1994-1-b-math The limit of \(f(x) = x/\sin x\) as \(x\rightarrow 0\) is
1997-1-1-math The sum of the infinite series \(\displaystyle 3 + 1 + \frac {1}{3} + \left (\frac {1}{3}\right )^2+\cdots +\left (\frac {1}{3}\right )^n\) is
1997-1-2-math \(\displaystyle \lim _{x\rightarrow \infty }\frac {x^3+1}{2x^2+80x+1}\) is
1997-1-3-math The value of \(\displaystyle \int _0^{5\pi }(2-\sin x)dx\) is
1998-1-3-math \(\displaystyle \lim _{x\rightarrow 0}\frac {x-\sin 2x}{x+\sin 3x}\) has the value:
1999-1-4-math The harmonic series \(\displaystyle \sum _{n=1}^\infty \frac {1}{n^p}\)
2000-1-2-math For an even function \(f(x)\),
2003-3-math The Taylor series expansion of the function: \(f(x) = x/(1+x)\) around \(x=0\) is
2004-2-math The function \(f(x) = 3x(x-2)\) has a
2006-4-math The derivative of \(|x|\) with respect to \(x\) when \(x\ne 0\) is
2008-3-math The limit of \(\dfrac {\sin x}{x}\) as \(x\rightarrow \infty \) is
2011-11-math Which ONE of the following functions \(y(x)\) has the slope of its tangent equal to \(\dfrac {ax}{y}\) ? Note: \(a\) and \(b\) are real constants.
2012-3-math For the function \(f(t)=e^{-t/\tau }\), the Taylor series approximation for \(t\ll \tau \) is
2013-4-math Evaluate: \(\displaystyle \ \int \frac {dx}{e^x-1} \)
(Note: \(C\) is a constant of integration.)
2014-4-math If \(f(x)\) is a real and continuous function of \(x\), the Taylor series expansion of \(f(x)\) about its minima will NEVER have a term containing
EC-2018-S1-6-math Consider \(p(s) = s^3 + a_2 s^2 + a_1 s + a_0\) with all real coefficients. It is known that its derivative \(p'(s)\) has no real roots. The number of real roots of \(p(s)\) is
ME-2004-1-math If \(x = a(\theta + \sin {\theta })\) and \(y = a(1-\cos {\theta })\) then \(\dfrac {dy}{dx}\) will be equal to
ME-2010-5-math The function \(y = |2 - 3x|\)
2017-1-math The value of \(\displaystyle \lim _{x\rightarrow 0}\frac {\tan (x)}{x}\) is ____________
1995-2-a-math A function \(f(x)=12x-x^3\) has maximum value at \(x=\)
1995-2-b-math \(\displaystyle \lim _{x\rightarrow 0}\frac {\tanh x}{x} =\)
1995-2-c-math The second order Taylor series expansion for a function \(f(x) = x^2\) at \(x=1\) is
1995-2-d-math The average value of function \(f(x) = x^3\) in the interval \(0\le x\le 2\) is
1997-2-2-math The cubic equation \(x^3-x+10=0\) has a root in the interval
1997-2-3-math The Fourier series of the function \[ f(x) = \left \{ \begin {array}{ll} 1 & 0\le x < \pi \\ -1 & -\pi < x < 0\end {array} \right . \] extended periodically, \(f(x+2\pi )=f(x)\), is
1998-2-1-math The function \(z = (x-1)^2 - 2y^2\) has
1998-2-3-math The integral \(\displaystyle \frac {dx}{x^p}\) is convergent for
2000-2-1-math The line integral of \(\displaystyle \int _C \left \{ \frac {y}{x^2+y^2}dx - \frac {x}{x^2+y^2}dy \right \} \), where \(C\) is the unit circle around the origin traversed once in the counter-clockwise direction, is
2001-2-1-math The function \(f(x,y) = x^2+y^2-xy-x-y+5\) has the
2002-2-1-math The coefficient of \(x^2\) in the Taylor series of \(\cos ^2x\) about 0 is
2004-36-math Value of the integral \(\displaystyle \int _{-2}^2\frac {dx}{x^2} \) is
2004-38-math The value of \(\displaystyle \lim _{x\rightarrow 9}\frac {\sqrt {x}-3}{x^2-81}\) is
2005-34-math In the limit \(x\rightarrow 0\), what is the limiting value of the function \(F(x)\) given below? \[ F(x) = \frac {1-\cos (2x)}{x^2} \]
2005-37-math In the domain \(-\infty <x<\infty \), the function \(y(x)=x^3e^{-x}\) has
2006-23-math With \(y=e^{ax}\), if the sum \[ S = \frac {dy}{dx} + \frac {d^2y}{dx^2} + \cdots + \frac {d^ny}{dx^n} \] approaches \(2y\) as \(n\rightarrow \infty \), then the value of \(a\) is
2006-25-math The liquid surface in a cylindrical bucket of radius \(R\) rotating about its axis acquires a parabolic profile given by the equation \(y=a+br^2\), where \(y\) is the height of the liquid surface from the bottom of the bucket at a radial distance \(r\)
from the bucket axis. If the liquid has density \(\rho \), then the mass of the liquid in the bucket is
2007-26-math The family of curves that is orthogonal to \(xy=c\) is
2008-24-math The first terms of the Taylor series expansion of \(\cos x\) about the point \(x=0\) are
2009-21-math The value of the limit \[ \lim _{x\rightarrow \pi /2}\frac {\cos x}{(x-\pi /2)^3} \] is
2010-30-math For a function \(g(x)\), if \(g(0)=0\) and \(g'(0)=2\), then \[ \lim _{x\rightarrow 0}\int _0^{g(x)}\frac {2t}{x}dt \] is equal to
2011-28-math The value of the improper integral \(\displaystyle \int _{-\infty }^{\infty }\frac {dx}{(1+x^2)}\)  is
2012-27-math If \(a\) is a constant, then the value of the integral \(\ \displaystyle a^2\!\!\int _0^\infty \!\!xe^{-ax}dx\) is
CE-2011-28-math What is the value of definite integral, \(\displaystyle \int _0^a\dfrac {\sqrt {x}}{\sqrt {x} + \sqrt {a-x}} dx\) ?
EE-2009-38-math
A cubic polynomial with real coefficients
ME-2004-34-math The volume of an object expressed in spherical co-ordinates is given by \[I = \int _0^{2\pi } \int _0^{\pi /3} \int _0^1 r^2\sin(\phi) dr d\phi d\theta \] The value of integral is
ME-2007-21-math If \(y = x + \sqrt {x+\sqrt {x+\sqrt {x +\cdots \infty }}}\), then \(y(2)\)=?
2016-29-math The Lagrange mean-value theorem is satisfied for \(f(x)=x^3+5\), in the interval \((1,4)\) at a value (rounded off to the second decimal place) of \(x\) equal to ____________
2017-26-math For the initial value problem \[ \frac {dx}{dt} = \sin (t), \qquad x(0) = 0 \] the value of \(x\) at \(t=\pi /3\), is ____________
1994-2-a-math The Taylor's series expansion of \(f(x)\) around \(x=a\) is ------------------------
1994-2-b-math For a differential function \(f(x)\) to have a maximum, \(\dfrac {df}{dx}\) should be ------------------------
1994-2-d-math The integral of \(x\sin x\) is ------------------------
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