A hydrogen atom in its ground state is irradiated by light of wavelength 970Å. Taking $hc/e =1.237\times 10^{-6} eVm$ and the ground state energy of hydrogen atom as - 13.6 eV the number of lines present in the emission spectrum is

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1

A hydrogen atom in its ground state is irradiated by light of wavelength 970Å. Taking $hc/e =1.237\times 10^{-6} eVm$ and the ground state energy of hydrogen atom as - 13.6 eV the number of lines present in the emission spectrum is

2

Consider two solid spheres P and Q each of density 8 gm cm^{-3} and diameters 1 cm and 0.5 cm, respectively. Sphere P is dropped into a liquid of density 0.8 gm cm^{-3} and viscosity η = 3 poiseulles. Sphere Q is dropped into a liquid of density 1.6 gm cm^{-3} and viscosity η = 2 poiseulles. The ratio of the terminal velocities of P and Q is

3

A metal is heated in a furnace where a sensor is kept above the metal surface to read the power radiated (P) by the metal. The sensor has a scale that displays \log(P/P_{0}) where P_{0} is a constant. When the metal surface is at a temperature of 487^{0}C, the sensor shows a value 1. Assume that the emissivity of the metallic surface remains constant. What is the value displayed by the sensor when the temperature of the metal surface is raised to 2767^{0}C?

4

A frame of the reference that is accelerated with respect to an inertial frame of reference is called a non- inertial frame of reference. A coordinate system fixed on a circular disc rotating about a fixed axis with a constant angular velocity ω is an example of a non-inertial frame of reference. The relationship between the force $\overrightarrow{F}_{rot}$ experienced by a particle of mass m moving on the rotating disc and the force $\overrightarrow{F}_{in}$ experienced by the particle in an inertial frame of reference is,

$\overrightarrow{F}_{rot}= \overrightarrow{F_{in}}+2m(\overrightarrow{v}_{rot}\times \overrightarrow{ \omega})+ m(\overrightarrow{\omega}\times \overrightarrow{r})\times\overrightarrow{\omega}$

where, V_{rot} is the velocity of the particle in the rotating frame of reference and r is the position vector of the particle with respect to center of the disc

Now, consider a smooth slot along a diameter of a disc of radius R rotating counter-clockwise with a constant angular speed ω about its vertical axis through its centre. We assign a coordinate system with the origin at the centre of the disc, the X-axis along the slot, the Y-axis perpendicular to the slot and Z-axis along the rotation axis($(\omega =\omega\hat{k})$. A small block of mass m is gently placed in the slot at $r=(\frac{R}{2})\hat{i}$ at t=0 and is constrained to move only along the slot

The net reaction of the disc on the block is

5

For the following electrochemical cell at 298 K,

$Pt(s) | H_{2}(g,1bar) | H^{+}(aq,1M) $

$|| M^{4+}(aq), M^{2+}(aq)| Pt (s)$

$E_{cell}$=0.092 V

$when \frac{[M^{2+}(aq)]}{[M^{4+}(aq)]}=10^{x}$

Given: $E^{0}_{M^{4+/}M^{2^{+}}}=0.15V:$

$2.303 \frac{RT}{F}=0.059 V$

The value of x is

6

Reagent(s) which can be used to bring about the following transformation is (are)

7

Football teams $T_{1}$ and $T_{2}$ have to play two games against each other. It is assumed that the outcomes of the two games are independent. The probabilities of $T_{1}$ winning, drawing and losing a game against $T_{2}$ are $\frac{1}{2},\frac{1}{6}and \frac{1}{3}$ ,respectively. Each team gets 3 points for a win, 1 point for a draw and 0 point for a loss in a game. Let X and Y denote the total points scored by teams $T_{1}$ and $T_{2}$, respectively, after two games

P(X=Y) is

8

Let $F_{1}(x_{1},0)$ and $F_{2}(x_{2},0)$ , for $x_{1}<0$ and $x_{2}>0$ , be the foci of the ellipise $\frac{x^{2}}{9}+\frac{y^{2}}{8}=1$ . Suppose a parabola having vertex at the origin and focus at $F_{2}$ intersects the ellipse at the point M in the first quadrant and at point N in the fourth quadrant

The orthocentre of $\triangle F_{1}MN$ is

9

Let $F_{1}(x_{1},0)$ and $F_{2}(x_{2},0)$ , for $x_{1}<0$ and $x_{2}>0$ , be the foci of the ellipise $\frac{x^{2}}{9}+\frac{y^{2}}{8}=1$ . Suppose a parabola having vertex at the origin and focus at $F_{2}$ intersects the ellipse at the point M in the first quadrant and at point N in the fourth quadrant

If the tangents to the ellipse at M and N meet at R and the normal to the parabola at M meets the X-axis at Q, then the ratio of area of $\triangle MQR$ to area of the quadrilateral $MF_{1}NF_{2}$ is

10

One mole of an ideal gas at 300K in thermal contact with surroundings expands isothermally from 1.0 L to 2.0 L against a constant pressure of 3.0 atm. In this process, the change in entropy of surrounds ( $\triangle S_{surr}$ ) in J $K^{-1}$ is (1L atm=101.3 J)

11

The compound(s) with two lone pairs of electrons on the central atom is (are

12

The mole fraction of a solute in a solution is 0.1, At 298K , molarity of this solution is the same as its molality . Density of this solution at 298K is 2.0 g cm^{-3} . The ratio of the molecular weights of the solute and solvent, $(\frac{m_{solute}}{m_{solvent}})$ is.......

13

In a $\triangle XYZ$ , let x,y,z be the lengths of sides opposite to the angle X,Y,Z respectively and 2s=x+y+z. If $\frac{s-x}{4}=\frac{s-y}{3}=\frac{s-z}{2}$ and area of incircle of the $\triangle XYZ$ is $\frac{8\pi}{3}$ then

14

Let m be the smallesr positive integer such that the coefficient of $x^{2}$ in the expansion of $(1+x)^{2}+(1+x)^{3}+.....+(1+x)^{49}+(1+mx)^{50}$ is

$\left(3n+1\right)^{51}C_{3}$ for some positive integer n, Then the value of n is

15

The total number of distincts $x \epsilon [0,1]$ for which $\int_{0}^{x} \frac{t^{2}}{1+t^{4}}dt=2x-1$ is

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