1
GATE PI 2016
+1
-0.3
At $$x=0,$$ the function is $$f\left( x \right) = \left| {\sin {{2\pi x} \over L}} \right|\left( { - \infty < x < \infty ,L > 0} \right)$$
A
continuous and differentiable.
B
not continuous and not differentiable.
C
not continuous but differentiable.
D
continuous but not differentiable.
2
GATE PI 2016
+2
-0.6
The range of values of $$k$$ for which the function $$\,\,f\left( x \right) = \left( {{k^2} - 4} \right){x^2} + 6{x^3} + 8{x^4}$$ has a local maxima at point $$x=0$$ is
A
$$k < - 2\,\,\,or\,\,\,k > 2$$
B
$$k \le - 2\,\,\,or\,\,\,k \ge 2$$
C
$$- 2 < k < 2$$
D
$$- 2 \le k \le 2$$
3
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
$${{\partial f} \over {\partial y}} = {{\partial g} \over {\partial x}}$$
4
GATE PI 2016
Numerical
+2
-0
$$\mathop {\lim }\limits_{x \to 0} \left( {{{{e^{5x}} - 1} \over x}} \right)$$ is equal to ________.
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