Let $\alpha ,\beta \in R$ be such that $\underset{x\to 0}{\lim }\frac{x^2\sin \left(\beta x\right)}{\alpha x-\sin x}=1.$ Then $6\left(\alpha +\beta \right)$ equals

Consider the region \(R=\left\{(x, y) \in \mathbb{R} \times \mathbb{R}: x \geq 0\right.\) and \(\left.y^{2} \leq 4-x\right\}\). Let \(\mathcal{F}\) be the family of all circles that are contained in \(R\) and have centers on the \(x\)-axis. Let \(C\) be the circle that has largest radius among the circles in \(\mathcal{F}\). Let \((\alpha, \beta)\) be a point where the circle \(C\) meets the curve \(y^{2}=4-x\).
The radius of the circles $C$ is____.

Three numbers are chosen at random, one after another with replacement, from the set \(S=\{1,2,3, \ldots, 100\}\). Let \(p_{1}\) be the probability that the maximum of chosen numbers is at least 81 and \(p_{2}\) be the probability that the minimum of chosen numbers is at most 40 .
The value of $\frac{625}{4}{p}_1$ is

For positive integer $n$, define $f\left(n\right)=n+\frac{16+5n-3n^2}{4n+3n^2}+\frac{32+n-3n^2}{8n+3n^2}+\frac{48-3n-3n^2}{12n+3n^2}+\dots +\frac{25n-7n^2}{7n^2}$. Then, the value of $\underset{n\to \infty }{\lim }f\left(n\right)$ is equal to

The probability that a missile hits a target successfully is $0.75.$ In order to destroy the target completely, at least three successful hits are required. Then the minimum number of missiles that have to be fired so that the probability of completely destroying the target is not less than $0.95,$ is ___________

A student appears for a quiz consisting of only true-false type questions and answers all the questions. The student knows the answers of some questions and guesses the answers for the remaining questions. Whenever the student knows the answer of a question, he gives the correct answer. Assume that the probability of the student giving the correct answer for a question, given that he has guessed it, is \(\frac{1}{2}\). Also assume that the probability of the answer for a question being guessed, given that the student's answer is correct, is \(\frac{1}{6}\). Then the probability that the student knows the answer of a randomly chosen question is

Let \(\alpha, \beta\) and \(\gamma\) be real numbers such that the system of linear equations
\[
\begin{array}{c}
x+2 y+3 z=\alpha \\
4 x+5 y+6 z=\beta \\
7 x+8 y+9 z=\gamma-1
\end{array}
\]
is consistent. Let \(|M|\) represent the determinant of the matrix
\[
M=\left[\begin{array}{ccc}
\alpha & 2 & \gamma \\
\beta & 1 & 0 \\
-1 & 0 & 1
\end{array}\right]
\]
Let \(P\) be the plane containing all those \((\alpha, \beta, \gamma)\) for which the above system of linear equations is consistent, and \(D\) be the square of the distance of the point \((0,1,0)\) from the plane \(P\).
The value of $\left|M\right|$ is

One of the products formed from the reaction of permanganate ion with iodide ion in neutral aqueous medium is

Bleaching powder and bleach solution are produced on a large scale and used in several household products. The effectiveness of bleach solution is often measured by iodometry.
Question:
\(25 \mathrm{~mL}\) of household solution was mixed with \(30 \mathrm{~mL}\) of \(0.50\) M KI and \(10 \mathrm{~mL}\) of \(4 \mathrm{~N}\) acetic acid. In the titration of the liberated iodine, \(48 \mathrm{~mL}\) of \(0.25 \mathrm{~N~} \mathrm{Na}_{2} \mathrm{~S}_{2} \mathrm{O}_{3}\) was used to reach the end point. The molarity of the household bleach solution is

The entropy versus temperature plot for phases $\alpha$ and $\beta$ at $1$ bar pressure is given. ${\mathrm{S}}_{\mathrm{T}}$ and ${\mathrm{S}}_0$ are entropies of the phases at temperatures $\mathrm{T}$ and $0 \mathrm{K}$, respectively. The transition temperature for $\alpha$ to $\beta$ phase change is $600 \mathrm{K}$ and ${\mathrm{C}}_{\mathrm{p},\mathrm{\beta }}-{\mathrm{C}}_{\mathrm{p},\mathrm{\alpha }}=1 \mathrm{J} {\mathrm{mol}}^{-1} {\mathrm{K}}^{-1}$. Assume $\left({\mathrm{C}}_{\mathrm{p},\mathrm{\beta }}-{\mathrm{C}}_{\mathrm{p},\mathrm{\alpha }}\right)$ is independent of temperature in the range of $200$ to $700 \mathrm{K}.{\mathrm{C}}_{\mathrm{p},\mathrm{\alpha }}$ and ${\mathrm{C}}_{\mathrm{p},\mathrm{\beta }}$ are heat capacities of $\alpha$ and $\beta$ phases, respectively.
The value of entropy change, ${\mathrm{S}}_{\beta }-{\mathrm{S}}_{\alpha }$ (in ${\mathrm{Jmol}}^{-1} {\mathrm{K}}^{-1}$ ), at $300 \mathrm{K}$ is [Use: $\ln 2=0.69$0. Given: ${\mathrm{S}}_{\beta }-{\mathrm{S}}_{\alpha }=0$ at $0 \mathrm{K}$]

A resistance of \(2 \Omega\) is connected across one gap of a meter-bridge (the length of the wire is \(100 \mathrm{~cm}\) ) and an unknown resistance, greater than \(2 \Omega\), is connected across the other gap. When these resistances are interchanged, the balance point shifts by \(20 \mathrm{~cm}\). Neglecting any corrections, the unknown resistance is

Let $X=\left\{\left(x, y\right)\in \mathrm{\mathbb{Z}}\times \mathrm{\mathbb{Z}} : \frac{x^2}{8}+\frac{y^2}{20}<1 \mathrm{and} y^2<5\mathrm{x}\right\}$. Three distinct point $P, Q$ and $R$ are randomly chosen from $X$. Then the probability that $P, Q$ and $R$ form a triangle whose area is a positive integer, 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:
What is the disadvantage of this system?