GATE ME
Engineering Mathematics
Calculus
Previous Years Questions

## Marks 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
The value of $$\mathop {\lim }\limits_{x \to 0} \left( {{{{x^3} - \sin \left( x \right)} \over x}} \right)$$ is
$$\mathop {Lt}\limits_{x \to 0} {{{{\log }_e}\left( {1 + 4x} \right)} \over {{e^{3x}} - 1}}$$ is equal to
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
The value of $$\mathop {Lim}\limits_{x \to 0} \left( {{{ - \sin x} \over {2\sin x + x\cos x}}} \right)\,\,\,$$ is __________.
At $$x=0,$$ the function $$f\left( x \right) = \left| x \right|$$ has
The value of $$\mathop {Lim}\limits_{x \to 0} \,{{1 - \cos \left( {{x^2}} \right)} \over {2{x^4}}}$$ is
The value of the integral $$\int\limits_0^2 {{{{{\left( {x - 1} \right)}^2}\sin \left( {x - 1} \right)} \over {{{\left( {x - 1} \right)}^2} + \cos \le...$$\mathop {Lt}\limits_{x \to 0} \left( {{{{e^{2x}} - 1} \over {\sin \left( {4x} \right)}}} \right)\,\,$$is equal to If a function is continuous at a point,$$\mathop {Lt}\limits_{x \to 0} {{x - \sin x} \over {1 - \cos x}}$$is The value of the definite integral$$\int_1^e {\sqrt x \ln \left( x \right)dx} $$is The area enclosed between the straight line$$y=x$$and the parabola$$y = {x^2}$$in the$$x-y$$plane is Consider the function$$f\left( x \right) = \left| x \right|$$in the interval$$\,\, - 1 \le x \le 1.\,\,\,$$At the point$$x=0, f(x)$$is$$\,\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$$has A series expansion for the function$$\sin \theta $$is _______. What is$$\mathop {Lim}\limits_{\theta \to 0} {{\sin \theta } \over \theta }\,\,$$equal to ? If$$f(x)$$is even function and a is a positive real number , then$$\int\limits_{ - a}^a {f\left( x \right)dx\,\,} $$equals ________. The parabolic are$$y = \sqrt x ,1 \le x \le 2$$is revolved around the$$x$$-axis. The volume of the solid of revolution is The function$$y = \left| {2 - 3x} \right|$$The value of the integral$$\int\limits_{ - a}^a {{{dx} \over {1 + {x^2}}}} $$The area enclosed between the curves$${y^2} = 4x\,\,$$and$${{x^2} = 4y}$$is The distance between the origin and the point nearest to it on the surface$$\,\,{z^2} = 1 + xy\,\,$$is In the Taylor series expansion of$${e^x}$$about$$x=2,$$the coefficient of$$\,\,{\left( {x - 2} \right)^4}\,\,$$is The value of$$\,\,\mathop {Lim}\limits_{x \to 8} {{{x^{1/3}} - 2} \over {x - 8}}\,\,$$is The minimum value of function$$\,\,y = {x^2}\,\,$$in the interval$$\,\,\left[ {1,5} \right]\,\,$$is$$\int\limits_{ - a}^a {\left[ {{{\sin }^6}\,x + {{\sin }^7}\,x} \right]dx} $$is equal to Changing the order of integration in the double integral$${\rm I} = \int\limits_0^8 {\int\limits_{{\raise0.5ex\hbox{$x$} \kern-0.1em/\k...
If $$\,\,\,x = a\left( {\theta + Sin\theta } \right)$$ and $$y = a\left( {1 - Cos\theta } \right)$$ then $$\,\,{{dy} \over {dx}} = \,\_\_\_\_\_.$$
Value of the function $$\mathop {Lim}\limits_{x \to a} \,{\left( {x - a} \right)^{x - a}}$$ is _______.
Area bounded by the curve $$y = {x^2}$$ and the lines $$x=4$$ and $$y=0$$ is given by
If a function is continuous at a point its first derivative
The area bounded by the parabola $$2y = {x^2}$$ and the lines $$x=y-4$$ is equal to _________.
The value of $$\int\limits_0^\infty {{e^{ - {y^3}}}.{y^{1/2}}}$$ dy is _________.
If $$H(x, y)$$ is homogeneous function of degree $$n$$ then $$x{{\partial H} \over {\partial x}} + y{{\partial H} \over {\partial y}} = nH$$
$$\mathop {Lim}\limits_{x \to 0} {{x\left( {{e^x} - 1} \right) + 2\left( {\cos x - 1} \right)} \over {x\left( {1 - \cos x} \right)}} = \_\_\_\_\_\_.$$...
The function $$f\left( {x,y} \right) = {x^2}y - 3xy + 2y + x$$ has

## Marks 2

A parametric curve defined by $$x = \cos \left( {{{\pi u} \over 2}} \right),y = \sin \left( {{{\pi u} \over 2}} \right)\,\,$$ in the range $$0 \le u \...$$\mathop {Lt}\limits_{x \to \infty } \left( {\sqrt {{x^2} + x - 1} - x} \right)$$is Consider the function$$f\left( x \right) = 2{x^3} - 3{x^2}\,\,$$in the domain$$\,\left[ { - 1,2} \right].$$The global minimum of$$f(x)$$is _____... Consider a spatial curve in three -dimensional space given in parametric form by$$\,\,x\left( t \right)\,\, = \,\,\cos t,\,\,\,y\left( t \right)\,\, ...
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...
The value of the integral $$\,\int\limits_0^2 {\int\limits_0^x {{e^{x + y}}\,\,dy} }$$ $$dx$$ 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...
The infinite series $${\,f\left( x \right) = x - {{{x^3}} \over {3!}} + {{{x^5}} \over {5!}} - {{{x^7}} \over {7!}} + - - - \,\,}$$ Converges to
Consider the shaded triangular region $$P$$ shown in the figure. What is $$\int\!\!\!\int\limits_p {xy\,dx\,dy\,?}$$ ...
Which of the following integrals is unbounded?
The length of the curve $$\,y = {2 \over 3}{x^{3/2}}$$ between $$x=0$$ & $$x=1$$ is
Let $$\,\,f = {y^x}.$$ What is $$\,\,{{{\partial ^2}f} \over {\partial x\partial y}}\,\,$$ at $$x=2,$$ $$y=1$$?
If $$\,\,\,y = x + \sqrt {x + \sqrt {x + \sqrt {x + .....\alpha } } } \,\,\,$$ then $$y(2)=$$ __________.
$$\mathop {Lim}\limits_{x \to 0} {{{e^x} - \left( {1 + x + {{{x^2}} \over 2}} \right)} \over {{x^3}}} =$$
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 $$\,... The volume of an object expressed in spherical co-ordinates is given by$$V = \int\limits_0^{2\pi } {\int\limits_0^{{\raise0.5ex\hbox{\$\scriptstyle \...
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