GATE ME

Calculus

Engineering Mathematics

(Past Years Questions)

Marks 1

More
The value of $$\mathop {\lim }\limits_{x \to 0} \left( {{{{x^3} - \sin \left( x \right)} \over x}} \right)$$ is
GATE ME 2017 Set 1
$$\mathop {Lt}\limits_{x \to 0} {{{{\log }_e}\left( {1 + 4x} \right)} \over {{e^{3x}} - 1}}$$ is equal to
GATE ME 2016 Set 3
The values of $$x$$ for which the function $$f\left( x \right) = {{{x^2} - 3x - 4} \over {{x^2} + 3x - 4}}$$ is NOT con...
GATE ME 2016 Set 2
The value of $$\mathop {Lim}\limits_{x \to 0} \left( {{{ - \sin x} \over {2\sin x + x\cos x}}} \right)\,\,\,$$ is ______...
GATE ME 2015 Set 3
At $$x=0,$$ the function $$f\left( x \right) = \left| x \right|$$ has
GATE ME 2015 Set 2
The value of $$\mathop {Lim}\limits_{x \to 0} \,{{1 - \cos \left( {{x^2}} \right)} \over {2{x^4}}}$$ is
GATE ME 2015 Set 1
The value of the integral $$\int\limits_0^2 {{{{{\left( {x - 1} \right)}^2}\sin \left( {x - 1} \right)} \over {{{\left( ...
GATE ME 2014 Set 4
If a function is continuous at a point,
GATE ME 2014 Set 3
$$\mathop {Lt}\limits_{x \to 0} \left( {{{{e^{2x}} - 1} \over {\sin \left( {4x} \right)}}} \right)\,\,$$ is equal to
GATE ME 2014 Set 2
$$\mathop {Lt}\limits_{x \to 0} {{x - \sin x} \over {1 - \cos x}}$$ is
GATE ME 2014 Set 1
The value of the definite integral $$\int_1^e {\sqrt x \ln \left( x \right)dx} $$ is
GATE ME 2013
The area enclosed between the straight line $$y=x$$ and the parabola $$y = {x^2}$$ in the $$x-y$$ plane is
GATE ME 2012
Consider the function $$f\left( x \right) = \left| x \right|$$ in the interval $$\,\, - 1 \le x \le 1.\,\,\,$$ At the po...
GATE ME 2012
$$\,\mathop {Lim}\limits_{x \to 0} \left( {{{1 - \cos x} \over {{x^2}}}} \right)$$ is
GATE ME 2012
At $$x=0,$$ the function $$f\left( x \right) = {x^3} + 1$$ has
GATE ME 2012
A series expansion for the function $$\sin \theta $$ is _______.
GATE ME 2011
If $$f(x)$$ is even function and a is a positive real number , then $$\int\limits_{ - a}^a {f\left( x \right)dx\,\,} $...
GATE ME 2011
What is $$\mathop {Lim}\limits_{\theta \to 0} {{\sin \theta } \over \theta }\,\,$$ equal to ?
GATE ME 2011
The parabolic are $$y = \sqrt x ,1 \le x \le 2$$ is revolved around the $$x$$-axis. The volume of the solid of revolutio...
GATE ME 2010
The value of the integral $$\int\limits_{ - a}^a {{{dx} \over {1 + {x^2}}}} $$
GATE ME 2010
The function $$y = \left| {2 - 3x} \right|$$
GATE ME 2010
The distance between the origin and the point nearest to it on the surface $$\,\,{z^2} = 1 + xy\,\,$$ is
GATE ME 2009
The area enclosed between the curves $${y^2} = 4x\,\,$$ and $${{x^2} = 4y}$$ is
GATE ME 2009
In the Taylor series expansion of $${e^x}$$ about $$x=2,$$ the coefficient of $$\,\,{\left( {x - 2} \right)^4}\,\,$$ is
GATE ME 2008
The value of $$\,\,\mathop {Lim}\limits_{x \to 8} {{{x^{1/3}} - 2} \over {x - 8}}\,\,$$ is
GATE ME 2008
The minimum value of function $$\,\,y = {x^2}\,\,$$ in the interval $$\,\,\left[ {1,5} \right]\,\,$$ is
GATE ME 2007
$$\int\limits_{ - a}^a {\left[ {{{\sin }^6}\,x + {{\sin }^7}\,x} \right]dx} $$ is equal to
GATE ME 2005
Changing the order of integration in the double integral $${\rm I} = \int\limits_0^8 {\int\limits_{{\raise0.5ex\hbox{$\...
GATE ME 2005
If $$\,\,\,x = a\left( {\theta + Sin\theta } \right)$$ and $$y = a\left( {1 - Cos\theta } \right)$$ then $$\,\,{{dy} \o...
GATE ME 2004
Value of the function $$\mathop {Lim}\limits_{x \to a} \,{\left( {x - a} \right)^{x - a}}$$ is _______.
GATE ME 1999
Area bounded by the curve $$y = {x^2}$$ and the lines $$x=4$$ and $$y=0$$ is given by
GATE ME 1997
If a function is continuous at a point its first derivative
GATE ME 1996
The area bounded by the parabola $$2y = {x^2}$$ and the lines $$x=y-4$$ is equal to _________.
GATE ME 1995
The value of $$\int\limits_0^\infty {{e^{ - {y^3}}}.{y^{1/2}}} $$ dy is _________.
GATE ME 1994
If $$H(x, y)$$ is homogeneous function of degree $$n$$ then $$x{{\partial H} \over {\partial x}} + y{{\partial H} \over ...
GATE ME 1994
The function $$f\left( {x,y} \right) = {x^2}y - 3xy + 2y + x$$ has
GATE ME 1993
$$\mathop {Lim}\limits_{x \to 0} {{x\left( {{e^x} - 1} \right) + 2\left( {\cos x - 1} \right)} \over {x\left( {1 - \cos ...
GATE ME 1993

Marks 2

More
A parametric curve defined by $$x = \cos \left( {{{\pi u} \over 2}} \right),y = \sin \left( {{{\pi u} \over 2}} \right)\...
GATE ME 2017 Set 1
$$\mathop {Lt}\limits_{x \to \infty } \left( {\sqrt {{x^2} + x - 1} - x} \right)$$ is
GATE ME 2016 Set 3
Consider the function $$f\left( x \right) = 2{x^3} - 3{x^2}\,\,$$ in the domain $$\,\left[ { - 1,2} \right].$$ The globa...
GATE ME 2016 Set 1
Consider a spatial curve in three -dimensional space given in parametric form by $$\,\,x\left( t \right)\,\, = \,\,\cos ...
GATE ME 2015 Set 1
Consider an ant crawling along the curve $$\,{\left( {x - 2} \right)^2} + {y^2} = 4,$$ where $$x$$ and $$y$$ are in mete...
GATE ME 2015 Set 1
The value of the integral $$\,\int\limits_0^2 {\int\limits_0^x {{e^{x + y}}\,\,dy} } $$ $$dx$$ is
GATE ME 2014 Set 4
A political party orders an arch for the entrance to the ground in which the annual convention is being held. The profil...
GATE ME 2012
The infinite series $${\,f\left( x \right) = x - {{{x^3}} \over {3!}} + {{{x^5}} \over {5!}} - {{{x^7}} \over {7!}} + -...
GATE ME 2010
Consider the shaded triangular region $$P$$ shown in the figure. What is $$\int\!\!\!\int\limits_p {xy\,dx\,dy\,?} $$ ...
GATE ME 2008
Which of the following integrals is unbounded?
GATE ME 2008
The length of the curve $$\,y = {2 \over 3}{x^{3/2}}$$ between $$x=0$$ & $$x=1$$ is
GATE ME 2008
Let $$\,\,f = {y^x}.$$ What is $$\,\,{{{\partial ^2}f} \over {\partial x\partial y}}\,\,$$ at $$x=2,$$ $$y=1$$?
GATE ME 2008
$$\mathop {Lim}\limits_{x \to 0} {{{e^x} - \left( {1 + x + {{{x^2}} \over 2}} \right)} \over {{x^3}}} = $$
GATE ME 2007
If $$\,\,\,y = x + \sqrt {x + \sqrt {x + \sqrt {x + .....\alpha } } } \,\,\,$$ then $$y(2)=$$ __________.
GATE ME 2007
By a change of variables $$x(u, v) = uv,$$ $$\,\,y\left( {u,v} \right) = {v \over u}$$ in a double integral, the integr...
GATE ME 2005
The volume of an object expressed in spherical co-ordinates is given by $$V = \int\limits_0^{2\pi } {\int\limits_0^{{\r...
GATE ME 2004

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