When Uranium is bombarded with neutrons, it undergoes fission. The fission reaction can be written as  \({}_{92}{U^{235}} + {}_0{n^1} \to {}_{56}B{a^{141}} + {}_{36}K{r^{92}} + 3x + Q{\rm{( energy)}}\) where three particles named \(x\) are produced and energy \(Q\) is released. What is the name of the particle \(x\) ?

This question has Statement \(- 1\) and Statement \(-2\). Of the four choices given after the Statements, choose the one that best describes the two Statements.
Statement \(1\) : The internal energy of a perfect gas is entirely kinetic and depends only on absolute temperature of the gas and not on its pressure or volume.
Statement \(2\) : A perfect gas is heated keeping pressure constant and later at constant volume. For the same amount of heat the temperature of the gas at constant pressure is lower than that at constant volume.

A hole is drilled in a metal sheet. At \(27^{\circ} C\), the diameter of hole is 5 cm . When the sheet is heated to \(177^{\circ} C\), the change in the diameter of hole is \(d \times 10^{-3} cm\). The value of \(d\) will be ________ , if coefficient of linear expansion of the metal is \(1.6 \times 10^{-5} /{ }^{\circ} C\)

The following diagram shows a Zener diode as a voltage regulator. The Zener diode is rated at \(V _{ Z }=5 V\) and the desired current in load is 5 mA . The unregulated voltage source can supply upto 25 V . Considering the Zener diode can withstand four times of the load current, the value of resistor \(R _{ S }\) (shown in circuit) should be ___________ \(\Omega\).

A particle of mass \(M\) is moving in a circle of fixed radius \(R\) in such a way that its centripetal acceleration at time \(t\) is given by \(n^2Rt^2\) where \(n\) is a constant. The Power delivered to the particle by the force acting on it, is

A Spherical ball of radius \(1 mm\) and density \(10.5 g / cc\) is dropped in glycerine of coefficient of viscosity \(9.8\) poise and density \(1.5 g / cc\). Viscous force on the ball when it attains constant velocity is \(3696 \times 10^{-x} N\). The value of \(x\) is \(\text { (Given, } g =9.8 m / s ^2 \text { and } \pi=\frac{22}{7} \text { ) }\)

A bullet of mass \(50 \mathrm{~g}\) is fired with a speed \(100 \mathrm{~m} / \mathrm{s}\) on a plywood and emerges with \(40 \mathrm{~m} / \mathrm{s}\). The percentage loss of kinetic energy is _______.

All five letter words are made using all the letters \(\mathrm{A}, \mathrm{B}, \mathrm{C}, \mathrm{D}, \mathrm{E}\) and arranged as in an English dictionary with serial numbers. Let the word at serial number \(n\) be denoted by \(W_n\). Let the probability \(\mathrm{P}\left(\mathrm{W}_{\mathrm{n}}\right)\) of choosing the word \(\mathrm{W}_{\mathrm{n}}\) satisfy \(\mathrm{P}\left(\mathrm{W}_{\mathrm{n}}\right)=2 \mathrm{P}\left(\mathrm{W}_{\mathrm{n}-1}\right), \mathrm{n} \gt 1\).
If \(\mathrm{P}(\mathrm{CDBEA})=\frac{2^\alpha}{2^\beta-1}, \alpha, \beta \in \mathbb{N}\), then \(\alpha+\beta\) is equal to : _______

The electric field of an electromagnetic wave in free space is \(\overrightarrow{\mathrm{E}}=57 \cos \left[7.5 \times 10^6 \mathrm{t}-5 \times 10^{-3}(3 x+4 y)\right](4 \hat{i}-3 \hat{j}) N / C\). The associated magnetic field in Tesla is _______.

Let \(A\) be a point on the line \(\vec r = \left( {1 - 3\mu } \right)\hat i + \left( {\mu  - 1} \right)\hat j + \left( {2 + 5\mu } \right)\hat k\) and \(B(3, 2, 6)\) be a point in the space. Then the value of \(\mu \) for which the vector \(\overrightarrow {AB} \) is parallel to the plane \(x -4y +3z = 1\) is

Let \(f:(0, \infty) \rightarrow R\) and \(F(x)=\int_0^x t f(t) d t\). If \(F\left(x^2\right)=\) \(x^4+x^5\), then \(\sum_{r=1}^{12} f\left(r^2\right)\) is equal to :

The vertices of a hyperbola \(H\) are \((\pm 6,0)\) and its eccentricity is \(\frac{\sqrt{5}}{2}\). Let \(N\) be the normal to \(H\) at a point in the first quadrant and parallel to the line \(\sqrt{2} x + y =2 \sqrt{2}\). If \(d\) is the length of the line segment of \(N\) between \(H\) and the \(y\)-axis then \(d ^2\) is equal to \(............\).

Let the mean and variance of 7 observations 2, 4, 10, x, 12, 14, y, \( x>y \) be 8 and 16 respectively. Two numbers are chosen from {1, 2, 3, x-4, y, 5} one after another without replacement, then the probability, that the smaller number among the two chosen numbers is less than 4, is: