GATE ECE
Engineering Mathematics
Calculus
Previous Years Questions

## Marks 1

Consider the two-dimensional vector field $$\overrightarrow F (x,y) - x\overrightarrow i + y\overrightarrow j$$, where $$\overrightarrow i$$ and $$... The function f(x) = 8loge x$$-$$x2 + 3 attains its minimum over the interval [1, e] at x = __________. (Here loge x is the natural logarithm of x.)... The value of the integral$$\int\!\!\!\int\limits_D {3({x^2} + {y^2})dx\,dy} $$, where D is the shaded triangular region shown in the diagram, is ____... The families of curves represented by the solution of the equation$${{dy} \over {dx}} = - {\left( {{x \over y}} \right)^n}$$for n = –1 and n = 1 re... Let$$f\left( {x,y} \right) = {{a{x^2} + b{y^2}} \over {xy}}$$, where$$a$$and$$b$$are constants. If$${{\partial f} \over {\partial x}} = {{\parti...
Taylor series expansion of $$f\left( x \right) = \int\limits_0^x {{e^{ - \left( {{{{t^2}} \over 2}} \right)}}} dt$$ around 𝑥 = 0 has the form f(x) = ...
The integral $$\int\limits_0^1 {{{dx} \over {\sqrt {\left( {1 - x} \right)} }}}$$ is equal ________.
As $$x$$ varies from $$- 1$$ to $$3,$$ which of the following describes the behavior of the function $$f\left( x \right) = {x^3} - 3{x^2} + 1?$$
How many distinct values of $$x$$ satisfy the equation $$sin(x)=x/2,$$ where $$x$$ is in radians ?
Given the following statements about a function $$f:R \to R,$$ select the right option: $$P:$$ If $$f(x)$$ is continuous at $$x = {x_0},$$ then it is...
The contour on the $$x-y$$ plane, where the partial derivative of $${x^2} + {y^2}$$ with respect to $$y$$ is equal to the partial derivative of $$6y+4... The value of$$\sum\limits_{n = 0}^\infty {n{{\left( {{1 \over 2}} \right)}^n}\,\,} $$is _______. A function$$f\left( x \right) = 1 - {x^2} + {x^3}\,\,$$is defined in the closed interval$$\left[ { - 1,1} \right].$$The value of$$x,$$in the ope... The series$$\sum\limits_{n = 0}^\infty {{1 \over {n!}}\,} $$converges to The maximum value of the function$$\,f\left( x \right) = \ln \left( {1 + x} \right) - x$$(where$$x > - 1$$) occurs at$$x=$$________. If$$z=xyln(xy),$$then For$$0 \le t < \infty ,$$the maximum value of the function$$f\left( t \right) = {e^{ - t}} - 2{e^{ - 2t}}\,$$occurs at The value of$$\mathop {\lim }\limits_{x \to \infty } {\left( {1 + {1 \over x}} \right)^x}\,\,$$is If$$\,{e^y} = {x^{1/x}}\,\,$$then$$y$$has a For real values of$$x,$$the minimum value of function$$f\left( x \right) = {e^x} + {e^{ - x}}\,\,$$is Which of the following function would have only odd powers of$$x$$in its Taylor series expansion about the point$$x=0$$?$$\mathop {Lim}\limits_{\theta \to 0} {{\sin \left( {\theta /2} \right)} \over \theta }\,\,\,$$is The following plot shows a function$$y$$which varies linearly with$$x$$. The value of the integral$$\,\,{\rm I} = \int\limits_1^2 {y\,dx\,\,} $$... For the function$${e^{ - x}},$$the linear approximation around$$x=2$$is For$$\left| x \right| < < 1,\,\cot \,h\left( x \right)\,\,\,$$can be approximated as The value of the integral$$1 = {1 \over {\sqrt {2\pi } }}\,\,\int\limits_0^\infty {{e^{ - {\raise0.5ex\hbox{$\scriptstyle {{x^2}}$} \kern-0.1em/\ker...
The curve given by the equation $${x^2} + {y^2} = 3axy$$ is
By reversing the order of integration $$\int\limits_0^2 {\int\limits_{{x^2}}^{2x} {f\left( {x,y} \right)dy\,dx} }$$ may be represented as ______.
The third term in the taylor's series expansion of $${e^x}$$ about $$'a'$$ would be ________.
The function $$y = {x^2} + {{250} \over x}$$ at $$x=5$$ attains

## Marks 2

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), the value (correct to two deci...
The values of the integrals $$\int\limits_0^1 {\left( {\int\limits_0^1 {{{x - y} \over {{{\left( {x + y} \right)}^3}}}dy} } \right)} dx\,\,$$ and $$\,... The minimum value of the function$$f\left( x \right) = {1 \over 3}x\left( {{x^2} - 3} \right)\,\,$$in the interval$$ - 100 \le x \le 100$$oc... Let$$\,\,f\left( x \right) = {e^{x + {x^2}}}\,\,$$for real$$x.$$From among the following. Choose the Taylor series approximation of$$f(x)$$... A three dimensional region$$R$$of finite volume is described by$$\,\,{x^2} + {y^2} \le {z^3},\,\,\,0 \le z \le 1$$Where$$x, y, z$$are real. Th... A triangle in the$$xy-$$plane is bounded by the straight lines$$2x=3y, y=0$$and$$x=3.$$The volume above the triangle and under the plane$$x+y+z=...
The integral $$\,\,{1 \over {2\pi }}\int {\int_D {\left( {x + y + 10} \right)dxdy\,\,} }$$ where $$D$$ denotes the disc: $${x^2} + {y^2} \le 4,$$ eva...
The region specified by $$\left\{ {\left( {\rho ,\varphi ,{\rm Z}} \right):3 \le \rho \le 5,\,\,{\pi \over 8} \le \phi \le {\pi \over 4},\,\,3 \l... The value of the integral$$\int_{ - \infty }^\infty {12\,\,\cos \left( {2\pi t} \right){{\sin \left( {4\pi t} \right)} \over {4\pi t}}} dt\,\,$$is ... The maximum area (in square units) of a rectangle whose vertices lie on the ellipse$${x^2} + 4{y^2} = 1\,\,$$is Which one of the following graphs describes the function?$$f\left( x \right) = {e^{ - x}}\left( {{x^2} + x + 1} \right)\,?$$For a right angled triangle, if the sum of the lengths of the hypotenuse and a side is kept constant, in order to have maximum area of the triangle, t... The maximum value of$$f\left( x \right) = 2{x^3} - 9{x^2} + 12x - 3$$in the interval$$\,0 \le x \le 3$$is __________. The Taylor series expansion of$$3sinx+2cosx$$is The volume under the surface$$z\left( {x,y} \right) = x + y$$and above the triangle in the$$xy$$plane defined by$$\left\{ {0 \le y \le x} \righ...
The Taylor series expansion of $$\,\,{{\sin x} \over {x - \pi }}\,\,$$ at $$x = \pi$$ is given by
The value of the integral of the function $$\,\,g\left( {x,y} \right) = 4{x^3} + 10{y^4}\,\,$$ along the straight line segment from the point $$(0,0)... In the Taylor series expansion of$${e^x} + \sin x$$about the point$$x = \pi ,$$the coefficient of$${\left( {x = \pi } \right)^2}$$is Consider the function$$\,f\left( x \right) = {x^2} - x - 2.\,$$The maximum value of$$f(x)$$in the closed interval$$\left[ { - 4,4} \right]\,\int\limits_0^{{\raise0.5ex\hbox{$\scriptstyle \pi$} \kern-0.1em/\kern-0.15em \lower0.25ex\hbox{$\scriptstyle 2$}}} {\int\limits_0^{{\raise0.5ex\hb...
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Joint Entrance Examination