List of practice Questions

The set of all points of differentiability of the function $f(x)=e^{-|x|}$ is

If there is an error of 3% in the volume of a sphere then the percentage error in its radius is

The interval in which $f(x) = 2x^{2} - \log x$ increases is

The function $y = xe^{x}$ has}

$Lt_{x \to 0}\frac{e^{x^2}-cos x}{sin^2x}=$

Which of the following functions have finite number of points of discontinuity?

$Lt_{x \to 0}(\frac{|x|}{x}+x+2)=$

For \(a \in \mathbb{R}\), consider the real valued function defined on \((-1,1)\): \[ f(x)= \begin{cases} \dfrac{(1+x)^{1/3}-(1+2x)^{1/4}}{x}, & x\neq0,\\ a,& x=0. \end{cases} \] If \(f\) is differentiable at \(x=0\), then value of \(a+f'(0)\) is:

\[ \lim_{x\to0} \frac{|x|\log_e(1+|\sin2x|)} {x^2(|x|+3)} \]

Find the acute angle at which curves \[ y=(x-2)^2 \] and \[ y=-4+6x-x^2 \] intersect.

Let \(f:\mathbb{R}\to\mathbb{R}\), \(f(x)=|x+1|e^{-x^2}\), then which of the following option is true?

Find the value of \[ \lim_{x\to\infty} \frac{\sqrt{x}} {\sqrt{x+\sqrt{x+\sqrt{x}}}} \]

Let \(f:[0,\infty)\to\mathbb{R}\) be defined by \[ f(x)=\frac{3x^2+4x+1}{x^2+3x+2}. \] Then the value of \((f^{-1})'(2)\) is:

Evaluate \[ \int_{0}^{\frac{\pi}{4}}\frac{dx}{\cos^4x}. \]

If \(f:[0,\infty)\to\mathbb R\), \[ f(x)=\frac{x^2-1}{x^2+1}, \] then find \[ \int_{-1}^{1}f^{-1}(y)\,dy. \]

Find the value of \[ \lim_{x\to 0}\frac{x^2+2\cos x-2}{x\sin 3x} \]

Arrange the following in ascending order based on their values:
A. \(\displaystyle \int_0^{\frac{\pi}{2}}\frac{1}{1+\sin x}\,dx\),
B. \(\displaystyle \int_1^2 x^2\,dx\),
C. \(\displaystyle \int_0^{\frac{\pi}{2}}\sin x\,dx\),
D. \(\displaystyle \int_{-\frac{\pi}{2}}^{\frac{\pi}{2}}\sin^3x\,dx\).

The function \(f(x)=|x-2|+|x|+|x+2|\) is not differentiable at which points?

Which of the following is the variance of the data? \[ 6,\ 8,\ 10,\ 12,\ 14,\ 16,\ 18,\ 20,\ 22,\ 24 \]

Arrange the following in increasing order as per the last digit:
A. \(2^{444}\),
B. \(17^{10}\),
C. \(13^{10}+2\),
D. \(1^1+2^2+3^3+\cdots+100^{100}\),
E. \(11!+2\).

Arrange the following in ascending order based on their values:
A. \(\displaystyle \int_0^{\frac{\pi}{2}}\frac{1}{1+\sin x}\,dx\),
B. \(\displaystyle \int_1^2 x^2\,dx\),
C. \(\displaystyle \int_0^{\frac{\pi}{2}}\sin x\,dx\),
D. \(\displaystyle \int_{-\frac{\pi}{2}}^{\frac{\pi}{2}}\sin^3x\,dx\).

The function \(f(x)=|x-2|+|x|+|x+2|\) is not differentiable at which points?

Which of the following is equation of hyperbola with vertices \((\pm 2,0)\) and foci \((\pm 3,0)\)?

The normal at the point \((1,1)\) on the curve \(2y+x^2=3\) is?