Let the point $\mathrm{B}$ be the reflection of the point $\mathrm{A}\left(2,\mathrm{ }3\right)$ with respect to the line $8\mathrm{x}-6\mathrm{y}-23=0.$ Let ${\mathrm{T}}_{\mathrm{A}}$ and ${\mathrm{T}}_{\mathrm{B}}$ be circle of radii $2$ and $1$ with centres $\mathrm{A}$ and $\mathrm{B}$ respectively. Let $\mathrm{T}$ be a common tangent to the circle ${\mathrm{T}}_{\mathrm{A}}$ and ${\mathrm{T}}_{\mathrm{B}}$ such that both the circle are on the same side of $\mathrm{T}.$ If $\mathrm{C}$ is the point of intersection of $\mathrm{T}$ and the line passing through $\mathrm{A}$ and $\mathrm{B},$ then the length of the line segment $\mathrm{A}\mathrm{C}$ is ___________.

The product of all positive real values of $x$ satisfying the equation $x^{\left(16{\left({\log }_{5}x\right)}^3-68{\log }_{5}x\right)}=5^{-16}$ is

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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:
The sum \(V_1+V_2+\ldots+V_n\) is

let $f_1 :R\to R, f_2 :\left[0, \infty \right)\to R, f_3 :R\to R and f_4 :R\to [0, \infty )$ be defined by
$f_1\left(x\right)=\left\{\begin{array}{ccc}\left|x\right|& if& x<0\\ e^x& if& x\ge 0\end{array}\right. ;f_2\left(x\right)=x^2;f_3\left(x\right)=\left\{\begin{array}{ccc}\sin x& if& x<0\\ x& if& x\ge 0\end{array}\right. and f_4\left(x\right)=\left\{\begin{array}{ccc}{f}_2\left(f_1\left(x\right)\right)& if& x<0\\ f_2\left(f_1\left(x\right)\right)-1& if& x\ge 0\end{array}\right.$
 

	List – I		List – II
(A)	$f_4$ is	(P)	Onto but not one – one
(B)	$f_3$ is	(Q)	neither continuous nor one-one
(C)	$f_2$ of $f_1$ is	(R)	differentiable but not one-one
(D)	$f_2$ is	(S)	continuous and one-one

Let \(H_1, H_2, \ldots, H_n\) be mutually exclusive events with \(P\left(H_i\right)>0, i=1,2, \ldots, n\). Let \(E\) be any other event with \(0 < P(E) < 1\).
Statement I \(P\left(H_i / E\right)>P\left(E / H_i\right) P\left(H_i\right)\) for \(i=1,2, \ldots, n\).
Statement II \(\sum_{i=1}^n P\left(H_i\right)=1\)

The area enclosed by the curves $y=\sin x+\cos x$ and $\text{y}=\left|\text{cos x}-\text{sin x}\right|$ over the interval $\left[0\text{,}\frac{\pi }{2}\right]$ is

If \(f(x)=\min \left\{1, x^2, x^3\right\}\), then

The correct statement about the following disaccharide is

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A special metal \(S\) conducts electricity without any resistance. A closed wire loop, made of \(S\), does not allow any change in flux through itself by inducing a suitable current to generate a compensating flux. The induced current in the loop cannot decay due to its zero resistance. This current gives rise to a magnetic moment which in turn repels the source of magnetic field or flux. Consider such a loop, of radius \(a\), with its center at the origin. A magnetic dipole of moment \(m\) is brought along the axis of this loop from infinity to a point at distance \(r(\gg a)\) from the center of the loop with its north pole always facing the loop, as shown in the figure below.

The magnitude of magnetic field of a dipole \(m\), at a point on its axis at distance \(r\), is \(\frac{\mu_{0}}{2 \pi} \frac{m}{r^{3}}\), where \(\mu_{0}\) is the permeability of free space. The magnitude of the force between two magnetic dipoles with moments, \(m_{1}\) and \(m_{2}\), separated by a distance \(r\) on the common axis, with their north poles facing each other, is \(\frac{k m_{1} m_{2}}{r^{4}}\), where \(k\) is a constant of appropriate dimensions. The direction of this force is along the line joining the two dipoles.

Question :
When the dipole m is placed at a distance r from the centre of the loop (as shown in the figure), the current induced in the loop will be proportional to:

The correct statement(s) about O₃ is(are)

When benzene sulphonic acid and \(p\)-nitrophenol are treated with \(\mathrm{NaHCO}_3\), the gases released respectively are

To find the distance d over which a signal can be seen clearly in foggy conditions, a railways engineer uses dimensional analysis and assumes that the distance depends on the mass density $\mathrm{\rho }$ of the fog, intensity (power/area) S of the light from the signal and its frequency f. The engineer finds that d is proportional to ${\mathrm{S}}^{\frac{1}{\mathrm{n}}}$ . The value of n is

The filament of a light bulb has surface area $64 {\mathrm{mm}}^2.$ The filament can be considered as a black body at temperature $2500 \mathrm{K}$ emitting radiation like a point source when viewed from far. At night the light bulb is observed from a distance of $100 \mathrm{m}$. Assume the pupil of the eyes of the observer to be circular with radius $3 \mathrm{mm}$. Then: (Take Stefan-Boltzmann constant $=5.67\times {10}^{-8} \mathrm{W} {\mathrm{m}}^{-2} {\mathrm{K}}^{-4},$ Wiens' displacement constant $=2.90\times {10}^{-3} \mathrm{m} \mathrm{K}$, Planck's constant $=6.60\times {10}^{-34} \mathrm{J} \mathrm{s}$, speed of light in vacuum $=3.00\times {10}^8 \mathrm{m}{\mathrm{s}}^{-1}$)