/\left( {1 + {s^2}} \right)$$ for all $$s>0$$
D
$${e^{ - \pi s}}/\left( {1 + {s^2}} \right)$$ for all $$s > 0$$
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