Let $f\left(x\right)=\frac{1-x(1+\left|1-x\right|)}{|1-x|}\cos \left(\frac{1}{1-x}\right) \mathrm{for} x\ne 1,$ then

The number of points in $\left(-\infty ,\infty \right)$, for which $x^2-x\sin x-\cos x=0$, is:

Let $E, F$ and $G$ be three events having probabilities $P\left(E\right)=\frac{1}{8},P\left(F\right)=\frac{1}{6}$ and $P\left(G\right)=\frac{1}{4},$ and let $P(E\cap F\cap G)=\frac{1}{10}$
For any event $H$, if $H^c$ denotes its complement, then which of the following statements is (are) TRUE ?

Suppose $a, b$ denote the distinct real roots of the quadratic polynomial $x^2+20x-2020$ and suppose $c, d$ denote the distinct complex roots of the quadratic polynomial $x^2-20x+2020$. Then the value of $ac\left(a-c\right)+ad\left(a-d\right)+bc\left(b-c\right)+bd\left(b-d\right)$

Let $f:\left(0, \pi \right)\to \mathbb{R}$ be a twice differentiable function such that $\underset{t\to x}{\lim }\frac{f\left(x\right)\sin t-f\left(t\right)\sin x}{t-x}={\sin }^{2}x$ for all $x\in \left(0, \pi \right)$. If $f\left(\frac{\pi }{6}\right)=-\frac{\pi }{12}$, then which of the following statement(s) is (are) TRUE ?

From a point $P\left(\lambda , \lambda , \lambda \right),$ perpendiculars PQ and PR are drawn respectively on the lines $y=x, z=1$ and $y= -x, z= -1$ . If P is such that $\angle QPR$ is a right angle, then the possible value (s) of $\lambda$ is (are)

Match List I with List II and select the correct answer using the code given below the lists :
 

	List - I		List - II
A.	${\left(\frac{1}{{\text{y}}^2}\left(\frac{\text{cos}\left({\text{tan}}^{-1}\text{y}\right)+\text{y sin}\left({\text{tan}}^{-1}\text{y}\right)}{\text{cot}\left({\text{sin}}^{-1}\text{y}\right)+\text{tan}\left({\text{sin}}^{-1}\text{y}\right)}\right)+{\text{y}}^4\right)}^{1/2}takes\; value$	P.	$\frac{1}{2}\sqrt{\frac{5}{3}}$
B.	​If cosx + cosy + cosz = 0 = sinx + siny + sinz then possible value of cos $\frac{\text{x}-\text{y}}{2}is$	Q.	$\sqrt{2}$
C.	If cos $\left(\frac{\pi }{4}-\text{x}\right)$ cos2x + sinx sin 2x secx = cosx sin2x secx + cos $\left(\frac{\pi }{4}+\text{x}\right)$ cos2x then possible value of secx is	R.	$\frac{1}{2}$
D.	If cot $\left({\text{sin}}^{-1}\sqrt{1-{\text{x}}^2}\right)=\text{sin}\left({\text{tan}}^{-1}\left(\text{x}\sqrt{6}\right)\right)\text{,}x\ne 0\text{,}$  then possible value of x is	S.	1

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Let \(U_1\) and \(U_2\) be two urns such that \(U_1\) contains 3 white and 2 red balls and \(U_2\) contains only 1 white ball. A fair coin is tossed. If head appears then 1 ball is drawn at random from \(U_1\) and put into \(U_2\). However, if tail appears then 2 balls are drawn at random from \(U_1\) and put into \(U_2\). Now, 1 ball is drawn at random from \(U_2\).Question:
Given that the drawn ball from \(U_2\) is white, the probability that head appeared on the coin is

Consider a Vernier callipers in which each 1 cm on the main scale is divided into 8 equal divisions and a screw gauge with 100 divisions on its circular scale. In the Vernier callipers, 5 divisions of the Vernier scale coincide with 4 divisions on the main scale and in the screw gauge, one complete rotation of the circular scale moves it by two divisions on the linear scale. Then:

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In a mixture of \(\mathrm{H}-\mathrm{H}^{+}\)gas \(\left(\mathrm{He}^{+}\right.\)is singly ionised \(\mathrm{He}\) atom), \(\mathrm{H}\) atoms and \(\mathrm{He}^{+}\)ions are excited to their respective first excited states. Subsequently, \(\mathrm{H}\) atoms transfer their total excitation energy to \(\mathrm{He}^{+}\)ions (by collisions). Assume that the Bohr model of atom is exactly valid.
Question:
The quantum number \(n\) of the state finally populated in \(\mathrm{He}^{+}\)ions is

The hyperconjugative stabilities of tert-butyl cation and 2-butene, respectively, due to 

Vessel-1 contains \(\mathbf{w}_2 \mathrm{~g}\) of a non-volatile solute \(\mathbf{X}\) dissolved in \(\mathbf{w}_1 \mathrm{~g}\) of water. Vessel- 2 contains \(\mathbf{w}_2 \mathrm{~g}\) of another non-volatile solute \(\mathbf{Y}\) dissolved in \(\mathbf{w}_1 \mathrm{~g}\) of water. Both the vessels are at the same temperature and pressure. The molar mass of \(\mathbf{X}\) is \(80 \%\) of that of \(\mathbf{Y}\). The van't Hoff factor for \(\mathbf{X}\) is 1.2 times of that of \(\mathbf{Y}\) for their respective concentrations.
The elevation of boiling point for solution in Vessel-1 is ________. \(\%\) of the solution in Vessel-2.

Which of the following liberates ${\mathrm{O}}_2$ upon hydrolysis?