GATE PI
Engineering Mathematics
Calculus
Previous Years Questions

## Marks 1

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

## Marks 2

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...
$$\mathop {\lim }\limits_{x \to 0} \left( {{{{e^{5x}} - 1} \over x}} \right)$$ is equal to ________.
The curve $$\,\,y = {x^4}\,\,$$
The value of $$\mathop {\lim }\limits_{\left( {x,y} \right) \to \left( {0,0} \right)} {{{x^2} - xy} \over {\sqrt x - \sqrt y }}$$ is
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...
If $$f\left( x \right) = \sin \left| x \right|\,\,$$ then the value of $${{df} \over {dx}}\,\,$$ at $$\,\,x = {{ - \pi } \over 4}\,\,$$ is
The integral $$\,\,{1 \over {\sqrt {2\pi } }}\int\limits_{ - \infty }^\infty {{e^{{{ - {x^2}} \over 2}}}} dx\,\,$$ is equal to
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