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Let \(A\) be the set of all \(3 \times 3\) symmetric matrices all of whose entries are either 0 or 1 . Five of these entries are 1 and four of them are 0 .Question:
The number of matrices \(A\) in \(A\) for which the system of linear equations \(A\left[\begin{array}{l}x \\ y \\ z\end{array}\right]=\left[\begin{array}{l}1 \\ 0 \\ 0\end{array}\right]\) is is inconsistent, is

If \(\lim _{x \rightarrow \infty}\left(\frac{x^{2}+x+1}{x+1}-a x-b\right)=4\), then

For a real number $\alpha ,$ if the system $\left[\begin{array}{ccc}1& \alpha & {\alpha }^2\\ \alpha & 1& \alpha \\ {\alpha }^2& \alpha & 1\end{array}\right] \left[\begin{array}{c}x\\ y\\ z\end{array}\right]=\left[\begin{array}{c}1\\ -1\\ 1\end{array}\right]$ of linear equations, has infinitely many solutions, then $1+\alpha +{\alpha }^2=$

Let $E_1=${$x\in \mathrm{ℝ}: x\ne 1$ and $\frac{x}{x-1}>0$} and $E_2=${$x\in E_1: {\sin }^{-1}\left({\log }_e\left(\frac{x}{x-1}\right)\right)$ is a real number} (Here the inverse trigonometric function ${\sin }^{-1}\left(x\right)$ assumes values in $\left[-\frac{\mathrm{\pi }}{2}. \frac{\mathrm{\pi }}{2}\right]$.)
Let $f:E_1\to \mathrm{ℝ}$ be the function defined by $f\left(x\right)={\log }_e\left(\frac{x}{x-1}\right)$
And $g:E_2\to \mathbb{R}$ be the function defined by $g\left(x\right)={\sin }^{-1}\left({\log }_e\left(\frac{x}{x-1}\right)\right)$ .

LIST-I	LIST-II
A. The range of $f$ is	P. $\left(-\infty ,\frac{1}{1-e}\right]\cup \left[\frac{e}{e-1}, \infty \right)$
B. The range of $g$ contains	Q. (0, 1)
C. The domain of $f$ contains	R. $\left[-\frac{1}{2},\frac{1}{2}\right]$
D. The domain of $g$ is	S. $\left(-\infty ,0\right)\cup \left(0, \infty \right)$
	T. $\left(-\infty ,\frac{e}{e-1}\right]$
	U. $\left(-\infty ,0\right)\cup \left(\frac{1}{2},\frac{e}{e-1}\right]$

The correct option is:

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

Let \(P Q R\) be a triangle of area \(\Delta\) with \(a=2, b=\frac{7}{2}\) and \(c=\frac{5}{2} ;\) where \(a, b\), and \(c\) are the lengths of the sides of the triangle opposite to the angles at \(P, Q\) and \(R\) respectively. Then \(\frac{2 \sin P-\sin 2 P}{2 \sin P+\sin 2 P}\) equals.

If \(\vec{a}\) and \(\vec{b}\) are vectors such that \(|\vec{a}+\vec{b}|=\sqrt{29}\) and \(\vec{a} \times(2 \hat{i}+3 \hat{j}+4 \hat{k})=(2 \hat{i}+3 \hat{j}+4 \hat{k}) \times \vec{b}\), then a possible value of \((\vec{a}+\vec{b}) \cdot(-7 \hat{i}+2 \hat{j}+3 \hat{k})\) is

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Properties such as boiling point, freezing point and vapour pressure of a pure solvent change when solute molecules are added to get homogeneous solution. These are called colligative properties. Applications of colligative properties are very useful in day-to-day life. One of its examples is the use of ethylene glycol and water mixture as anti-freezing liquid in the radiator automobiles.
A solution \(M\) is prepared by mixing ethanol and water. The mole fraction of ethanol in the mixtrue is \(0.9\)
Given Freezing point depression constant of water \(\left(k^{\text {water }}\right)=1.86 \mathrm{~K} \mathrm{~kg} \mathrm{~mol}^{-1}\)
Freezing point depression constant of ethanol \(\left(k_f^{\text {ethanol }}\right)=2.0 \mathrm{Kkg} \mathrm{mol}^{-1}\)
Boiling point elevation constant of water \(\left(k_b^{\text {water }}\right)=0.52 \mathrm{Kkg} \mathrm{mol}^{-1}\)
Boiling point elevation constant of ethanol \(\left(k_b^{\text {ethanol }}\right)=1.2 \mathrm{Kkg} \mathrm{mol}^{-1}\)
Standard freezing point of water \(=273 \mathrm{~K}\)
Standard freezing point of ethanol \(=155.7 \mathrm{~K}\)
Standard boiling point of water \(=373 \mathrm{~K}\)
Standard boiling point of ethanol \(=351.5 \mathrm{~K}\)
Vapour pressure of pure water \(=328 \mathrm{~mm}\) of \(\mathrm{Hg}\)
Vapour pressure of pure ethanol \(=40 \mathrm{~mm}\) of \(\mathrm{Hg}\)
Molecular weight of water \(=18 \mathrm{~g} \mathrm{~mol}^{-1}\)
Molecular weight of ethanol \(=46 \mathrm{~g} \mathrm{~mol}^{-1}\)
In answering the following questions, consider the solutions to be ideal dilute solutions and solutes to be non-volatile and non-dissociative.

Question:
The vapour pressure of the solution \(M\) is

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There are \(n\) urns each containing \((n+1)\) balls such that the ith urn contains \(i\) white balls and \((n+1-i)\) red balls. Let \(u_i\) be the event of selecting ith urn, \(i=1,2,3, \ldots, n\) and \(W\) denotes the event of getting a white balls.Question:
If \(P\left(u_i\right)=c\), where \(c\) is a constant, then \(P\left(u_i / W\right)\) is equal to

The number of real solutions of the equation ${\sin }^{-1}\left(\sum _{i=1}^{\infty }{x}^{i+1}-x\sum _{i=1}^{\infty }{\left(\frac{x}{2}\right)}^i\right)=\frac{\pi }{2}-{\cos }^{-1}\left(\sum _{i=1}^{\infty }{\left(-\frac{x}{2}\right)}^i-\sum _{i=1}^{\infty }{\left(-x\right)}^i\right)$ lying in the interval $\left(-\frac{1}{2},\frac{1}{2}\right)$ is____.
(Here, the inverse trigonometric functions ${\sin }^{-1}x and {\cos }^{-1}x$ assume values in $\left[-\frac{\pi }{2},\frac{\pi }{2}\right] \mathrm{a}\mathrm{n}\mathrm{d} \left[0,\pi \right]$
respectively.)

The total number of lone pairs of electrons in ${\mathrm{N}}_2{\mathrm{O}}_3$ is

A parallel beam of light is incident from air at an angle $\alpha$ on the side PQ of a right angled triangular prism of refractive index $n=\sqrt{2}$ . Light undergoes total internal reflection in the prism at the face PR when $\alpha$ has a minimum value of ${45}^o.$ The angle $\theta$ of the prism is:

The species formed on fluorination of phosphorus pentachloride in a polar organic solvent are :