A square loop of side 2.0 cm is placed inside a long solenoid that has 50 turns per centimetre and carries a sinusoidally varying current of amplitude 2.5 A and angular frequency $700 \mathrm { rad } \mathrm {~s} ^ { - 1 }$. The central axes of the loop and solenoid coincide. The amplitude of the emf induced in the loop is $x \times 10 ^ { - 4 } \mathrm {~V}$. The value of $x$ is ( Take, $\pi = \frac { 22 } { 7 }$)

An insulated copper wire of 100 turns is wrapped around a wooden cylindrical core of the cross-sectional area $24 \mathrm {~cm} ^ { 2 }$. The two ends of the wire are connected to a resistor. The total resistance in the circuit is $12 \Omega$. If an externally applied uniform magnetic field in the core along its axis changes from 1.5 T in one direction to 1.5 T in the opposite direction, the charge flowing through a point in the circuit during the change of magnetic field will be $\_\_\_\_$ mC.

An ideal transformer with purely resistive load operates at 12 kV on the primary side. It supplies electrical energy to a number of nearby houses at 120 V. The average rate of energy consumption in the houses served by the transformer is 60 kW. The value of resistive load $\left(R_s\right)$ required in the secondary circuit will be $\_\_\_\_$ $\mathrm{m}\Omega$.

A point source of light is placed at the centre of curvature of a hemispherical surface. The source emits a power of 24 W. The radius of curvature of hemisphere is 10 cm and the inner surface is completely reflecting. The force on the hemisphere due to the light falling on it is $\_\_\_\_$ $\times 10^{-8}~\mathrm{N}$.

A point object $O$ is placed in front of two thin symmetrical coaxial convex lenses $L _ { 1 }$ and $L _ { 2 }$ with focal length 24 cm and 9 cm respectively. The distance between two lenses is 10 cm and the object is placed 6 cm away from lens $L _ { 1 }$ as shown in the figure. The distance between the object and the image formed by the system of two lenses is $\_\_\_\_$ cm

A bi convex lens of focal length 10 cm is cut in two identical parts along a plane perpendicular to the principal axis. The power of each lens after cut is $\_\_\_\_$ D.

A pole is vertically submerged in swimming pool, such that it gives a length of shadow 2.15 m within water when sunlight is incident at an angle of $30^{\circ}$ with the surface of water. If swimming pool is filled to a height of 1.5 m, then the height of the pole above the water surface in centimeters is $\left(n_w = \frac{4}{3}\right)$ $\_\_\_\_$.

The radius of curvature of each surface of a convex lens having refractive index 1.8 is 20 cm. The lens is now immersed in a liquid of refractive index 1.5. The ratio of power of lens in air to its power in the liquid will be $x : 1$. The value of $x$ is

In a screw gauge, there are 100 divisions on the circular scale and the main scale moves by 0.5 mm on a complete rotation of the circular scale. The zero of circular scale lies 6 divisions below the line of graduation when two studs are brought in contact with each other. When a wire is placed between the studs, 4 linear scale divisions are clearly visible while $46^{\text{th}}$ division of the circular scale coincide with the reference line. The diameter of the wire is $\_\_\_\_$ $\times 10^{-2}~\mathrm{mm}$.

If $917 \AA$ be the lowest wavelength of Lyman series then the lowest wavelength of Balmer series will be $\_\_\_\_$ $\AA$.

An atom absorbs a photon of wavelength 500 nm and emits another photon of wavelength 600 nm. The net energy absorbed by the atom in this process is $n \times 10 ^ { - 4 } \mathrm { eV }$. The value of $n$ is [Assume the atom to be stationary during the absorption and emission process] (Take $h = 6.6 \times 10 ^ { - 34 } \mathrm {~J} \cdot \mathrm {~s}$ and $c = 3 \times 10 ^ { 8 } \mathrm {~m} \mathrm {~s} ^ { - 1 }$).

In an amplitude modulation, a modulating signal having amplitude of $X \mathrm {~V}$ is superimposed with a carrier signal of amplitude $Y \mathrm {~V}$ in first case. Then, in second case, the same modulating signal is superimposed with different carrier signal of amplitude $2Y \mathrm {~V}$. The ratio of modulation index in the two case respectively will be :
(1) $1 : 2$
(2) $1 : 1$
(3) $2 : 1$
(4) $4 : 1$

The radius of fifth orbit of $\mathrm{Li}^{++}$ is $\_\_\_\_$ $\times 10^{-12}$ m. Take: radius of hydrogen atom $= 0.51\AA$

A monochromatic light is incident on a hydrogen sample in ground state. Hydrogen atoms absorb a fraction of light and subsequently emit radiation of six different wavelengths. The frequency of incident light is $x \times 10^{15}$ Hz. The value of $x$ is $\_\_\_\_$ (Given $h = 4.25 \times 10^{-15}$ eVs)

A solution of sugar is obtained by mixing 200 g of its $25\%$ solution and 500 g of its $40\%$ solution (both by mass). The mass percentage of the resulting sugar solution is $\_\_\_\_$ (Nearest integer)

Let $S = \{ 1,2,3,5,7,10,11 \}$. The number of non-empty subsets of $S$ that have the sum of all elements a multiple of 3 , is $\_\_\_\_$ .

Consider: S1: $p \Rightarrow q \vee (p \wedge \sim q)$ is a tautology. S2: $\sim p \Rightarrow (\sim q \wedge \sim p) \vee q$ is a contradiction. Then
(1) only S2 is correct
(2) both S1 and S2 are correct
(3) both S1 and S2 are wrong
(4) only S1 is correct

The negation of the expression $q \vee ((\sim q) \wedge p)$ is equivalent to
(1) $(\sim p) \wedge (\sim q)$
(2) $p \wedge (\sim q)$
(3) $(\sim p) \vee (\sim q)$
(4) $(\sim p) \vee q$

Let $R$ be a relation on $\mathbb{N} \times \mathbb{N}$ defined by $(a,b)\, R\, (c,d)$ if and only if $ad(b-c) = bc(a-d)$. Then $R$ is
(1) symmetric but neither reflexive nor transitive
(2) transitive but neither reflexive nor symmetric
(3) reflexive and symmetric but not transitive
(4) symmetric and transitive but not reflexive

The negation of the statement $(p \vee q) \wedge (q \vee \sim r)$ is
(1) $p \vee r \wedge \sim q$
(2) $(\sim p) \vee r \wedge \sim q$
(3) $\sim p \vee \sim q \vee \sim r$
(4) $\sim p \vee \sim q \wedge \sim r$

The remainder when $( 2023 ) ^ { 2023 }$ is divided by 35 is $\_\_\_\_$.

Let $R$ be a relation on $\mathbb{R}$, given by $R = \{(a, b) : 3a - 3b + \sqrt{7}$ is an irrational number$\}$. Then $R$ is
(1) Reflexive but neither symmetric nor transitive
(2) Reflexive and transitive but not symmetric
(3) Reflexive and symmetric but not transitive
(4) An equivalence relation

A triangle is formed by $X$-axis, $Y$-axis and the line $3 x + 4 y = 60$. Then the number of points $P ( a , b )$ which lie strictly inside the triangle, where $a$ is an integer and $b$ is a multiple of $a$, is $\_\_\_\_$.

Let $S$ be the set of all solutions of the equation $\cos^{-1}(2x) - 2\cos^{-1}\left(\sqrt{1 - x^2}\right) = \pi$, $x \in \left[-\frac{1}{2}, \frac{1}{2}\right]$. Then $\sum_{x \in S} 2\sin^{-1}(x^2 - 1)$ is equal to
(1) 0
(2) $\frac{-2\pi}{3}$
(3) $\pi - \sin^{-1}\frac{\sqrt{3}}{4}$
(4) $\pi - 2\sin^{-1}\frac{\sqrt{3}}{4}$

The statement $( p \wedge ( \sim q ) ) \Rightarrow ( p \Rightarrow ( \sim q ) )$ is
(1) equivalent to $( \sim p ) \vee ( \sim q )$
(2) a tautology
(3) equivalent to $p \vee q$
(4) a contradiction