GATE CE
Engineering Mathematics
Calculus
Previous Years Questions

## Marks 1

Let $$\,\,W = f\left( {x,y} \right),\,\,$$ where $$x$$ and $$y$$ are functions of $$t.$$ Then, according to the chain rule, $${{dw} \over {dt}}$$ is e...
$$\mathop {Lim}\limits_{x \to 0} \left( {{{\tan x} \over {{x^2} - x}}} \right)$$ is equal to _________.
Let $$x$$ be a continuous variable defined over the interval $$\left( { - \infty ,\infty } \right)$$, and $$f\left( x \right) = {e^{ - x - {e^{ - x}}}...$$\mathop {\lim }\limits_{x \to \infty } {\left( {1 + {1 \over x}} \right)^{2x}}\,\,$$is equal to Given$$i = \sqrt { - 1} ,$$the value of the definite integral,$$\,{\rm I} = \int\limits_0^{\pi /2} {{{\cos x + \sin x} \over {\cos x - i\,\sin x}}d...
With reference to the conventional Cartesian $$(x,y)$$ coordinate system, the vertices of a triangle have the following coordinates: $$\,\left( {{x_1...$$\,\,\mathop {Lim}\limits_{x \to \infty } \left( {{{x + \sin x} \over x}} \right)\,\,$$equal to The solution$$\int\limits_0^{\pi /4} {{{\cos }^4}3\theta {{\sin }^3}\,6\theta d\theta \,\,} $$is : The infinite series$$1 + x + {{{x^2}} \over {2!}} + {{{x^3}} \over {3!}} + {{{x^4}} \over {4!}} + ........$$corresponds to What should be the value of$$\lambda $$such that the function defined below is continuous at$$x = {\pi \over 2}$$?$$f\left( x \right) = \left\{ {...
The $$\mathop {Lim}\limits_{x \to 0} {{\sin \left( {{2 \over 3}x} \right)} \over x}\,\,\,$$ is
The value of the function, $$f\left( x \right) = \mathop {Lim}\limits_{x \to 0} {{{x^3} + {x^2}} \over {2{x^3} - 7{x^2}}}\,\,\,$$ is
The following function has local minima at which value of $$x,$$ $$f\left( x \right) = x\sqrt {5 - {x^2}}$$
The value of the following definite integral in $$\int\limits_{{\raise0.5ex\hbox{{ - \pi }} \kern-0.1em/\kern-0.15em \lower0.25ex\hbox{... Limit of the following series as$$x$$approaches$${\pi \over 2}$$is$$f\left( x \right) = x - {{{x^3}} \over {3!}} + {{{x^5}} \over {5!}} - {{{x^...
Consider the following integral $$\mathop {Lim}\limits_{x \to 0} \int\limits_1^a {{x^{ - 4}}} dx$$ ________.
Limit of the function, $$\mathop {Lim}\limits_{n \to \infty } {n \over {\sqrt {{n^2} + n} }}$$ is _______.
Number of inflection points for the curve $$\,\,\,y = x + 2{x^4}\,\,\,\,$$ is_______.
The function $$f\left( x \right) = {e^x}$$ is _________.
A discontinuous real function can be expressed as
The Taylor's series expansion of sin $$x$$ is ______.
The continuous function $$f(x, y)$$ is said to have saddle point at $$(a, b)$$ if
If $$y = \left| x \right|$$ for $$x < 0$$ and $$y=x$$ for $$x \ge 0$$ then
If $$\varphi \left( x \right) = \int\limits_0^{{x^2}} {\sqrt t \,dt\,}$$ then $${{d\varphi } \over {dx}} = \_\_\_\_\_\_\_.$$
The function $$f\left( x \right) = {x^3} - 6{x^2} + 9x + 25$$ has
The function $$f\left( x \right) = \left| {x + 1} \right|$$ on the interval $$\left[ { - 2,0} \right]$$ is __________.
The value of $$\varepsilon$$ in the mean value theoram of $$f\left( b \right) - f\left( a \right) = \left( {b - a} \right)\,\,f'\left( \varepsilon \... ## Marks 2 The tangent to the curve represented by$$y=xlnx$$is required to have$${45^ \circ }$$inclination with the$$x-$$axis. The coordinates of ... Consider the following definite integral$$${\rm I} = \int\limits_0^1 {{{{{\left( {{{\sin }^{ - 1}}x} \right)}^2}} \over {\sqrt {1 - {x^2}} }}dx} $$... The expression$$\mathop {Lim}\limits_{a \to 0} \,{{{x^a} - 1} \over a}\,\,$$is equal to What is the value of the definite integral?$$\,\,\int\limits_0^a {{{\sqrt x } \over {\sqrt x + \sqrt {a - x} }}dx\,\,} $$? A parabolic cable is held between two supports at the same level. The horizontal span between the supports is$$L.$$The sag at the mid-span is$$h.$$... Given a function$$f\left( {x,y} \right) = 4{x^2} + 6{y^2} - 8x - 4y + 8,$$the optimal values of$$f(x,y)$$is The function$$f\left( x \right) = 2{x^3} - 3{x^2} - 36x + 2\,\,\,$$has its maxima at Limit of the following sequence as$$n \to \infty \,\,\,$$is$$\,\,\,{x_n} = {n^{{1 \over n}}}$$The value of the following improper integral is$$\,\int\limits_0^1 {x\,\log \,x\,dx} = \_\_\_\_\_.$$The Taylor series expansion of sin$$x$$about$$x = {\pi \over 6}$$is given by Limit of the function$$f\left( x \right) = {{1 - {a^4}} \over {{x^4}}}\,\,as\,\,x \to \infty $$is given by If$$f\left( {x,y,z} \right) = {\left( {{x^2} + {y^2} + {z^2}} \right)^{{\raise0.5ex\hbox{$\scriptstyle { - 1}\$} \kern-0.1em/\kern-0.15em \lower...
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Joint Entrance Examination