1
GATE EE 2016 Set 2
+2
-0.6
Let the probability density function of a random variable $$X,$$ be given as: $${f_x}\left( x \right) = {3 \over 2}{e^{ - 3x}}u\left( x \right) + a{e^{4x}}u\left( { - x} \right)$$\$
where $$u(x)$$ is the unit step function. Then the value of $$'a'$$ and Prob $$\left\{ {X \le 0} \right\},$$ respectively, are
A
$$2,{1 \over 2}$$
B
$$4,{1 \over 2}$$
C
$$2,{1 \over 4}$$
D
$$4,{1 \over 4}$$
2
GATE EE 2016 Set 2
Numerical
+2
-0
Let $$y(x)$$ be the solution of the differential equation $$\,\,{{{d^2}y} \over {d{x^2}}} - 4{{dy} \over {dx}} + 4y = 0\,\,$$ with initial conditions $$y(0)=0$$ and $$\,\,{\left. {{{dy} \over {dx}}} \right|_{x = 0}} = 1.\,\,$$ Then the value of $$y(1)$$ is __________.
3
GATE EE 2016 Set 2
+1
-0.3
Consider the function $$f\left( z \right) = z + {z^ * }$$ where $$z$$ is a complex variable and $${z^ * }$$ denotes its complex conjugate. Which one of the following is TRUE?
A
$$f(z)$$ is both continuous and analytic
B
$$f(z)$$ is continuous but not analytic
C
$$f(z)$$ is not continuous but is analytic
D
$$f(z)$$ is neither continuous nor analytic
4
GATE EE 2016 Set 2
+1
-0.3
The solution of the differential equation, for
$$t > 0,\,\,y''\left( t \right) + 2y'\left( t \right) + y\left( t \right) = 0$$ with initial conditions $$y\left( 0 \right) = 0$$ and $$y'\left( 0 \right) = 1,$$ is $$\left[ {u\left( t \right)} \right.$$ denotes the unit step function$$\left. \, \right]$$,
A
$$t{e^{ - t}}\,u\left( t \right)$$
B
$$\left( {{e^{ - t}} - t{e^{ - t}}} \right)u\left( t \right)$$
C
$$\left( { - {e^{ - t}} + t{e^{ - t}}} \right)u\left( t \right)$$
D
$${e^{ - t}}u\left( t \right)$$
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