Calculus · Engineering Mathematics · GATE ME

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Marks 1

GATE ME 2024
Let $f(.)$ be a twice differentiable function from $ \mathbb{R}^{2} \rightarrow \mathbb{R}$. If $P, \mathbf{x}_{0} \in \mathbb{R}^{2}$ where $\vert \v...
GATE ME 2020 Set 1
Define [x] as the greatest integer less than or equal to x, for each x ϵ (-∞, ∞). If y = [x], then area under y for x ϵ [1,4] is
GATE ME 2017 Set 1
The value of $$\mathop {\lim }\limits_{x \to 0} \left( {{{{x^3} - \sin \left( x \right)} \over x}} \right)$$ is
GATE ME 2016 Set 2
The values of $$x$$ for which the function $$f\left( x \right) = {{{x^2} - 3x - 4} \over {{x^2} + 3x - 4}}$$ is NOT continuous are
GATE ME 2016 Set 3
$$\mathop {Lt}\limits_{x \to 0} {{{{\log }_e}\left( {1 + 4x} \right)} \over {{e^{3x}} - 1}}$$ is equal to
GATE ME 2015 Set 3
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 2
At $$x=0,$$ the function $$f\left( x \right) = \left| x \right|$$ has
GATE ME 2015 Set 1
The value of $$\mathop {Lim}\limits_{x \to 0} \,{{1 - \cos \left( {{x^2}} \right)} \over {2{x^4}}}$$ is
GATE ME 2014 Set 2
$$\mathop {Lt}\limits_{x \to 0} \left( {{{{e^{2x}} - 1} \over {\sin \left( {4x} \right)}}} \right)\,\,$$ is equal to
GATE ME 2014 Set 1
$$\mathop {Lt}\limits_{x \to 0} {{x - \sin x} \over {1 - \cos x}}$$ is
GATE ME 2014 Set 4
The value of the integral $$\int\limits_0^2 {{{{{\left( {x - 1} \right)}^2}\sin \left( {x - 1} \right)} \over {{{\left( {x - 1} \right)}^2} + \cos \le...
GATE ME 2014 Set 3
If a function is continuous at a point,
GATE ME 2013
The value of the definite integral $$\int_1^e {\sqrt x \ln \left( x \right)dx} $$ is
GATE ME 2012
The area enclosed between the straight line $$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.\,\,\,$$ At the point $$x=0, f(x)$$ is
GATE ME 2012
$$\,\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$$ has
GATE ME 2011
A series expansion for the function $$\sin \theta $$ is _______.
GATE ME 2011
What is $$\mathop {Lim}\limits_{\theta \to 0} {{\sin \theta } \over \theta }\,\,$$ equal to ?
GATE ME 2011
If $$f(x)$$ is even function and a is a positive real number , then $$\int\limits_{ - a}^a {f\left( x \right)dx\,\,} $$ equals ________.
GATE ME 2010
The parabolic are $$y = \sqrt x ,1 \le x \le 2$$ is revolved around the $$x$$-axis. The volume of the solid of revolution is
GATE ME 2010
The function $$y = \left| {2 - 3x} \right|$$
GATE ME 2010
The value of the integral $$\int\limits_{ - a}^a {{{dx} \over {1 + {x^2}}}} $$
GATE ME 2009
The area enclosed between the curves $${y^2} = 4x\,\,$$ and $${{x^2} = 4y}$$ is
GATE ME 2009
The distance between the origin and the point nearest to it on the surface $$\,\,{z^2} = 1 + xy\,\,$$ is
GATE ME 2008
In the Taylor series expansion of $${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}}\,\,$$ is
GATE ME 2007
The minimum value of function $$\,\,y = {x^2}\,\,$$ in the interval $$\,\,\left[ {1,5} \right]\,\,$$ is
GATE ME 2005
$$\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{$\scriptstyle x$} \kern-0.1em/\k...
GATE ME 2004
If $$\,\,\,x = a\left( {\theta + Sin\theta } \right)$$ and $$y = a\left( {1 - Cos\theta } \right)$$ then $$\,\,{{dy} \over {dx}} = \,\_\_\_\_\_.$$
GATE ME 1999
Value of the function $$\mathop {Lim}\limits_{x \to a} \,{\left( {x - a} \right)^{x - a}}$$ is _______.
GATE ME 1997
Area bounded by the curve $$y = {x^2}$$ and the lines $$x=4$$ and $$y=0$$ is given by
GATE ME 1996
If a function is continuous at a point its first derivative
GATE ME 1995
The area bounded by the parabola $$2y = {x^2}$$ and the lines $$x=y-4$$ is equal to _________.
GATE ME 1994
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 $$x{{\partial H} \over {\partial x}} + y{{\partial H} \over {\partial y}} = nH$$
GATE ME 1993
$$\mathop {Lim}\limits_{x \to 0} {{x\left( {{e^x} - 1} \right) + 2\left( {\cos x - 1} \right)} \over {x\left( {1 - \cos x} \right)}} = \_\_\_\_\_\_.$$...
GATE ME 1993
The function $$f\left( {x,y} \right) = {x^2}y - 3xy + 2y + x$$ has

Marks 2

GATE ME 2024
If the value of the double integral$\int_{x=3}^{4} \int_{y=1}^{2} \frac{dydx}{(x + y)^2}$is $\log_e(\frac{a}{24})$, then $a$ is __________ (answer in ...
GATE ME 2017 Set 1
A parametric curve defined by $$x = \cos \left( {{{\pi u} \over 2}} \right),y = \sin \left( {{{\pi u} \over 2}} \right)\,\,$$ in the range $$0 \le u \...
GATE ME 2016 Set 1
Consider the function $$f\left( x \right) = 2{x^3} - 3{x^2}\,\,$$ in the domain $$\,\left[ { - 1,2} \right].$$ The global minimum of $$f(x)$$ is _____...
GATE ME 2016 Set 3
$$\mathop {Lt}\limits_{x \to \infty } \left( {\sqrt {{x^2} + x - 1} - x} \right)$$ is
GATE ME 2015 Set 1
Consider a spatial curve in three -dimensional space given in parametric form by $$\,\,x\left( t \right)\,\, = \,\,\cos t,\,\,\,y\left( t \right)\,\, ...
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 meters. The ant starts at the poin...
GATE ME 2014 Set 4
The value of the integral $$\,\int\limits_0^2 {\int\limits_0^x {{e^{x + y}}\,\,dy} } $$ $$dx$$ is
GATE ME 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 ME 2010
The infinite series $${\,f\left( x \right) = x - {{{x^3}} \over {3!}} + {{{x^5}} \over {5!}} - {{{x^7}} \over {7!}} + - - - \,\,}$$ Converges to
GATE ME 2008
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 2007
If $$\,\,\,y = x + \sqrt {x + \sqrt {x + \sqrt {x + .....\alpha } } } \,\,\,$$ then $$y(2)=$$ __________.
GATE ME 2007
$$\mathop {Lim}\limits_{x \to 0} {{{e^x} - \left( {1 + x + {{{x^2}} \over 2}} \right)} \over {{x^3}}} = $$
GATE ME 2005
By a change of variables $$x(u, v) = uv,$$ $$\,\,y\left( {u,v} \right) = {v \over u}$$ in a double integral, the integral $$f(x, y)$$ changes to $$\,...
GATE ME 2004
The volume of an object expressed in spherical co-ordinates is given by $$V = \int\limits_0^{2\pi } {\int\limits_0^{{\raise0.5ex\hbox{$\scriptstyle \...
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