Let the curve C be the mirror image of the parabola ${\mathrm{y}}^{\mathrm{2 }}$$=4\mathrm{x}$ with respect to the line $\mathrm{x}+\mathrm{y}+4=0.$ If A and B are the points of intersection of C with the line $\mathrm{y}=\mathrm{ }-5,$ then the distance between A and B is

Let $S=\left(0,1\right)\cup \left(1, 2\right)\cup \left(3, 4\right)$ and $T=\left\{0, 1, 2, 3\right\}$. Then which of the following statements is(are) true?

Let $O$ be the origin and let $PQR$ be an arbitrary triangle. The point$S$ is such that $\overrightarrow{OP}.\overrightarrow{OQ}+\overrightarrow{OR}.\overrightarrow{OS}=\overrightarrow{OR}.\overrightarrow{OP}+\overrightarrow{OQ}.\overrightarrow{OS}=\overrightarrow{OQ}.\overrightarrow{OR}+\overrightarrow{OP}.\overrightarrow{OS}$ then triangle $PQR$ has $S$ as its

Let \(f_{1}:(0, \infty) \rightarrow \mathbb{R}\) and \(f_{2}:(0, \infty) \rightarrow \mathbb{R}\) be defined by

\[

f_{1}(x)=\int_{0}^{x} \prod_{j=1}^{21}(t-j)^{j} d t, \quad x>0

\]

and

\(

f_{2}(x)=98(x-1)^{50}-600(x-1)^{49}+2450, \quad x>0

\)

where, for any positive integer \(n\) and real numbers \(a_{1}, a_{2}, \ldots, a_{n}, \prod_{i=1}^{n} a_{i}\) denotes the product of \(a_{1}, a_{2}, \ldots, a_{n} .\) Let \(m_{i}\) and \(n_{i}\), respectively, denote the number of points of local minima and the number of points of local maxima of function \(f_{i}, i=1,2\), in the interval \((0, \infty)\).

The value of $2m_1+3n_1+m_1{n}_1$ is _____.

For the function \(f(x)=x \cos \frac{1}{x}, x \geq 1\),

The point \(P\) is the intersection of the straight line joining the points \(Q(2,3,5)\) and \(R(1,-1,4)\) with the plane \(5 x-4 y-\) \(z=1\). If \(S\) is the foot of the perpendicular drawn from the point \(T(2,1,4)\) to \(Q R\), then the length of the line segment \(P S\) is

The total number of distinct $x\in R$ for which $\left|\begin{array}{ccc}x& x^2& 1+x^3\\ 2x& 4x^2& 1+8x^3\\ 3x& 9x^2& 1+27x^3\end{array}\right|=10$ is

The value of
\(\sec ^{-1}\left(\frac{1}{4} \sum_{k=0}^{10} \sec (\frac{7 \pi}{12}+\frac{k \pi}{2}\right) \sec (\frac{7 \pi}{12}\) \(+\frac{(k+1) \pi}{2}))\) in the interval \(\left[-\frac{\pi}{4}, \frac{3 \pi}{4}\right]\) equals

According to kinetic theory of gases

A positive point charge of \(10^{-8} \mathrm{C}\) is kept at a distance of 20 cm from the center of a neutral conducting sphere of radius 10 cm . The sphere is then grounded and the charge on the sphere is measured. The grounding is then removed and subsequently the point charge is moved by a distance of 10 cm further away from the center of the sphere along the radial direction. Taking \(\frac{1}{4 \pi \epsilon_0}=9 \times 10^9 \mathrm{Nm}^2 / \mathrm{C}^2\) (where \(\epsilon_0\) is the permittivity of free space), which of the following
statements is/are correct: :

A plot of the number of neutrons $(\text{N})$ against the number of protons $(\text{P})$ of the stable nuclei exhibits upwards deviation from linearity for atomic number, $\text{Z}>20$. For an unstable nucleus having $\frac{\text{N}}{\text{P}}$ ratio less than $1$, the possible mode(s) of decay is (are):

Paragraph :
When a particle of mass \(m\) moves on the \(x\)-axis in a potential of the form \(V(x)=k x^2\), it performs simple harmonic motion.


The corresponding time period is proportional to \(\sqrt{\frac{m}{k}}\), as can be seen easily using dimensional analysis. However, the motion of a particle can be periodic even when its potential energy increases on both sides of \(x=0\) in a way different from \(k x^2\) and its total energy is such that the particle does not escape to infinity. Consider a particle of mass \(\mathrm{m}\) moving on the \(x\)-axis. Its potential energy is \(V(x)=\alpha x^4(\alpha>0)\) for \(|x|\) near the origin and becomes a constant equal to \(V_0\) for \(|x| \geq X_0\) (see figure).
Question :
The acceleration of this particle for \(|x|>X_0\) is

Paragraph:
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?