For the two cells having same EMF E and internal resistance r, the current passing through the external resistor 6 \(\Omega\) is same when both the cells are connected either in parallel or in series. The value of internal resistance r is ___________ \(\Omega\).

Given below are two statements: one is labelled as Assertion \(A\) and the other is labelled as Reason \(R\). \(Assertion\) \(A\) : Number of photons increases with increase in frequency of light. \(Reason\) \(R\) : Maximum kinetic energy of emitted electrons increases with the frequency of incident radiation. In the light of the above statements, choose the most appropriate answer from the options given below :

A vehicle travels \(4\,km\) with speed of \(3\,km / h\) and another \(4\,km\) with speed of \(5\,km / h\), then its average speed is \(..........km/h\)

A hydraulic press containing water has two arms with diameters as mentioned in the figure. A force of \(10 \mathrm{~N}\) is applied on the surface of water in the thinner arm. The force required to be applied on the surface of water in the thicker arm to maintain equilibrium of water is _______ \({N}\).

Identify the correct statements :
A. Effective capacitance of a series combination of capacitors is always smaller than the smallest capacitance of the capacitor in the combination.
B. When a dielectric medium is placed between the charged plates of a capacitor, displacement of charges cannot occur due to insulation property of dielectric.
C. Increasing of area of capacitor plate or decreasing of thickness of dielectric is an alternate method to increase the capacitance.
D. For a point charge, concentric spherical shells centered at the location of the charge are equipotential surfaces.
Choose the correct answer from the options given below.

An electromagnetic wave of frequency \(3\, GHz\) enters a dielectric medium of relative electric permittivity \(2.25\) from vacuum. The wavelength of this wave in that medium will be \(.......\,\times 10^{-2} \, cm\)

In the figure shown \(ABC\) is a uniform wire . If centre of mass of wire lies vertically below point \(A\) , then \(\frac{{BC}}{{AB}}\) is close to 

A force \(\overrightarrow{ F }=(\hat{ i }+2 \hat{ j }+3 \hat{ k }) N\) acts at a point \((4 \hat{ i }+3 \hat{ j }-\hat{ k }) m \cdot\) Then the magnitude of torque about the point \((\hat{i}+2 \hat{j}+\hat{k}) m\) will be \(\sqrt{ x } N - m .\) The value of \(x\) is\(........\)

Given below are two statements: Statement \(I :\) Maximum power is dissipated in a circuit containing an inductor, a capacitor and a resistor connected in series with an \(AC\) source, when resonance occurs Statement \(II :\) Maximum power is dissipated in a circuit containing pure resistor due to zero phase difference between current and voltage. In the light of the above statements, choose the correct answer from the options given below :

Let \(y=f(x)\) be the solution of the differential equation \(\frac{\mathrm{d} y}{\mathrm{~d} x}+\frac{x y}{x^2-1}=\frac{x^6+4 x}{\sqrt{1-x^2}},-1 \lt x \lt 1\) such that \(f(0)=0\). If \(6 \int_{-1 / 2}^{1 / 2} f(x) \mathrm{d} x=2 \pi-\alpha\) then \(\alpha^2\) is equal to _______ .

A stream of a positively charged particles having \(\frac{ q }{ m }=2 \times 10^{11} \frac{ C }{ kg }\) and velocity \(\overrightarrow{ v }_0=3 \times 10^7 \hat{ i ~ m} / s\) is deflected by an electric field \(1.8 \hat{ j } kV / m\). The electric field exists in a region of \(10 cm\) along \(x\) direction. Due to the electric field, the deflection of the charge particles in the \(y\) direction is \(...........mm\)

Let a function \(g:[0,4] \rightarrow R\) be defined as \(g ( x )=\left\{\begin{array}{ll}\max _{0 \leq t \leq x }\left\{ t ^{3}-6 t ^{2}+9 t -3\right\} & , 0 \leq x \leq 3 \\ 4- x & , 3 < x \leq 4\end{array}\right.\) then the number of points in the interval \((0,4)\) where \(g(x)\) is NOT differentiable, is \(.....\)

The equation of circle described on the chord \(3x + y+ 5\, = 0\) of the circle \(x^2 + y^2\, = 16\) as diameter is