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GATE ECE

Calculus

Engineering Mathematics

Previous Years Questions

Marks 1

More
The value of the integral $$\int\!\!\!\int\limits_D {3({x^2} + {y^2})dx\,dy} $$, where D is the shaded triangular region...
GATE ECE 2022
The function f(x) = 8loge x $$-$$ x2 + 3 attains its minimum over the interval [1, e] at x = __________. (Here loge x is...
GATE ECE 2022
Consider the two-dimensional vector field $$\overrightarrow F (x,y) - x\overrightarrow i + y\overrightarrow j $$, where...
GATE ECE 2022
The families of curves represented by the solution of the equation $${{dy} \over {dx}} = - {\left( {{x \over y}} \right...
GATE ECE 2019
Taylor series expansion of $$f\left( x \right) = \int\limits_0^x {{e^{ - \left( {{{{t^2}} \over 2}} \right)}}} dt$$ arou...
GATE ECE 2018
Let $$f\left( {x,y} \right) = {{a{x^2} + b{y^2}} \over {xy}}$$, where $$a$$ and $$b$$ are constants. If $${{\partial f} ...
GATE ECE 2018
How many distinct values of $$x$$ satisfy the equation $$sin(x)=x/2,$$ where $$x$$ is in radians ?
GATE ECE 2016 Set 2
The integral $$\int\limits_0^1 {{{dx} \over {\sqrt {\left( {1 - x} \right)} }}} $$ is equal ________.
GATE ECE 2016 Set 3
As $$x$$ varies from $$- 1$$ to $$3,$$ which of the following describes the behavior of the function $$f\left( x \righ...
GATE ECE 2016 Set 2
Given the following statements about a function $$f:R \to R,$$ select the right option: $$P:$$ If $$f(x)$$ is continuou...
GATE ECE 2016 Set 1
The contour on the $$x-y$$ plane, where the partial derivative of $${x^2} + {y^2}$$ with respect to $$y$$ is equal to th...
GATE ECE 2015 Set 3
The value of $$\sum\limits_{n = 0}^\infty {n{{\left( {{1 \over 2}} \right)}^n}\,\,} $$ is _______.
GATE ECE 2015 Set 3
A function $$f\left( x \right) = 1 - {x^2} + {x^3}\,\,$$ is defined in the closed interval $$\left[ { - 1,1} \right].$$ ...
GATE ECE 2015 Set 1
The series $$\sum\limits_{n = 0}^\infty {{1 \over {n!}}\,} $$ converges to
GATE ECE 2014 Set 4
If $$z=xy$$ $$ln(xy),$$ then
GATE ECE 2014 Set 3
The maximum value of the function $$\,f\left( x \right) = \ln \left( {1 + x} \right) - x$$ (where $$x > - 1$$ ) occu...
GATE ECE 2014 Set 3
The value of $$\mathop {\lim }\limits_{x \to \infty } {\left( {1 + {1 \over x}} \right)^x}\,\,$$ is
GATE ECE 2014 Set 2
For $$0 \le t < \infty ,$$ the maximum value of the function $$f\left( t \right) = {e^{ - t}} - 2{e^{ - 2t}}\,$$ occ...
GATE ECE 2014 Set 2
If $$\,{e^y} = {x^{1/x}}\,\,$$ then $$y$$ has a
GATE ECE 2010
Which of the following function would have only odd powers of $$x$$ in its Taylor series expansion about the point $$x=0...
GATE ECE 2008
For real values of $$x,$$ the minimum value of function $$f\left( x \right) = {e^x} + {e^{ - x}}\,\,$$ is
GATE ECE 2008
For $$\left| x \right| < < 1,\,\cot \,h\left( x \right)\,\,\,$$ can be approximated as
GATE ECE 2007
For the function $${e^{ - x}},$$ the linear approximation around $$x=2$$ is
GATE ECE 2007
The following plot shows a function $$y$$ which varies linearly with $$x$$. The value of the integral $$\,\,{\rm I} = \i...
GATE ECE 2007
$$\mathop {Lim}\limits_{\theta \to 0} {{\sin \left( {\theta /2} \right)} \over \theta }\,\,\,$$ is
GATE ECE 2007
The value of the integral $$1 = {1 \over {\sqrt {2\pi } }}\,\,\int\limits_0^\infty {{e^{ - {\raise0.5ex\hbox{$\scriptst...
GATE ECE 2005
The curve given by the equation $${x^2} + {y^2} = 3axy$$ is
GATE ECE 1997
The third term in the taylor's series expansion of $${e^x}$$ about $$'a'$$ would be ________.
GATE ECE 1995
By reversing the order of integration $$\int\limits_0^2 {\int\limits_{{x^2}}^{2x} {f\left( {x,y} \right)dy\,dx} } $$ may...
GATE ECE 1995
The function $$y = {x^2} + {{250} \over x}$$ at $$x=5$$ attains
GATE ECE 1994

Marks 2

More
Let r = x2 + y - z and z3 - xy + yz + y3 = 1. Assume that x and y are independent variables. At (x, y, z) = (2, -1, 1), ...
GATE ECE 2018
A three dimensional region $$R$$ of finite volume is described by $$\,\,{x^2} + {y^2} \le {z^3},\,\,\,0 \le z \le 1$$ W...
GATE ECE 2017 Set 1
Let $$\,\,f\left( x \right) = {e^{x + {x^2}}}\,\,$$ for real $$x.$$ From among the following. Choose the Taylor series a...
GATE ECE 2017 Set 1
The minimum value of the function $$f\left( x \right) = {1 \over 3}x\left( {{x^2} - 3} \right)\,\,$$ in the interval $$ ...
GATE ECE 2017 Set 2
The values of the integrals $$\int\limits_0^1 {\left( {\int\limits_0^1 {{{x - y} \over {{{\left( {x + y} \right)}^3}}}dy...
GATE ECE 2017 Set 2
A triangle in the $$xy-$$plane is bounded by the straight lines $$2x=3y, y=0$$ and $$x=3.$$ The volume above the triangl...
GATE ECE 2016 Set 3
The region specified by $$\left\{ {\left( {\rho ,\varphi ,{\rm Z}} \right):3 \le \rho \le 5,\,\,{\pi \over 8} \le \ph...
GATE ECE 2016 Set 1
The integral $$\,\,{1 \over {2\pi }}\int {\int_D {\left( {x + y + 10} \right)dxdy\,\,} } $$ where $$D$$ denotes the disc...
GATE ECE 2016 Set 1
The value of the integral $$\int_{ - \infty }^\infty {12\,\,\cos \left( {2\pi t} \right){{\sin \left( {4\pi t} \right)}...
GATE ECE 2015 Set 2
Which one of the following graphs describes the function? $$f\left( x \right) = {e^{ - x}}\left( {{x^2} + x + 1} \right)...
GATE ECE 2015 Set 1
The maximum area (in square units) of a rectangle whose vertices lie on the ellipse $${x^2} + 4{y^2} = 1\,\,$$ is
GATE ECE 2015 Set 1
The maximum value of $$f\left( x \right) = 2{x^3} - 9{x^2} + 12x - 3$$ in the interval $$\,0 \le x \le 3$$ is _________...
GATE ECE 2014 Set 3
For a right angled triangle, if the sum of the lengths of the hypotenuse and a side is kept constant, in order to have m...
GATE ECE 2014 Set 4
The volume under the surface $$z\left( {x,y} \right) = x + y$$ and above the triangle in the $$xy$$ plane defined by $...
GATE ECE 2014 Set 1
The Taylor series expansion of $$3$$ $$sin$$ $$x$$ $$+2cos$$ $$x$$ is
GATE ECE 2014 Set 1
The Taylor series expansion of $$\,\,{{\sin x} \over {x - \pi }}\,\,$$ at $$x = \pi $$ is given by
GATE ECE 2009
In the Taylor series expansion of $${e^x} + \sin x$$ about the point $$x = \pi ,$$ the coefficient of $${\left( {x = \p...
GATE ECE 2008
The value of the integral of the function $$\,\,g\left( {x,y} \right) = 4{x^3} + 10{y^4}\,\,$$ along the straight line ...
GATE ECE 2008
Consider the function $$\,f\left( x \right) = {x^2} - x - 2.\,$$ The maximum value of $$f(x)$$ in the closed interval $$...
GATE ECE 2007
$$\int\limits_0^{{\raise0.5ex\hbox{$\scriptstyle \pi $} \kern-0.1em/\kern-0.15em \lower0.25ex\hbox{$\scriptstyle 2$}}} {...
GATE ECE 2000

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