1
GATE PI 2016
+1
-0.3
For the two functions
$$f\left( {x,y} \right) = {x^3} - 3x{y^2}\,\,$$ and $$\,\,g\left( {x,y} \right) = 3{x^2}y - {y^3}\,\,$$

Which one of the following options is correct?

A
$${{\partial f} \over {\partial x}} = {{\partial g} \over {\partial x}}$$
B
$${{\partial f} \over {\partial x}} = - {{\partial g} \over {\partial y}}$$
C
$${{\partial f} \over {\partial y}} = - {{\partial g} \over {\partial x}}$$
D
$${{\partial f} \over {\partial y}} = {{\partial g} \over {\partial x}}$$
2
GATE PI 2016
Numerical
+2
-0
$$\mathop {\lim }\limits_{x \to 0} \left( {{{{e^{5x}} - 1} \over x}} \right)$$ is equal to ________.
3
GATE PI 2016
+2
-0.6
A normal random variable $$X$$ has the following probability density function $${f_x}\left( x \right) = {1 \over {\sqrt {8\pi } }}e{}^{ - \left\{ {{{{{\left( {x - 1} \right)}^2}} \over 8}} \right\}}, - \infty < x < \infty$$\$
Then $$\,\int\limits_1^\infty {{f_x}\left( x \right)dx = }$$
A
$$0$$
B
$${{1 \over 2}}$$
C
$${1 - {1 \over e}}$$
D
$$1$$
4
GATE PI 2016
+1
-0.3
A fair coin is tossed $$N$$ times. The probability that head does not turn up in any of the tosses is
A
$${\left( {{1 \over 2}} \right)^{N - 1}}$$
B
$$1 - {\left( {{1 \over 2}} \right)^{N - 1}}$$
C
$${\left( {{1 \over 2}} \right)^N}$$
D
$$1 - {\left( {{1 \over 2}} \right)^N}$$
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