Statement 1 For an observer looking out through the window of a fast moving train, the nearby objects appear to move in the opposite direction to the train, while the distant objects appear to be stationary.
and Statement 2 If the observer and the object are moving at velocities \(\overrightarrow{\mathbf{v}}_{\mathbf{1}}\) and \(\overrightarrow{\mathbf{v}_2}\) respectively with reference to a laboratory frame, the velocity of the object with respect to the observer is \(\overrightarrow{\mathbf{v}_2}-\overrightarrow{\mathbf{v}_1}\).

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Modern trains are based on Maglev technology in which trains are magnetically leviated, which runs its EDS Maglev system.
There are coils on both sides of wheels. Due to motion of train, current induces in the coil of track which levitate it. This is in accordance with Lenz's law. If trains lower down then due to Lenz's law a repulsive force increases due to which train gets uplifted and if it goes much high then there is a net downward force disc to gravity. The advantage of Maglev train is that there is no friction between the train and the track, thereby reducing power consumption and enabling the train to attain very high speeds.
Disadvantage of Maglev train is that as it slows down the electromagnetic forces decreases and it becomes difficult to keep it leviated and as it moves forward according to Lenz law, there is an electromagnetic drag force.
Question:
Which force causes the train to elevate up?

A liquid at ${30\mathrm{ }}^{\mathrm{o}}\mathrm{C}$ is poured very slowly into a Calorimeter that is at temperature of ${110\mathrm{ }}^{\mathrm{o}}\mathrm{C}.$ The boiling temperature of the liquid is ${80\mathrm{ }}^{\mathrm{o}}\mathrm{C}.$ It is found that the first $5\mathrm{ }\mathrm{g}\mathrm{m}$ of the liquid completely evaporates. After pouring another $80\mathrm{ }\mathrm{g}\mathrm{m}$ of the liquid the equilibrium temperature is found to be ${50\mathrm{ }}^{\mathrm{o}}\mathrm{C}.$ The ratio of the Latent heat of the liquid to its specific heat will be ______ ${\mathrm{ }}^{\mathrm{o}}\mathrm{C}.$
[Neglect the heat exchange with surrounding]

A massless rod \(B D\) is suspended by two identical massless strings \(A B\) and \(C D\) of equal lengths. A block of mass ' \(m\) ' is suspended point \(P\) such that \(B P\) is equal to ' \(x\) ', if the fundamental frequency of the left wire is twice the fundamental frequency of right wire, then the value of \(x\) is

Consider a disc rotating in the horizontal plane with a constant angular speed \(\omega\) about its centre \(\mathrm{O}\). The disc has a shaded region on one side of the diameter and an unshaded region on the other side as shown in the figure. When the disc is in the orientation as shown, two pebbles \(P\) and \(Q\) are simultaneously projected at an angle towards \(R\). The velocity of projection is in the \(y\) - \(z\) plane and is same for both pebbles with respect to the disc. Assume that (i) they land back on the disc before the disc has completed \(1 / 8\) rotation, (ii) their range is less than half the disc radius, and (iii) \(\omega\) remains constant throughout. Then

A student performs an experiment to determine the Young's modulus of a wire, exactly \(2 \mathrm{~m}\) long, by Searle's method. In a particular reading, the student measures the extension in the length of the wire to be \(0.8 \mathrm{~mm}\) with an uncertainty of \(\pm 0.05 \mathrm{~mm}\) at a load of exactly \(1.0 \mathrm{~kg}\). The student also measures the diameter of the wire to be \(0.4 \mathrm{~mm}\) with an uncertainty of \(\pm 0.01 \mathrm{~mm}\). Take \(g=9.8 \mathrm{~m} / \mathrm{s}^2\) (exact). The Young's modulus obtained from the reading is

There is a rectangular plate of mass \(M \mathrm{~kg}\) of dimensions \((a \times b)\). The plate is held in horizontal position by striking \(n\) small balls each of mass \(m\) per unit area per unit time. These are striking in the shaded half region of the plate. The balls are colliding elastically with velocity \(v\). What is \(v\) ? It is given \(n=100, M=3 \mathrm{~kg}, m=0.01 \mathrm{~kg}\); \(b=2 \mathrm{~m} ; a=1 \mathrm{~m} ; g=10 \mathrm{~m} / \mathrm{s}^2\).

Let $p,q,r$ be non-zero real numbers that are, respectively, the ${10}^{\mathrm{th} },{100}^{\mathrm{th} }$ and ${1000}^{\mathrm{th} }$ terms of a harmonic progression. Consider the system of linear equations
$x+y+z=1$ $10x+100y+1000z=0$ $qrx+pry+pqz=0$

	List-$\mathrm{I}$		List-$\mathrm{II}$
$\left(\mathrm{I}\right)$	If $\frac{q}{r}=10$, then the system of linear equations has	$\left(\mathrm{P}\right)$	$x=0,y=\frac{10}{9},z=-\frac{1}{9}$ as a solution
$\left(\mathrm{II}\right)$	If $\frac{p}{r}\ne 100$, then the system of linear equations has	$\left(\mathrm{Q}\right)$	$x=\frac{10}{9},y=-\frac{1}{9},z=0$ as a solution
$\left(\mathrm{III}\right)$	If $\frac{p}{q}\ne 10$, then the system of linear equations has	$\left(\mathrm{R}\right)$	infinitely many solutions
$\left(\mathrm{IV}\right)$	If $\frac{p}{q}=10$, then the system of linear equations has	$\left(\mathrm{S}\right)$	no solution
		$\left(\mathrm{T}\right)$	at least one solution

The correct option is:

A student is performing the experiment of resonance column. The diameter of the column tube is \(4 \mathrm{~cm}\). The frequency of the tuning fork is \(512 \mathrm{~Hz}\). The air temperature is \(38^{\circ} \mathrm{C}\) in which the speed of sound is \(336 \mathrm{~m} / \mathrm{s}\). The zero of the meter scale coincides with the top end of the resonance column tube. When the first resonance occurs, the reading of the water level in the column is

Consider the following frequency distribution :

Value	4	5	8	9	6	12	11
Frequency	5	\(f_1\)	\(f_2\)	2	1	1	3

Suppose that the sum of the frequencies is 19 and the median of this frequency distribution is 6. For the given frequency distribution, let \(\alpha\) denote the mean deviation about the mean, \(\beta\) denote the mean deviation about the median, and \(\sigma^2\) denote the variance.
Match each entry in List-I to the correct entry in List-II and choose the correct option.
\(\begin{array}{|l|l|l|l|} \hline & \text{LIST - I} & & \text{LIST - II} \\ \hline (P) & 7 f_1+9 f_2 \text{ is equal to} & (1) & 146 \\ \hline (Q) & 19 \alpha \text{ is equal to} & (2) & 47 \\ \hline (R) & 19 \beta \text{ is equal to} & (3) & 48 \\ \hline (S) & 19 \sigma^2 \text{ is equal to} & (4) & 145 \\ \hline & & (5) & 55 \\ \hline \end{array}\)

Tangents are drawn from the point \((17,7)\) to the circle \(x^2+y^2=169\).
Statement I The tangents are mutually perpendicular.
Statement II The locus of the points from which mutually perpendicular tangents can be drawn to the given circle is \(x^2+y^2=338\)

\(\frac{d^2 x}{d y^2}\) equals

Let a, b, c be positive integers such that $\frac{b}{a}$ is an integer. If a, b, c are in geometric progression and the arithmetic mean of a, b, c is b + 2, then the value of $\frac{a^2+a-14}{a+1}$ is ________