When two progressive waves \(y_1=4 \sin (2 x-6 t) \quad\) and \(y_2=3 \sin \left(2 x-6 t-\frac{\pi}{2}\right) \quad\) are superimposed, the amplitude of the resultant wave is

A heavy nucleus $Q$ of half-life $20$ minutes undergoes alpha-decay with probability of $60\%$  and beta-decay with probability of $40\%$. Initially, the number of $Q$ nuclei is $1000$. The number of alpha-decays of $Q$ in the first one hour is:

Two rectangular blocks, having identical dimensions, can be arranged either in configuration$\text{I}$ or in configuration $\text{I}\text{I}$ as shown in the figure. One of the blocks has thermal conductivity $\text{K}$ and the other $\text{2 K}$. The temperature difference between the ends along the x-axis is the same in both the configurations. It takes $\text{9 s}$ to transport a certain amount of heat from the hot end to cold end in the configuration $\text{I}$. The time to transport the same amount of heat in the configuration $\text{II}$ is

During Searle's experiment, zero of the Vernier scale lies between $3.20\times {10}^{-2}\mathrm{ }\mathrm{m}$ and $3.25\times {10}^{-2}\mathrm{ }\mathrm{m}$ of the main scale. The ${20}^{\mathrm{t}\mathrm{h}}$ division of the Vernier scale exactly coincides with one of the main scale divisions. When an additional load of 2 kg is applied to the wire, the zero of the Vernier scale still lies between $3.20\times {10}^{-2\mathrm{ }}\mathrm{m}$ and $3.25\times {10}^{-2}\mathrm{ }\mathrm{m}$ of the main scale but now the ${45}^{\mathrm{t}\mathrm{h}}$ division of Vernier scale coincides with one of the main scale divisions. The length of the thin metallic wire is 2 m and its cross-sectional area is $8\times {10}^{-7}\mathrm{ }{\mathrm{m}}^2$ . The least count of the Vernier scale is $1.0\times \mathrm{ }{10}^{-5}\mathrm{ }\mathrm{m}$ . The maximum percentage error in the Young's modulus of the wire is

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The key feature of Bohr's theory of spectrum of hydrogen atom is the quantization of angular momentum when an electron is revolving around a proton. We will extend this to a general rotational motion to find quantized rotational energy of a diatomic molecule assuming it to be rigid. The rule to be applied is Bohr's quantization condition.
Question:
It is found that the excitation frequency from ground to the first excited state of rotation for the \(\mathrm{CO}\) molecule is close to \(\frac{4}{\pi} \times 10^{11} \mathrm{~Hz}\). Then the moment of inertia of CO molecule about its centre of mass is close to (Take \(h=2 \pi \times 10^{-34} J-s\) )

An infinitely long thin non-conducting wire is parallel to the z - axis and carries a uniform line charge density $\lambda$ . It pierces a thin non-conducting spherical shell of radius $R$ in such a way that the arc $PQ$ subtends an angle ${120}^o$ at the centre $O$ of the spherical shell, as shown in the figure. The permittivity of free space is ${ϵ}_0$ . Which of the following statements is (are) true?

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The nuclear charge \((\mathrm{Ze})\) is non-uniformly distributed within a nucleus of radius \(R\). The charge density \(\rho(r)\) [charge per unit volume] is dependent only on the radial distance \(r\) from the centre of the nucleus as shown in figure. The electric field is only along the radial direction.

Question:
The electric field at \(r=R\) is

The solubility of a salt of weak acid $\left(\mathrm{AB}\right)\mathrm{ }\mathrm{at}\mathrm{ }\mathrm{pH}3\mathrm{ }\mathrm{is}\mathrm{ }\mathrm{Y}\times {10}^{-3}\mathrm{ }\mathrm{mol}\mathrm{ }{\mathrm{L}}^{-1}$ . The value of $\mathrm{Y}$ is ____. (Given that the value of solubility product of $\mathrm{AB}\left({\mathrm{K}}_{\mathrm{sp}}\right)=2\times {10}^{-10}\mathrm{ }$, and the value of ionization constant of $\mathrm{HB}\left(\mathrm{Ka}\right)=1\times {10}^{-8}$)

A block of mass $2\mathrm{M}$ is attached to a massless spring with spring-constant $\mathrm{k}.$ This block is connected to two other blocks of masses $\mathrm{M}$ and $2\mathrm{M}$ using two massless pulleys and strings. The accelerations of the blocks are ${\mathrm{a}}_1,{\mathrm{a}}_2$ and ${\mathrm{a}}_3$ as shown in figure. The system is released from rest with the spring in its unstretched state. The maximum extension of the spring is ${\mathrm{x}}_0.$ Which of the following option(s) is/are correct?
[ $\mathrm{g}$ is the acceleration due to gravity. Neglect friction]

A Conductor (shown in the figure) carrying constant current $\mathrm{I}$ is kept in the $\mathrm{x}$ - $\mathrm{y}$ plane in a uniform magnetic field $\overrightarrow{\mathrm{B}}$ . If $\mathrm{F}$ is the magnitude of the total magnetic force acting on the conductor, then the correct statement(s) is (are)

The concentration of \(R\) in the reaction \(R \longrightarrow P\) was measured as a function of time and the following data is obtained :
\(\begin{array}{lllll}{[R] \text { (molar) }} & 1.0 & 0.75 & 0.40 & 0.10 \\ t \text { (min) } & 0.0 & 0.05 & 0.12 & 0.18\end{array}\)
The order of the reaction is

An electric dipole is formed by two charges $+q$ and $-q$ located in $xy$-plane at $(0, 2) \mathrm{mm}$ and $(0, -2) \mathrm{mm}$, respectively, as shown in the figure. The electric potential at point $P(100, 100) \mathrm{mm}$ due to the dipole is $V_0$. The charges $+q$ and $–q$ are then moved to the points $(-1, 2) \mathrm{mm}$ and $(1, -2) \mathrm{mm}$, respectively. What is the value of electric potential at $$

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