If the initial velocity in horizontal direction of a projectile is unit vector \(\hat{i}\) and the equation of trajectory is \(y =5 x (1- x )\). The \(y\) component vector of the initial velocity is. (Take \(g=10\,m / s ^{2}\) )

Three charges \(+Q, q, +Q\) are placed respectively, at distance, \(0, \frac d2\) and \(d\) from the origin, on the \(x-\) axis. If the net force experienced by \(+Q\), placed at \(x = 0\), is zero, then value of \(q\) is

Which of the following statements are correct? \((A)\) Electric monopoles do not exist whereas magnetic monopoles exist. \((B)\) Magnetic field lines due to a solenoid at its ends and outside cannot be completely
straight and confined. \((C)\) Magnetic field lines are completely confined within a toroid. \((D)\) Magnetic field lines inside a bar magnet are not parallel. \((E)\) \(\chi=-1\) is the condition for a perfect diamagnetic material, where \(\chi\) is its
magnetic susceptibility. Choose the correct answer from the options given below:

A given object takes \(n\) times the time to slide down \(45^{\circ}\) rough inclined plane as it takes the time to slide down an identical perfectly smooth \(45^{\circ}\) inclined plane. The coefficient of kinetic friction between the object and the surface of inclined plane is _______.

A block moving horizontally on a smooth surface with a speed of \(40\, {m} / {s}\) splits into two parts with masses in the ratio of \(1: 2\). If the smaller part moves at \(60\, {m} / {s}\) in the same direction, then the fractional change in kinetic energy is :-

A physical quantity \(Q\) is related to four observables \(a, b, c, d\) as follows :
\(\mathrm{Q}=\frac{\mathrm{ab}{ }^4}{\mathrm{~cd}}\)

where, \(\mathrm{a}=(60 \pm 3) \mathrm{Pa} ; \mathrm{b}=(20 \pm 0.1) \mathrm{m} ; \mathrm{c}=(40 \pm 0.2) \mathrm{Nsm}^{-2}\) and \(\mathrm{d}=(50 \pm 0.1) \mathrm{m}\), then the percentage error in Q is \(\frac{x}{1000}\), where \(x=\) ________ .

A block of mass 1 kg , moving along x with speed \(\mathrm{v}_{\mathrm{i}}=10 \mathrm{~m} / \mathrm{s}\) enters a rough region ranging from \(\mathrm{x}=0.1 \mathrm{~m}\) to \(\mathrm{x}=1.9 \mathrm{~m}\). The retarding force acting on the block in this range is \(\mathrm{F}_{\mathrm{r}}=-\mathrm{kx} \mathrm{N}\), with \(\mathrm{k}=10 \mathrm{~N} / \mathrm{m}\). Then the final speed of the block as it crosses rough region is

If one wants to remove all the mass of the earth to infinity in order to break it up completely. The amount of energy that needs to be supplied will be \(\frac{x}{5}\, \frac{ GM ^{2}}{ R }\) where \(x\) is \(\quad\,..........\) (Round off to the Nearest Integer) (\(M\) is the mass of earth, \(R\) is the radius of earth, \(G\) is the gravitational constant)

A particle is moving in one dimension (along the x-axis) under the action of a variable force. Its position \( x \) (in meters) varies with time \( t \) (in seconds) according to the equation \( x = -3t^3 + 18t^2 + 16t + 16 \). The velocity of the particle when its acceleration becomes zero is _________ \( \text{m/s} \).

A liquid column of height \(0.04 \mathrm{~cm}\) balances excess pressure of soap bubble of certain radius. If density of liquid is \(8 \times 10^3 \mathrm{~kg} \mathrm{~m}^{-3}\) and surface tension of soap solution is \(0.28 \mathrm{Nm}^{-1}\), then diameter of the soap bubble is _______ \(\mathrm{cm}\). \(\text { (if } g=10 \mathrm{~ms}^{-2} \text { ) }\)

The root mean square speed of molecules of nitrogen gas at \(27^{\circ} C\) is approximately\(.......m/s\)(Given mass of a nitrogen molecule \(=4.6 \times 10^{-26}\,kg\) and take Boltzmann constant  \(k _{ B }=1.4 \times 10^{-23}\,JK ^{-1}\) )

Given below are two statements: one is labelled as Assertion (A) and the other is labelled as Reason (R).
Assertion (A): The electromagnetic wave exerts pressure on the surface on which they are allowed to fall.
Reason (R): There is no mass associated with the electromagnetic waves.
In the light of the above statements, choose the correct answer from the options given below :