Let a solution \(y=y(x)\) of the differential equation \(x \sqrt{x^2-1} d y-y \sqrt{y^2-1} d x=0\) satisfy \(y(2)=\frac{2}{\sqrt{3}}\)
Statement \(1 y(x)=\sec \left(\sec ^{-1} x-\frac{\pi}{6}\right)\)
Statement \(2 y(x)\) is given by \(\frac{1}{y}=\frac{2 \sqrt{3}}{x}-\sqrt{1-\frac{1}{x^2}}\)

Let \(\mathbb{R}\) denote the set of all real numbers. Let \(f: \mathbb{R} \rightarrow \mathbb{R}\) be a function such that \(f(x)>0\) for all \(x \in \mathbb{R}\), and \(f(x+y)=f(x) f(y)\) for all \(x, y \in \mathbb{R}\).
Let the real numbers \(\mathrm{a}_1, \mathrm{a}_2 \ldots, \mathrm{a}_{50}\) be in an arithmetic progression. If \(f\left(\mathrm{a}_{31}\right)=64 f\left(\mathrm{a}_{25}\right)\), and \(\sum_{i=1}^{50} f\left(a_i\right)=3\left(2^{25}+1\right)\) then the value of \(\sum_{i=6}^{30} f\left(a_i\right)\) is ________

\[
\text { Match the conics in Column I with the statements/expressions in Column II. }
\]

Consider the matrix, \(P=\left(\begin{array}{lll}2 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 3\end{array}\right)\)
Let the transpose of a matrix \(X\) be denoted by \(\mathrm{X}^T\). Then the number of \(3 \times 3\) invertible matrices Q with integer entries, such that \(Q^{-1}=Q^T\) and \(P Q=Q P\) is

Let $w=\frac{\sqrt{3}+\text{i}}{2}$ and $\text{P}=\left\{w^{\text{n}}\text{: }\text{n}=\text{1, 2, 3,}\dots \right\}$.Further ${\text{H}}_1=\left\{\text{z}\in \text{C}\text{ : }\text{Re z}>\frac{1}{2}\right\}$ and ${\text{H}}_2=\left\{\text{z}\in \text{C}\text{ : }\text{Re z}<\frac{-1}{2}\right\}$, where C is the set of all complex numbers. If ${\text{z}}_1\in \text{P}\cap {\text{H}}_1\text{, }{\text{z}}_2\in \text{P}\cap {\text{H}}_2$ and O represents the origin, then $\angle {\text{z}}_1\text{O}{\text{z}}_2=$

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Let \(V_r\) denotes the sum of the first \(r\) terms of an arithmetic progression \((A P)\) whose first term is \(r\) and the common difference is \((2 r-1)\). Let \(T_r=V_{r+1}-V_r-2\) and \(Q_r=T_{r+1}-T_r\) for \(r=1,2, \ldots\)
Question:
\(T_r\) is always

Let m and n be two positive integers greater than 1. If $\underset{\alpha \to 0}{\lim }\left(\frac{e^{\mathrm{c}\mathrm{o}\mathrm{s}\left({\alpha }^n \right)}-e}{{\alpha }^m}\right)= -\left(\frac{e}{2}\right)$ then the value of $\frac{m}{n}$ is

Six point charges, each of the same magnitude \(q\), are arranged in different manners as shown in Column-II. In each case, a point \(M\) and a line \(P Q\) passing through \(M\) are shown. Let \(E\) be the electric field and \(V\) be the electric potential at \(M\) (potential at infinity is zero) due to the given charge distribution when it is at rest. Now, the whole system is set into rotation with a constant angular velocity about the line \(P Q\). Let \(B\) the magnetic field at \(M\) and \(\mu\) be the magnetic moment of the system in this condition. Assume each rotating charge to be equivalent to a steady current.

Consider an experiment of tossing a coin repeatedly until the outcomes of two consecutive tosses are same. If the probability of a random toss resulting in head is $\frac{1}{3}$, then the probability that the experiment stops with head is

The correct order of the wavelength maxima of the absorption band in the ultraviolet-visible region for the given complexes is

Let \(S=\left\{A=\left(\begin{array}{lll}0 & 1 & c \\ 1 & a & d \\ 1 & b & e\end{array}\right): a, b, c, d, e \in\{0,1\}\right.\) and \(\left.|A| \in\{-1,1\}\right\}\), where \(|A|\) denotes the determinant of \(A\). Then the number of elements in \(S\) is

For $x\in R$, the number of real roots of the equation $3x^2-4\left|x^2-1\right|+x-1=0$ is

At 300 K , an ideal dilute solution of a macromolecule exerts osmotic pressure that is expressed in terms of the height ( h ) of the solution (density \(=1.00 \mathrm{~g} \mathrm{~cm}^{-3}\) ) where h is equal to 2.00 cm . If the concentration of the dilute solution of the macromolecule is \(2.00 \mathrm{~g} \mathrm{dm}^{-3}\), the molar mass of the macromolecule is calculated to be \(\boldsymbol{X} \times 10^4 \mathrm{~g} \mathrm{~mol}^{-1}\). The value of \(\boldsymbol{X}\) is \(\qquad\) .
Use : Universal gas constant \((\mathrm{R})=8.3 \mathrm{~J} \mathrm{~K}^{-1} \mathrm{~mol}^{-1}\) and acceleration due to gravity \((\mathrm{g})=10 \mathrm{~m} \mathrm{~s}^{-2}\)