Let \(X\) and \(Y\) be two events such that \(P(X \mid Y)=\frac{1}{2}\), \(P(Y / X)=\frac{1}{3}\) and \(P(X \cap Y)=\frac{1}{6}\). Which of the following is (are) correct?

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

Let ${\text{S}}_{\text{n}}=\underset{\text{k}=1}{\overset{4\text{n}}{\Sigma }}{\left(-1\right)}^{\frac{\text{k}\left(\text{k}+1\right)}{2}}{\text{k}}^2$. Then S_n can take value(s)

Let $f:R\to R$ be given by $f\left(x\right)$ $=\left\{\begin{array}{cc}{\mathrm{x}}^5+5{\mathrm{x}}^4+10{\mathrm{x}}^3+10{\mathrm{x}}^2+3\mathrm{x}+1,& \mathrm{x}<0;\\ {\mathrm{x}}^2-\mathrm{x}+1,& 0\le \mathrm{x}<1;\\ \frac{2}{3}{\mathrm{x}}^3-4{\mathrm{x}}^2+7\mathrm{x}-\frac{8}{3},& 1\le \mathrm{x}<3;\\ \left(\mathrm{x}-2\right){\mathrm{l}\mathrm{o}\mathrm{g}}_{\mathrm{e}}\left(\mathrm{x}-2\right)-\mathrm{x}+\frac{10}{3},& \mathrm{x}\ge 3\end{array}\right.$
Then which of the following options is/are correct?

Consider the lines $L_1\text{:}\frac{x-1}{2}=\frac{y}{-1}=\frac{z+3}{1}\text{,}{L}_2\text{:}\frac{x-4}{1}=\frac{y+3}{1}=\frac{z+3}{2}$ and the planes $P_1\text{:}7x+y+2z=3\text{,}{P}_2\text{:}3x+5y-6z=4\text{.}$ Let $ax+by+cz=d$ be the equation of the plane passing through the point of intersection of L₁ and L₂, and perpendicular to planes P₁ and P₂.
Match List- I with List - II and select the correct answer using the code given below the lists :
 

	List I		List II
A.	a =	P.	13
B.	b =	Q.	-3
C.	c =	R.	1
D.	d =	S.	-2

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Let \(M=\left\{(x, y) \in \mathbb{R} \times \mathbb{R}: x^{2}+y^{2} \leq r^{2}\right\}\),
where \(r>0 .\) Consider the geometric progression \(a_{n}=\frac{1}{2^{n-1}}, n=1,2,3, \ldots .\) Let \(S_{0}=0\) and, for \(n \geq 1\), let \(S_{n}\) denote the sum of the first \(n\) terms of this progression. For \(n \geq 1\), let \(C_{n}\) denote the circle with center \(\left(S_{n-1}, 0\right)\) and radius \(a_{n}\), and \(D_{n}\) denote the circle with center \(\left(S_{n-1}, S_{n-1}\right)\) and radius \(a_{n}\).

Question:
Consider \(M\) with \(r=\frac{\left(2^{199}-1\right) \sqrt{2}}{2^{198}} .\) The number of all those circles \(D_{n}\) that are inside \(M\) is

The tangent to the curve \(y=e^x\) drawn at the point \(\left(c, e^c\right)\) intersects the line joining the points \(\left(c-1, e^{c-1}\right)\) and \(\left(c+1, e^{c+1}\right)\)

A temperature difference can generate e.m.f. in some materials. Let \(S\) be the e.m.f. produced per unit temperature difference between the ends of a wire, \(\sigma\) the electrical conductivity and \(\kappa\) the thermal conductivity of the material of the wire. Taking \(M, L, T, I\) and \(K\) as dimensions of mass, length, time, current and temperature, respectively, the dimensional formula of the quantity \(Z=\frac{S^2 \sigma}{\kappa}\) is :-

A container of fixed volume has a mixture of one mole of hydrogen and one mole of helium in equilibrium at temperature $\mathrm{T}$ . Assuming the gases are ideal, the correct statement(s) is (are)

The crystalline form of borax has

The number of neutrons emitted when \({ }_{92}^{235} \mathrm{U}\) undergoes controlled nuclear fission to \({ }_{54}^{142} \mathrm{Xe}\) and \({ }_{38}^{90} \mathrm{Sr}\) is

Let $\mathrm{\Delta }\mathrm{P}\mathrm{Q}\mathrm{R}$ be a triangle. Let $\overrightarrow{\mathrm{a}}=\mathrm{ }\overrightarrow{\mathrm{Q}\mathrm{R}},\mathrm{ }\mathrm{ }\overrightarrow{\mathrm{b}}=\mathrm{ }\overrightarrow{\mathrm{R}\mathrm{P}}\mathrm{ }\mathrm{a}\mathrm{n}\mathrm{d}\mathrm{ }\overrightarrow{\mathrm{c}}=\mathrm{ }\overrightarrow{\mathrm{P}\mathrm{Q}}.$ If $\left|\overrightarrow{\mathrm{a}}\right|=12,\mathrm{ }\left|\overrightarrow{\mathrm{b}}\right|=4\sqrt{3}$ and $\overrightarrow{\mathrm{b}}\mathrm{ }\cdot \mathrm{ }\overrightarrow{\mathrm{c}}=24,$ then which of the following is (are) true ?

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\(RCONH _2\) is converted into \(RNH _2\) by means of Hofmann bromamide degradation.

In this reaction, \(R \mathrm{CONHBr}\) is formed from which this reaction has derived its name. Electron donating group at phenyl activates the reaction. Hofmann degradation reaction is an intramolecular reaction.
Question:
How can the conversion of (i) to (ii) be brought about?