Let \(f, g\) and \(h\) be real-valued functions defined on the interval \([0,1]\) by \(f(x)=e^{x^2}+e^{-x^2}, \quad g(x)=x e^{x^2}+e^{-x^2}\) and \(h(x)=x^2 e^{x^2}+e^{-x^2}\). If \(a, b\) and \(c\) denote respectively, the absolute maximum of \(f, g\) and \(h\) on \([0,1]\), then

Let $-\frac{\pi }{6}<\theta <-\frac{\pi }{12}$ . Suppose ${\alpha }_1$ and ${\beta }_1$ are the roots of the equation $x^2-2x\sec \theta +1=0$ and ${\alpha }_2$ and ${\beta }_2$ are the roots of the equation $x^2+2x\tan \theta -1=0.$ If ${\alpha }_1>{\beta }_1$ and ${\alpha }_2>{\beta }_2$ , then ${\alpha }_1+{\beta }_2$ equals

Paragraph:
Read the following passage and answer the questions.
Question:
The value of \(|U|\) is

Let $n_1<n_2<n_3<n_4<n_5$ be positive integers such that $n_1+n_2+n_3+n_4+n_5=20$ . Then the number of such distinct arrangements $\left(n_1, n_2, n_3, n_4, n_5\right)$ is _________

Let \(\omega=e^{i \pi / 3}\) and \(a, b, c, x, y, z\) be non-zero complex numbers such that \(a+b+c=x, a+b \omega+c \omega^2=y, a+b \omega^2+c \omega=z\).
Then, the value of \(\frac{|x|^2+|y|^2+|z|^2}{|a|^2+|b|^2+|c|^2}\) is

If \(a, b\) and \(c\) are the sides of a \(\triangle A B C\) such that \(x^2-2(a+b+c) x+3 \lambda(a b+b c+c a)=0\) has real roots, then

Consider the given data with frequency distribution $\begin{array}{lllllll}{x}_i& 3& 8& 11& 10& 5& 4\\ f_i& 5& 2& 3& 2& 4& 4\end{array}$ Match each entry in List-I to the correct entries in List-II.

	List-I		List-II
$\left(\mathrm{P}\right)$	The mean of the above data is	$\left(1\right)$	$2.5$
$\left(\mathrm{Q}\right)$	The median of the above data is	$\left(2\right)$	$5$
$\left(\mathrm{R}\right)$	The mean deviation about the mean of the above data is	$\left(3\right)$	$6$
$\left(\mathrm{S}\right)$	The mean deviation about the median of the above data is	$\left(4\right)$	$2.7$
		$\left(5\right)$	$2.4$

The correct option is

Let $\mathrm{A}\mathrm{P}\mathrm{ }\left(\mathrm{a};\mathrm{d}\right)$ denote the set of all the terms of an infinite arithmetic progression with first term a and common difference $\mathrm{d}>0.$ If $\mathrm{A}\mathrm{P}\left(1;3\right)\cap \mathrm{A}\mathrm{P}\left(2;5\right)\cap \mathrm{A}\mathrm{P}\left(3;7\right)=\mathrm{A}\mathrm{P}\left(\mathrm{a};\mathrm{d}\right)$ then $\mathrm{a}+\mathrm{d}$ equals ____

In the scheme given below, the total number of intramolecular aldol condensation products formed from ' \(Y\) ' is

The locus of the mid-point of the chord of contact of tangents drawn from points lying on the straight line \(4 x-\) \(5 y=20\) to the circle \(x^{2}+y^{2}=9\) is

Paragraph I: In a Young's double slit experiment, each of the two slits A and B, as shown in the figure, are oscillating about their fixed center and with a mean separation of \(0.8 \mathrm{~mm}\). The distance between the slits at time \(t\) is given by \(d=(0.8+0.04 \sin \omega t) \mathrm{mm}\), where \(\omega=0.08 \mathrm{rad} \mathrm{s}^{-1}\). The distance of the screen from the slits is \(1 \mathrm{~m}\) and the wavelength of the light used to illuminate the slits is \(6000 Ã…\). The interference pattern on the screen changes with time, while the central bright fringe (zeroth fringe) remains fixed at point \(O\).

The \(8^{\text {th }}\) bright fringe above the point \(\mathrm{O}\) oscillates with time between two extreme positions. The separation between these two extreme positions, in micrometer \((\mu \mathrm{m})\), is ___________ .

Let \(\alpha\) and \(\beta\) be the real numbers such that \(\lim _{x \rightarrow 0} \frac{1}{x^3}\left(\frac{\alpha}{2} \int_0^x \frac{1}{1-t^2} d t+\beta x \cos x\right)=2\). Then the value of \(\alpha+\beta\) is _______

Statement 1 For practical purposes, the earth is used as a reference at zero potential in electrical circuits.
and Statement 2 The electrical potential of a sphere of radius \(R\) with charge \(Q\) uniformly distributed on the surface is given by \(\frac{Q}{4 \pi \varepsilon_0 R}\).