The interval(s), where the function \(f(x)=\left\{\begin{array}{l}\frac{1-e^x}{e^{2 x}-1}: x \neq 0 \\ -\frac{1}{2}: x=0\end{array}\right.\) is increasing, is/are:

If the integral \(I=\int\left\{\log _e\left(\log _e x\right)^2+\frac{\alpha}{\log _e x}\right\} d x=x \log _e\left(\log _e x\right)^2+C\), where \(C\) is constant of integration. Then the value of \(\alpha\) is :

The value of p so that the lines \(\frac{x-1}{-3}=\frac{2 y-2}{2 p}=\frac{z-3}{2}\) and \(\frac{x-1}{-3 p}=\frac{y-1}{4}=\frac{6-z}{5}\) are at right angles is

Mr. Mittal invested Rs. 20, 000 in a mutual fund in the year 2019.
The value of the mutual fund increased to Rs. 32, 000 in the year 2024. The compound annual growth rate (CAGR) of his investment is :
[Given (1.6) \({ }^{1 / 5}=1.098\) ]

The value of \(\left|\begin{array}{ll}\sqrt{3} / 2 & 1 / 2 \\ \sqrt{3} / 2 & 1 / 2\end{array}\right|\)

Match List-I with List-II
The function \(f(x)=(x-1)(x+1)^2\) has

List-I	List-II
(A) A local maxima at \(x=\) __________	(I) \(\frac{1}{3}\)
(B) A local minima at x = __________	(II) \(0\)
(C) The local minimum value of \(f(x)=\) __________	(III) \(-1\)
(D) The local maximum value of \(f(x)=\) __________	(IV) \(-\frac{32}{27}\)

Choose the correct answer from the options given below :

If \(A=\left[a_{i j}\right]_{3 \times 2}\), where \(a_{i j}=i+j\), then

(A) A is a square matrix
(B) \(a_{21}+a_{32}=8\)
(C) Number of elements in A is 6
(D) Transpose of \(\mathrm{A}=\left[\begin{array}{ll}2 & 3 \\ 3 & 4 \\ 4 & 5\end{array}\right]\)
Choose the correct answer from the options given below:

Ethanol cannot be prepared from :

The thermal decomposition of HI on gold surface is an example of

Answer the question on basis of passage given below :
The noble gases have closed shell electronic configuration and are monatomic gases under normal conditions. The low boiling points of lighter noble gases are due to the weak dispersion forces between the atoms and the absence of other inter atomic interaction. The direct reaction of xenon with fluorine leads to a series of compounds with oxidation numbers \(+2, +4\) and \(+6\).
\(\text{XeF}_4\) reacts violently with water to give \(\text{XeO}_3\). The compound of Xenon exhibit stereoisomerism and their geometries can be deduced considering the total number of electrons in the valence shell.
\(X e F _4 \quad\) and \(X e F _6 \quad\) are expected to be :

The shortest wavelength in the Balmer series of a hydrogen atom is (Given: \(R=1.1 \times 10^7 m^{-1}\) )

If \(y=\tan ^{-1} x\) then \(\frac{d y}{d x}\) at \(y=\frac{\pi}{3}\) is equal to :

Read the following passage carefully and answer the given questions.
Theory of Evolution
The evolution of life is a gradual process through which simple organisms are transformed into complex life forms over billions of years. It involves genetic changes, adaptation to the environment, and the emergence of diversity in species. Fossil records provide evidence of extinct life forms and transitional stages. Evolution is influenced by factors such as mutation, genetic recombination, and environmental pressure, shaping the biodiversity we see today. Charles Darwin and Alfred Russel Wallace were pioneering naturalists who developed the theory of evolution by natural
selection. The voyage of the H.M.S. Beagle was significant in the development of Darwin's theory of evolution as-