GATE CE 2005 | GATE CE Year Wise Previous Years Questions

GATE CE 2005

MCQ (Single Correct Answer)

-0.6

Consider likely applicability of Cauchy's Integral theorem to evaluate the following integral counterclockwise around the unit circle C.

$$I\, = \,\oint\limits_C {\sec z\,dz} $$, z being a complex variable. The value of I will be

I = 0 ; Singularities set = $$\phi $$

I = 0 ; Singularities set = $$\left\{ { \pm {{\left( {2n + 1} \right)} \over 2}\pi \,\,;\,n = 0,1,2,.....} \right\}$$

I = 0 ; Singularities set = $$\left\{ { \pm \,n\pi \,\,;\,n = 0,1,2,.....} \right\}$$

None of the above

GATE CE 2005

MCQ (Single Correct Answer)

-0.3

Which one of the following is not true for the complex number z₁ and z₂ ?

$${{{z_1}} \over {{z_2}}} = {{{z_1}\overline {{z_2}} } \over {{{\left| {{z_2}} \right|}^2}}}$$

$$\left| {{z_1}\, + \,\,{z_2}} \right| \le \,\left| {{z_1}} \right|\, + \,\left| {{z_2}} \right|$$

$$\left| {{z_1}\, + \,\,{z_2}} \right| \le \,\left| {\left| {{z_1}} \right|\, - \,\left| {{z_2}} \right|} \right|$$

$${\left| {{z_1}\, + \,\,{z_2}} \right|^2}\, + \,{\left| {{z_1}\, - \,\,{z_2}} \right|^2} = \,\,2{\left| {{z_1}} \right|^2}\, + \,2{\left| {{z_2}} \right|^2}$$

GATE CE 2005

MCQ (Single Correct Answer)

-0.6

Laplace transform of $$f\left( t \right) = \cos \left( {pt + q} \right)$$ is

$${{s\,\cos \,q - p\,\sin \,q} \over {{s^2} + {p^2}}}$$

$${{s\,\sin \,q - p\,\cos \,q} \over {{s^2} + {p^2}}}$$

$${{s\,\sin \,q + p\,\cos \,q} \over {{s^2} + {p^2}}}$$

GATE CE 2005

MCQ (Single Correct Answer)

-0.3

The Laplace transform of a function $$f(t)$$ is $$$F\left( s \right) = {{5{s^2} + 23s + 6} \over {s\left( {{s^2} + 2s + 2} \right)}}$$$
As $$t \to \propto ,\,\,f\left( t \right)$$ approaches

$$3$$

$$5$$

$$17/2$$

$$ \propto $$

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