GATE CE

Calculus

Engineering Mathematics

(Past Years Questions)

Marks 1

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Let $$\,\,W = f\left( {x,y} \right),\,\,$$ where $$x$$ and $$y$$ are functions of $$t.$$ Then, according to the chain ru...
GATE CE 2017 Set 2
$$\mathop {Lim}\limits_{x \to 0} \left( {{{\tan x} \over {{x^2} - x}}} \right)$$ is equal to _________.
GATE CE 2017 Set 1
Let $$x$$ be a continuous variable defined over the interval $$\left( { - \infty ,\infty } \right)$$, and $$f\left( x \r...
GATE CE 2017 Set 1
Given $$i = \sqrt { - 1} ,$$ the value of the definite integral, $$\,{\rm I} = \int\limits_0^{\pi /2} {{{\cos x + \sin x...
GATE CE 2015 Set 2
$$\mathop {\lim }\limits_{x \to \infty } {\left( {1 + {1 \over x}} \right)^{2x}}\,\,$$ is equal to
GATE CE 2015 Set 2
$$\,\,\mathop {Lim}\limits_{x \to \infty } \left( {{{x + \sin x} \over x}} \right)\,\,$$ equal to
GATE CE 2014 Set 1
With reference to the conventional Cartesian $$(x,y)$$ coordinate system, the vertices of a triangle have the following...
GATE CE 2014 Set 1
The solution $$\int\limits_0^{\pi /4} {{{\cos }^4}3\theta {{\sin }^3}\,6\theta d\theta \,\,} $$ is :
GATE CE 2013
The infinite series $$1 + x + {{{x^2}} \over {2!}} + {{{x^3}} \over {3!}} + {{{x^4}} \over {4!}} + ........$$ correspond...
GATE CE 2012
What should be the value of $$\lambda $$ such that the function defined below is continuous at $$x = {\pi \over 2}$$? $...
GATE CE 2011
The $$\mathop {Lim}\limits_{x \to 0} {{\sin \left( {{2 \over 3}x} \right)} \over x}\,\,\,$$ is
GATE CE 2010
The value of the function, $$f\left( x \right) = \mathop {Lim}\limits_{x \to 0} {{{x^3} + {x^2}} \over {2{x^3} - 7{x^2}}...
GATE CE 2004
The following function has local minima at which value of $$x,$$ $$f\left( x \right) = x\sqrt {5 - {x^2}} $$
GATE CE 2002
The value of the following definite integral in $$\int\limits_{{\raise0.5ex\hbox{$\scriptstyle { - \pi }$} \kern-0.1em/\...
GATE CE 2002
Limit of the following series as $$x$$ approaches $${\pi \over 2}$$ is $$f\left( x \right) = x - {{{x^3}} \over {3!}} ...
GATE CE 2001
Consider the following integral $$\mathop {Lim}\limits_{x \to 0} \int\limits_1^a {{x^{ - 4}}} dx$$ ________.
GATE CE 2000
The function $$f\left( x \right) = {e^x}$$ is _________.
GATE CE 1999
Limit of the function, $$\mathop {Lim}\limits_{n \to \infty } {n \over {\sqrt {{n^2} + n} }}$$ is _______.
GATE CE 1999
Number of inflection points for the curve $$\,\,\,y = x + 2{x^4}\,\,\,\,$$ is_______.
GATE CE 1999
A discontinuous real function can be expressed as
GATE CE 1998
The Taylor's series expansion of sin $$x$$ is ______.
GATE CE 1998
The continuous function $$f(x, y)$$ is said to have saddle point at $$(a, b)$$ if
GATE CE 1998
If $$y = \left| x \right|$$ for $$x
GATE CE 1997
If $$\varphi \left( x \right) = \int\limits_0^{{x^2}} {\sqrt t \,dt\,} $$ then $${{d\varphi } \over {dx}} = \_\_\_\_\_\_...
GATE CE 1997
The function $$f\left( x \right) = {x^3} - 6{x^2} + 9x + 25$$ has
GATE CE 1995
The function $$f\left( x \right) = \left| {x + 1} \right|$$ on the interval $$\left[ { - 2,0} \right]$$ is __________.
GATE CE 1995
The value of $$\varepsilon $$ in the mean value theoram of $$f\left( b \right) - f\left( a \right) = \left( {b - a} \rig...
GATE CE 1994

Marks 2

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The tangent to the curve represented by $$y=x$$ $$ln$$ $$x$$ is required to have $${45^ \circ }$$ inclination with the $...
GATE CE 2017 Set 2
Consider the following definite integral $$${\rm I} = \int\limits_0^1 {{{{{\left( {{{\sin }^{ - 1}}x} \right)}^2}} \ove...
GATE CE 2017 Set 2
The expression $$\mathop {Lim}\limits_{a \to 0} \,{{{x^a} - 1} \over a}\,\,$$ is equal to
GATE CE 2014 Set 2
What is the value of the definite integral? $$\,\,\int\limits_0^a {{{\sqrt x } \over {\sqrt x + \sqrt {a - x} }}dx\,\,}...
GATE CE 2011
A parabolic cable is held between two supports at the same level. The horizontal span between the supports is $$L.$$ The...
GATE CE 2010
Given a function $$f\left( {x,y} \right) = 4{x^2} + 6{y^2} - 8x - 4y + 8,$$ the optimal values of $$f(x,y)$$ is
GATE CE 2010
The function $$f\left( x \right) = 2{x^3} - 3{x^2} - 36x + 2\,\,\,$$ has its maxima at
GATE CE 2004
Limit of the following sequence as $$n \to \infty $$ $$\,\,\,$$ is $$\,\,\,$$ $${x_n} = {n^{{1 \over n}}}$$
GATE CE 2002
The value of the following improper integral is $$\,\int\limits_0^1 {x\,\log \,x\,dx} = \_\_\_\_\_.$$
GATE CE 2002
If $$f\left( {x,y,z} \right) = $$ $${\left( {{x^2} + {y^2} + {z^2}} \right)^{{\raise0.5ex\hbox{$\scriptstyle { - 1}$} \...
GATE CE 2000
Limit of the function $$f\left( x \right) = {{1 - {a^4}} \over {{x^4}}}\,\,as\,\,x \to \infty $$ is given by
GATE CE 2000
The Taylor series expansion of sin $$x$$ about $$x = {\pi \over 6}$$ is given by
GATE CE 2000

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