If \(w=\alpha+i \beta\), where \(\beta \neq 0\) and \(z \neq 1\) satisfies the condition that \(\left(\frac{w-\bar{w} z}{1-z}\right)\) is purely real, then the set of values of \(z\) is

The edges of a parallelopiped are of unit length and are parallel to non-coplanar unit vectors \(\hat{\mathbf{a}}, \hat{\mathbf{b}}, \hat{\mathbf{c}}\) such that \(\hat{\mathbf{a}} \cdot \hat{\mathbf{b}}=\hat{\mathbf{b}} \cdot \hat{\mathbf{c}}=\hat{\mathbf{c}} \cdot \hat{\mathbf{a}}=\frac{1}{2}\). Then, the volume of the parallelopiped is

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 $P$ be the image of the point$(3,1,7)$ with respect to the plane $x-y+z=3.$ Then the equation of the plane passing through $P$ and containing the straight line $\frac{x}{1}=\frac{y}{2}=\frac{z}{1}$ is

Let \(P\left(x_1, y_1\right)\) and \(Q\left(x_2, y_2\right)\) be two distinct points on the ellipse
\(\frac{x^2}{9}+\frac{y^2}{4}=1\)
such that \(y_1>0\), and \(y_2>0\). Let C denote the circle \(x^2+y^2=9\), and \(M\) be the point \((3,0)\).
Suppose the line \(x=x_1\) intersects \(C\) at \(R\), and the line \(x=x_2\) intersects \(C\) at \(S\), such that the \(y\)-coordinates of \(R\) and \(S\) are positive. Let \(\angle R O M=\frac{\pi}{6}\) and \(\angle S O M=\frac{\pi}{3}\), where \(O\) denotes the origin \((0,0)\). Let \(|X Y|\) denote the length of the line segment \(X Y\).
Then which of the following statements is (are) TRUE?

Let \(a_1, a_2, a_3, \ldots ., a_{100}\) be an arithmetic progression with \(a_1=3\) and \(S_p=\sum_{i=1}^p a_i\), \(1 \leq p \leq 100\). For any integer \(n\) with \(1 \leq n \leq 20\), let \(m=5 n\). If \(\frac{S_m}{S_n}\) does not depend on \(n\), then \(a_2\) is

Suppose that all the terms of an arithmetic progression (A.P.) are natural numbers. If the ratio of the sum of the first seven terms to the sum of the first eleven terms is $6:11$ and the seventh terms lies in between $130$ and $140$, then the common difference of this A.P. is

The solubility of a salt of weak acid $\left(\mathrm{AB}\right)\mathrm{ }\mathrm{at}\mathrm{ }\mathrm{pH}3\mathrm{ }\mathrm{is}\mathrm{ }\mathrm{Y}\times {10}^{-3}\mathrm{ }\mathrm{mol}\mathrm{ }{\mathrm{L}}^{-1}$ . The value of $\mathrm{Y}$ is ____. (Given that the value of solubility product of $\mathrm{AB}\left({\mathrm{K}}_{\mathrm{sp}}\right)=2\times {10}^{-10}\mathrm{ }$, and the value of ionization constant of $\mathrm{HB}\left(\mathrm{Ka}\right)=1\times {10}^{-8}$)

Consider the following complex ions, P, Qand R.

The correct order of the complex ions, according to their spin-only magnetic moment values (in B.M.) is

Let $X$ be the set consisting of the first $2018$ terms of the arithmetic progression $\mathrm{1,6},11,\dots ,$ and $Y$ be the set consisting of the first $2018$ terms of the arithmetic progression $\mathrm{9,16,23},\dots$ . Then, the number of elements in the set $X\cup Y$ is___.

Three randomly chosen non negative integers $x,y,z$ are found to satisfy the equation  $x+y+z=10$ . Then the probability that $z$ is even, is
 

Paragraph:

A box \(B_{1}\) contains \(1\) white ball, \(3\) red balls and \(2\) black balls. Another box \(B_{2}\) contains \(2\) white balls, \(3\) red balls and \(4\) black balls. A third box \(B_{3}\) contains \(3\) white balls, \(4\) red balls and \(5\) black balls.


Question:

If \(1\) ball is drawn from each of the boxes \(B_{1}, B_{2}\) and \(B_{3}\), the probability that all \(3\) drawn balls are of the same colour is

Two spherical stars $A$ and $B$ have densities ${\rho }_A$ and ${\rho }_B$, respectively. $A$ and $B$ have the same radius, and their masses $M_A$ and $M_B$ are related by $M_B=2M_A$. Due to an interaction process, star $A$ loses some of its mass, so that its radius is halved, while its spherical shape is retained, and its density remains ${\rho }_A$. The entire mass lost by $A$ is deposited as a thick spherical shell on $B$ with the density of the shell being ${\rho }_A$. If $v_A$ and $v_B$ are the escape velocities from $A$ and $B$ after the interaction process, the ratio $\frac{v_B}{v_A}=\sqrt{\frac{10n}{{15}^{{\displaystyle \frac{1}{3}}}}}$. The value of $n$ is