The magnetic field existing in a region is given by $\vec { B } = 0.2 ( 1 + 2 x ) \hat { k } \mathrm {~T}$. A square loop of edge 50 cm carrying 0.5 A current is placed in $x - y$ plane with its edges parallel to the $x - y$ axes, as shown in figure. The magnitude of the net magnetic force experienced by the loop is $\_\_\_\_$ mN.

A 2 A current carrying straight metal wire of resistance $1\Omega$, resistivity $2 \times 10^{-6}\Omega\mathrm{m}$, area of cross-section $10\mathrm{~mm}^2$ and mass 500 g is suspended horizontally in mid air by applying a uniform magnetic field $\vec{B}$. The magnitude of $B$ is $\_\_\_\_$ $\times 10^{-1}\mathrm{~T}$ (given, $\mathrm{g} = 10\mathrm{~m/s}^2$).

A square loop of edge length 2 m carrying current of 2 A is placed with its edges parallel to the $x y$ axis. A magnetic field is passing through the $x - y$ plane and expressed as $\vec { B } = B _ { 0 } ( 1 + 4 x ) \hat { k }$, where $B _ { 0 } = 5 \mathrm {~T}$. The net magnetic force experienced by the loop is $\_\_\_\_$ N.

The magnetic flux $\phi$ (in weber) linked with a closed circuit of resistance $8\,\Omega$ varies with time (in seconds) as $\phi = 5t^2 - 36t + 1$. The induced current in the circuit at $t = 2$ s is $\_\_\_\_$ A.

A rectangular loop of sides 12 cm and 5 cm, with its sides parallel to the $x$-axis and $y$-axis respectively moves with a velocity of $5 \mathrm {~cm} \mathrm {~s} ^ { - 1 }$ in the positive $x$ axis direction, in a space containing a variable magnetic field in the positive $z$ direction. The field has a gradient of $10 ^ { - 3 } \mathrm {~T} \mathrm {~cm} ^ { - 1 }$ along the negative $x$ direction and it is decreasing with time at the rate of $10 ^ { - 3 } \mathrm {~T} \mathrm {~s} ^ { - 1 }$. If the resistance of the loop is $6 \mathrm {~m\Omega}$, the power dissipated by the loop as heat is $\_\_\_\_$ $\times 10 ^ { - 9 } \mathrm {~W}$.

An alternating current at any instant is given by $i = [ 6 + \sqrt { 56 } \sin ( 100 \pi t + \pi / 3 ) ]$ A. The $rms$ value of the current is $\_\_\_\_$ A.

An ac source is connected in given series LCR circuit. The rms potential difference across the capacitor of $20\mu\mathrm{F}$ is $\_\_\_\_$ V. $$\mathrm{V} = 50\sqrt{2}\sin 100t \text{ volt}$$

When a coil is connected across a 20 V dc supply, it draws a current of 5 A. When it is connected across $20 \mathrm {~V} , 50 \mathrm {~Hz}$ ac supply, it draws a current of 4 A. The self inductance of the coil is $\_\_\_\_$ mH. (Take $\pi = 3$)

The distance between object and its two times magnified real image as produced by a convex lens is 45 cm . The focal length of the lens used is $\_\_\_\_$ cm.

Light from a point source in air falls on a convex curved surface of radius 20 cm and refractive index 1.5. If the source is located at 100 cm from the convex surface, the image will be formed at $\_\_\_\_$ cm from the object.

The distance between object and its 3 times magnified virtual image as produced by a convex lens is 20 cm. The focal length of the lens used is $\_\_\_\_$ cm.

Two wavelengths $\lambda _ { 1 }$ and $\lambda _ { 2 }$ are used in Young's double slit experiment. $\lambda _ { 1 } = 450 \mathrm {~nm}$ and $\lambda _ { 2 } = 650 \mathrm {~nm}$. The minimum order of fringe produced by $\lambda _ { 2 }$ which overlaps with the fringe produced by $\lambda _ { 1 }$ is $n$. The value of $n$ is $\_\_\_\_$.

In Young's double slit experiment, carried out with light of wavelength $5000\,\text{\AA}$, the distance between the slits is 0.3 mm and the screen is at 200 cm from the slits. The central maximum is at $x = 0\mathrm{~cm}$. The value of $x$ for third maxima is $\_\_\_\_$ mm.

In a Young's double slit experiment, the intensity at a point is $\left( \frac { 1 } { 4 } \right) ^ { \text {th} }$ of the maximum intensity, the minimum distance of the point from the central maximum is $\_\_\_\_$ $\mu \mathrm { m }$. (Given: $\lambda = 600 \mathrm {~nm} , \mathrm {~d} = 1.0 \mathrm {~mm} , \mathrm { D } = 1.0 \mathrm {~m}$)

A nucleus has mass number $A_1$ and volume $V_1$. Another nucleus has mass number $A_2$ and volume $V_2$. If relation between mass number is $A_2 = 4A_1$, then $\dfrac{V_2}{V_1} =$ $\_\_\_\_$.

The radius of a nucleus of mass number 64 is 4.8 fermi. Then the mass number of another nucleus having radius of 4 fermi is $\frac { 1000 } { x }$, where $x$ is $\_\_\_\_$.

If three helium nuclei combine to form a carbon nucleus then the energy released in this reaction is $\_\_\_\_$ $\times 10^{-2}\mathrm{~MeV}$. (Given $1\mathrm{~u} = 931\mathrm{~MeV}/\mathrm{c}^2$, atomic mass of helium $= 4.002603\mathrm{~u}$)

The potential for the given half cell at 298 K is $-\ldots\ldots\ldots . . \times 10 ^ { - 2 } \mathrm {~V}$. $2 \mathrm { H } _ { (\mathrm { aq }) } ^ { + } + 2 \mathrm { e } ^ { - } \rightarrow \mathrm { H } _ { 2 } (\mathrm {~g})$ $\mathrm { H } ^ { + } = 1 \mathrm { M } , \mathrm { P } _ { \mathrm { H } _ { 2 } } = 2 \mathrm {~atm}$ (Given $2.303 RT / F = 0.06$ V, $\log 2 = 0.3$)

The ratio of $\frac { {}^{14}\mathrm { C } } { {}^{12}\mathrm { C } }$ in a piece of wood is $\frac { 1 } { 8 }$ part that of atmosphere. If half life of ${}^{ 14 } \mathrm { C }$ is 5730 years, the age of wood sample is $\_\_\_\_$ years.

Let $S = \{ 1,2,3 , \ldots , 10 \}$. Suppose $M$ is the set of all the subsets of $S$, then the relation $R = \{ ( A , B ) : A \cap B \neq \phi ; A , B \in M \}$ is:
(1) symmetric and reflexive only
(2) reflexive only
(3) symmetric and transitive only
(4) symmetric only

The function $\mathrm { f } : \mathrm { N } - \{ 1 \} \rightarrow \mathrm { N }$; defined by $f ( \mathrm { n } ) =$ the highest prime factor of $n$, is:
(1) both one-one and onto
(2) one-one only
(3) onto only
(4) neither one-one nor onto

The figure below is a schematic diagram of a building block, where $A B C$ is a right triangle with $\angle A C B = 90 ^ { \circ } , \overline { A C } = 5 , \overline { B C } = 6$ , and both $A D E B$ and $A D F C$ are rectangles. Select the correct options.
(1) Cutting this block along plane $A C E$ yields two tetrahedra
(2) The acute angle between planes $A D E B$ and $A D F C$ is greater than $45 ^ { \circ }$
(3) $\angle C E B < \angle A E B$
(4) $\tan \angle A E C < \sin \angle C E B$
(5) $\angle C E B < \angle A E C$

There is a wooden block where $ACFD$ and $ABED$ are two congruent isosceles trapezoids, and $BCFE$ is a rectangle. Let the projection of point $A$ on line $BC$ be $M$ and its projection on plane $BCFE$ be $P$. Given that $\overline{AD} = 30$, $\overline{CF} = 40$, $\overline{AP} = 15$, and $\overline{BC} = 10$. Place plane $BCFE$ on a horizontal table, and call any plane parallel to $BCFE$ a horizontal plane. Using the fact that the projection of $\overline{AD}$ on plane $BCFE$ has length 30, find $\tan \angle AMP$. (Fill-in-the-blank question, 2 points)

There is a wooden block where $ACFD$ and $ABED$ are two congruent isosceles trapezoids, and $BCFE$ is a rectangle. Let the projection of point $A$ on line $BC$ be $M$ and its projection on plane $BCFE$ be $P$. Given that $\overline{AD} = 30$, $\overline{CF} = 40$, $\overline{AP} = 15$, and $\overline{BC} = 10$. Place plane $BCFE$ on a horizontal table, and call any plane parallel to $BCFE$ a horizontal plane. Let $Q$ be a point on $\overline{FC}$ such that $\overrightarrow{AQ}$ is parallel to $\overrightarrow{DF}$. Using the fact that $\triangle ABC$ and $\triangle ACQ$ are congruent triangles, prove that if a horizontal plane $W$ lies between $A$ and $P$ and is at distance $x$ from $A$, then the rectangular region formed by the intersection of $W$ with this wooden block has area $20x + \frac{4}{9}x^2$. (Non-multiple choice question, 4 points)

A liquid crystal display consists of red, green, and blue LED bulbs. The lighting cycle rules for each color bulb are as follows:
Red: ``On for 3 seconds, then off for 1 second, then on for 2 seconds'' Green: ``On for 6 seconds, then off for 2 seconds'' Blue: ``On for $k$ seconds, then off for ($15 - k$) seconds'', where $k$ is a positive integer. If at a certain moment all three colors of bulbs simultaneously begin their respective cycles, and the display always has lights on, with the switching time between on and off being negligibly short, then the minimum value of $k$ is