# Calculus · Engineering Mathematics · GATE PI

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## Marks 1

GATE PI 2016
At $$x=0,$$ the function is $$f\left( x \right) = \left| {\sin {{2\pi x} \over L}} \right|\left( { - \infty < x < \infty ,L > 0} \right)$$
GATE PI 2016
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 follo...
GATE PI 2015
The function $$f\left( x \right) = {x^2} = x + x + x + ....x$$ times, is defined
GATE PI 2012
The area enclosed between the straight line $$y=x$$ and the parabola $$y = {x^2}$$ in the $$x-y$$ plane is
GATE PI 2012
Consider the function $$f\left( x \right) = \left| x \right|$$ in the interval $$\,\, - 1 \le x \le 1.\,\,\,$$ At the point $$x=0, f(x)$$ is
GATE PI 2012
$$\,\mathop {Lim}\limits_{x \to 0} \left( {{{1 - \cos x} \over {{x^2}}}} \right)$$ is
GATE PI 2012
At $$x=0,$$ the function $$f\left( x \right) = {x^3} + 1$$ has
GATE PI 2009
The total derivative of the function $$'xy'$$ is
GATE PI 2008
The value of the integral $$\,\,\int\limits_{ - \pi /2}^{\pi /2} {\left( {x\,\cos \,x} \right)dx\,\,}$$ is
GATE PI 2008
The value of the expansion $$\mathop {Lim}\limits_{x \to 0} \left[ {{{\sin \left( x \right)} \over {{e^x}X}}} \right]\,\,$$ is
GATE PI 2007
For the function $$\,\,f\left( {x,y} \right) = {x^2} - {y^2}\,\,$$ defined on $${R^2},$$ the point $$(0,0)$$ is
GATE PI 2007
What is the value of $$\mathop {Lim}\limits_{x \to \pi /4} {{\cos x - \sin x} \over {x - \pi /4}}\,\,$$
GATE PI 1995
Given $$y = \int\limits_1^{{x^2}} {\cos t\,\,\,dt,}$$ $${\,\,\,}$$ then $${{dy} \over {dx}} = \_\_\_\_\_.$$

## Marks 2

GATE PI 2016
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 a...
GATE PI 2016
$$\mathop {\lim }\limits_{x \to 0} \left( {{{{e^{5x}} - 1} \over x}} \right)$$ is equal to ________.
GATE PI 2015
The value of $$\mathop {\lim }\limits_{\left( {x,y} \right) \to \left( {0,0} \right)} {{{x^2} - xy} \over {\sqrt x - \sqrt y }}$$ is
GATE PI 2015
The curve $$\,\,y = {x^4}\,\,$$
GATE PI 2012
A political party orders an arch for the entrance to the ground in which the annual convention is being held. The profile of the arch follows the equa...
GATE PI 2010
If $$f\left( x \right) = \sin \left| x \right|\,\,$$ then the value of $${{df} \over {dx}}\,\,$$ at $$\,\,x = {{ - \pi } \over 4}\,\,$$ is
GATE PI 2010
The integral $$\,\,{1 \over {\sqrt {2\pi } }}\int\limits_{ - \infty }^\infty {{e^{{{ - {x^2}} \over 2}}}} dx\,\,$$ is equal to
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