GATE EE
Engineering Mathematics
Calculus
Previous Years Questions

## Marks 1

Let $$f(x) = \int\limits_0^x {{e^t}(t - 1)(t - 2)dt}$$. Then f(x) decreases in the interval.
Let R be a region in the first quadrant of the xy plane enclosed by a closed curve C considered in counter-clockwise direction. Which of the following...
Let $$\,{y^2} - 2y + 1 = x$$ and $$\,\sqrt x + y = 5.\,\,$$ The value of $$\,x + \sqrt y \,\,$$ equals ________. (Given the answer up to three decima...
Consider a function $$f\left( {x,y,z} \right)$$ given by $$f\left( {x,y,z} \right) = \left( {{x^2} + {y^2} - 2{z^2}} \right)\left( {{y^2} + {z^2}} \r... Let$$x$$and$$y$$be integers satisfying the following equations$$$2{x^2} + {y^2} = 34$x + 2y = 11$$The value of$$(x+y)$$is _________.... Let$${\rm I} = c\int {\int {_Rx{y^2}dxdy,\,\,} } $$where$$R$$is the region shown in the figure and$$c = 6 \times {10^{ - 4}}.\,\,$$The value of ... The maximum value attained by the function$$f(x)=x(x-1) (x-2)$$in the interval$$\left[ {1,2} \right]$$is _________. A particle, starting from origin at$$t=0s,$$is traveling along$$x$$-axis with velocity$$v = {\pi \over 2}\cos \left( {{\pi \over 2}t} \righ...
Minimum of the real valued function $$f\left( x \right) = {\left( {x - 1} \right)^{2/3}}$$ occurs at $$x$$ equal to
Let $$f\left( x \right) = x{e^{ - x}}.$$ The maximum value of the function in the interval $$\left( {0,\infty } \right)$$ is
A function $$y = 5{x^2} + 10x\,\,$$ is defined over an open interval $$x=(1,2).$$ At least at one point in this interval, $${{dy} \over {dx}}$$ is exa...
Roots of the algebraic equation $${x^3} + {x^2} + x + 1 = 0$$ are
The function $$f\left( x \right) = 2x - {x^2} + 3\,\,$$ has
At $$t=0,$$ the function $$f\left( t \right) = {{\sin t} \over t}\,\,$$ has
Consider the function $$f\left( x \right) = {\left( {{x^2} - 4} \right)^2}$$ where $$x$$ is a real number. Then the function has
For the function $$f\left( x \right) = {x^2}{e^{ - x}},$$ the maximum occurs when $$x$$ is equal to
If $$S = \int\limits_1^\infty {{x^{ - 3}}dx}$$ then $$S$$ has the value
The area enclosed between the parabola $$y = {x^2}$$ and the straight line $$y=x$$ is _______.
$$\mathop {Lim}\limits_{\theta \to 0} \,{{\sin \,m\,\theta } \over \theta },$$ where $$m$$ is an integer, is one of the following :
$$\mathop {Lim}\limits_{x \to \infty } \,x\sin {\raise0.5ex\hbox{1} \kern-0.1em/\kern-0.15em \lower0.25ex\hbox{x}} = \_\... If$$f(0)=2$$and$$f'\left( x \right) = {1 \over {5 - {x^2}}},$$then the lower and upper bounds of$$f(1)$$estimated by the mean value theorem are... The integration of$$\int {{\mathop{\rm logx}\nolimits} \,dx} $$has the value The volume generated by revolving the area bounded by the parabola$${y^2} = 8x$$and the line$$x=2$$about$$y$$-axis is ## Marks 2 Let$$g\left( x \right) = \left\{ {\matrix{ { - x} & {x \le 1} \cr {x + 1} & {x \ge 1} \cr } } \right.$$and$$f\left( x \right) ...
A function $$f(x)$$ is defined as $$f\left( x \right) = \left\{ {\matrix{ {{e^x},x < 1} \cr {\ln x + a{x^2} + bx,x \ge 1} \cr } \,\,,\... The value of the integral$$\,\,2\int_{ - \infty }^\infty {\left( {{{\sin \,2\pi t} \over {\pi t}}} \right)} dt\,\,$$is equal to Let$$\,\,S = \sum\limits_{n = 0}^\infty {n{\alpha ^n}} \,\,$$where$$\,\,\left| \alpha \right| < 1.\,\,$$The value of$$\alpha $$in the range... The volume enclosed by the surface$$f\left( {x,y} \right) = {e^x}$$over the triangle bounded by the lines$$x=y;x=0;y=1$$in the$$xy$$pl... To evaluate the double integral$$\int\limits_0^8 {\left( {\int\limits_{y/2}^{\left( {y/2} \right) + 1} {\left( {{{2x - y} \over 2}} \right)dx} } \ri...
The minimum value of the function $$f\left( x \right) = {x^3} - 3{x^2} - 24x + 100$$ in the interval $$\left[ { - 3,3} \right]$$ is
The maximum value of $$f\left( x \right) = {x^3} - 9{x^2} + 24x + 5$$ in the interval $$\left[ {1,6} \right]$$ is
The value of the quantity, where $$P = \int\limits_0^1 {x{e^x}\,dx\,\,\,}$$ is
If $$(x, y)$$ is continuous function defined over $$\left( {x,y} \right) \in \left[ {0,1} \right] \times \left[ {0,1} \right].\,\,\,$$ Given two cons...
The integral $$\,\,{1 \over {2\pi }}\int\limits_0^{2\Pi } {Sin\left( {t - \tau } \right)\cos \tau \,d\tau \,\,\,}$$ equals
The expression $$V = \int\limits_0^H {\pi {R^2}{{\left( {1 - {h \over H}} \right)}^2}dh\,\,\,}$$ for the volume of a cone is equal to _________.
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Joint Entrance Examination