A metal target with atomic number \(Z=46\) is bombarded with a high energy electron beam. The emission of X-rays from the target is analyzed. The ratio \(r\) of the wavelengths of the \(K_\alpha\)-line and the cut-off is found to be \(r=2\). If the same electron beam bombards another metal target with \(Z=41\), the value of \(r\) will be

In a particular system of units, a physical quantity can be expressed in terms of the electric charge $e$, electron mass $m_e$. Planck's constant $h$, and Coulomb's constant $k=\frac{1}{4\pi {\epsilon }_0}$, where ${\epsilon }_0$ is the permittivity of vacuum. In terms of these physical constants, the dimension of the magnetic field is $\left[B\right]={\left[e\right]}^{\alpha }{\left[m_e\right]}^{\beta }{\left[h\right]}^{\gamma }{\left[k\right]}^{\delta }$. The value of $\alpha +\beta +\gamma +\delta$ is _______.

Four charges ${\mathrm{Q}}_1,\mathrm{}{\mathrm{Q}}_2,\mathrm{}{\mathrm{Q}}_3$ and ${\mathrm{Q}}_4$ of same magnitude are fixed along the x axis at $\mathrm{x}=-2\mathrm{a},\mathrm{}-\mathrm{a},\mathrm{}+\mathrm{a}$ and $+2\mathrm{a}$ , respectively. A positive charge q is placed on the positive y axis at a distance $\mathrm{b}\mathrm{}>\mathrm{}0$. Four options of the signs of these charges are given in List I. The direction of the forces on the charge q is given in List II. Match List I with List II and select the correct answer using the code given below the lists.

	List I		List II
A.	${\mathrm{Q}}_1,\mathrm{}{\mathrm{Q}}_2,\mathrm{}{\mathrm{Q}}_3\mathrm{}{\mathrm{Q}}_4$ all positive	$\mathrm{P}.$	$+\mathrm{x}$
B.	${\mathrm{Q}}_1,\mathrm{}{\mathrm{Q}}_2$ positive; ${\mathrm{Q}}_3,\mathrm{}{\mathrm{Q}}_4$ negative	Q.	$-\mathrm{x}$
C.	${\mathrm{Q}}_1,\mathrm{}{\mathrm{Q}}_4$ positive; ${\mathrm{Q}}_2,\mathrm{}{\mathrm{Q}}_3$ negative	$\mathrm{R}.$	$+\mathrm{y}$
D.	${\mathrm{Q}}_1,\mathrm{}{\mathrm{Q}}_3$ positive; ${\mathrm{Q}}_2,\mathrm{}{\mathrm{Q}}_4$ negative	$\mathrm{S}.$	$-\mathrm{y}$

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\(P_{14-16}\) : Paragraph for Questions Nos. 14 to 16 A fixed thermally conducting cylinder has a radius \(R\) and height \(L_0\). The cylinder is open at its bottom and has a small hole at its top. A piston of mass \(M\) is held at a distance \(L\) from the top surface as shown in the figure. The atmospheric pressure is \(p_0\).

Question :
The piston is taken completely out of the cylinder. The hole at the top is sealed. A water tank is brought below the cylinder and put in a position so that the water surface in the tank is at the same level as the top of the cylinder as shown in the figure. The density of the water is \(\rho\). In equilibrium, the height \(H\) of the water column in the cylinder satisfies

Four identical thin, square metal sheets, \(S_1, S_2, S_3\) and \(S_4\), each of side \(a\) are kept parallel to each other with equal distance \(d(\ll a)\) between them, as shown in the figure. Let \(C_0=\varepsilon_0 a^2 / d\), where \(\varepsilon_0\) is the permittivity of free space.

Match the quantities mentioned in List-I with their values in List-II and choose the correct option.

A container with $1 \mathrm{kg}$ of water in it is kept in sunlight, which causes the water to get warmer than the surroundings. The average energy per unit time per unit area received due to the sunlight is $700 \mathrm{W} {\mathrm{m}}^{-2}$ and it is absorbed by the water over an effective area of $0.05 {\mathrm{m}}^2.$ Assuming that the heat loss from the water to the surroundings is governed by Newton's law of cooling, the difference (in ${}^{\circ }\mathrm{C}$) in the temperature of water and the surroundings after a long time will be_________. (Ignore effect of the container, and take constant for Newton's law of cooling $=0.001 {\mathrm{s}}^{-1},$ Heat capacity of water $=4200 \mathrm{J} {\mathrm{kg}}^{-1} {\mathrm{K}}^{-1}$)

The radius of the orbit of an electron in a Hydrogen-like atom is \(4.5 a _0\), where \(a _0\) is the Bohr radius. Its orbital angular momentum is \(\frac{3 h}{2 \pi}\). It is given that \(h\) is Planck's constant and R is Rydberg constant. the possible wavelength \((s)\), when the atom de-excites, is (are)

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

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Tollen's reagent is used for the detection of aldehyde when a solution of \(\mathrm{AgNO}_3\) is added to glucose with \(\mathrm{NH}_4 \mathrm{OH}\) then gluconic acid is formed
\(\mathrm{Ag}^{+}+e^{-} \rightarrow \mathrm{Ag} E_{\text {red }}^{\circ}=0.8 \mathrm{~V} \)
\( \mathrm{C}_6 \mathrm{H}_{12} \mathrm{O}_6+\mathrm{H}_2 \mathrm{O} \rightarrow \text { Gluconic acid }\)\(\left(\mathrm{C}_6 \mathrm{H}_{12} \mathrm{O}_7\right)+2 \mathrm{H}^{+}+2 e^{-} ; -E_{\mathrm{red}}^{\circ}=\) \(-0.05 \mathrm{~V} \)
\( \mathrm{Ag}\left(\mathrm{NH}_3\right)_2^{+}+e^{-} \rightarrow \mathrm{Ag}(s)+2 \mathrm{NH}_3 ;\) \(E_{\mathrm{red}}^{\circ}=0.337 \mathrm{~V}\)
[Use \(2.303 \times \frac{R T}{F}=0.0592\) and \(\frac{F}{R T}=38.92\) at \(298 \mathrm{~K}\) ]
Question:
Ammonia is always added in this reaction. Which of the following must be incorrect?

Let $f\left(x\right)=\sin \left(\pi \cos x\right)$ and $g\left(x\right)=\cos \left(2\pi \sin x\right)$ be two functions defined for $x>0.$ Define the following sets whose elements are written in the increasing order:
$X=\left\{x:f\left(x\right)=0\right\}, Y=\left\{x:f^{\text{'}}\left(x\right)=0\right\}$
$Z=\left\{x:g\left(x\right)=0\right\}, W=\left\{x:g^{\text{'}}\left(x\right)=0\right\}$
List- $I$ contains the sets $X, Y, Z$ and $W.$ List- $\mathrm{I}\mathrm{I}$ contains some information regarding these sets.
 

List- $\mathrm{I}$	List- $\mathrm{I}\mathrm{I}$
$\left(I\right)$	$X$	$\left(P\right)$	$\supseteq \left\{\frac{\pi }{2},\frac{3\pi }{2}, 4\pi , 7\pi \right\}$
$\left(II\right)$	$Y$	$\left(Q\right)$	an arithmetic progression
$\left(III\right)$	$Z$	$\left(R\right)$	NOT an arithmetic progression
$\left(IV\right)$	$W$	$\left(S\right)$	$\supseteq \left\{\frac{\pi }{6},\frac{7\pi }{6},\frac{13\pi }{6}\right\}$
		$\left(T\right)$	$\supseteq \left\{\frac{\pi }{3},\frac{2\pi }{3}, \pi \right\}$
		$\left(U\right)$	$\supseteq \left\{\frac{\pi }{6},\frac{3\pi }{4}\right\}$

Which of the following is the only correct combination?

Two soap bubbles \(A\) and \(B\) are kept in a closed chamber where the air is maintained at pressure \(8 \mathrm{Nm}^{-2}\). The radii of bubbles \(A\) and \(B\) are \(2 \mathrm{~cm}\), respectively. Surface tension of the soap-water used to make bubbles is \(0.04\) \(\mathrm{Nm}^{-1}\). Find the ratio \(\frac{n_B}{n_A}\), where \(n_A\) and \(n_B\) are the number of moles of air in bubbles \(A\) and \(B\), respectively. [Neglect the effect of gravity]

The number of solutions of the pair of equations \(2 \sin ^2 \theta-\cos 2 \theta=0\) and \(2 \cos ^2 \theta-3 \sin \theta=0\) in the interval \([0,2 \pi]\) is

Let $S$ be the of all column matrices $\left[\begin{array}{c}{b}_1\\ b_2\\ b_3\end{array}\right]$ such that $b_1, b_2, b_3\in \mathrm{ℝ}$ and the system of equations (in real variables)
$\begin{array}{c}-x+2y+5z=b_1\\ 2x-4y+3z=b_2\\ x-2y+2z=b_3\end{array}$
has at least one solution. Then, which of the following system(s) (in real variables) has (have) at least one solution of each $\left[\begin{array}{c}{b}_1\\ b_2\\ b_3\end{array}\right]ϵ S$ ?