The value of current \(i_{1}\) flowing from \(A\) to \(C\) in the circuit diagram is\(.......A\)

A uniformly thick wheel with moment of inertia \(I\) and radius \(R\) is free to rotate about its centre of mass (see fig). A massless string is wrapped over its rim and two blocks of masses \(\mathrm{m}_{1}\) and \(\mathrm{m}_{2}\left(\mathrm{m}_{1}>\mathrm{m}_{2}\right)\) are attached to the ends of the string. The system is released from rest. The angular speed of the wheel when \(\mathrm{m}_{1}\) descents by a distance \(h\) is

Three conductors of same length having thermal conductivity \(\mathrm{k}_1, \mathrm{k}_2\) and \(\mathrm{k}_3\) are connected as shown in figure.

Area of cross sections of \(1^{\text {st }}\) and \(2^{\text {nd }}\) conductor are same and for \(3^{\text {rd }}\) conductor it is double of the \(1^{\text {st }}\) conductor. The temperatures are given in the figure. In steady state condition, the value of \(\theta\) is _______ \({ }^{\circ} \mathrm{C}\).
(Given : \(\mathrm{k}_1=60 \mathrm{Js}^{-1} \mathrm{~m}^{-1} \mathrm{~K}^{-1}, \mathrm{k}_2=120 \mathrm{Js}^{-1}\mathrm{~m}^{-1}\) \( \mathrm{~K}^{-1}, \mathrm{k}_3=135 \mathrm{Js}^{-1} \mathrm{~m}^{-1} \mathrm{~K}^{-1}\) )

Two cells of emf \(2\, E\) and \(E\) with internal resistance \(r _{1}\) and \(r _{2}\) respectively are connected in series to an external resistor \(R\) (see \(figure\)). The value of \(R ,\) at which the potential difference across the terminals of the first cell becomes zero is

A body is moving in a low circular orbit about a planet of mass \(M\) and radius \(R\). The radius of the orbit can be taken to be \(R\) it self. Then the ratio of the speed of this body in the orbit to the escape velocity from the planet is

A capacitor of \(50 \,\mu {F}\) is connected in a circuit as shown in figure. The charge on the upper plate of the capacitor is \(......\,\mu {C} .\)

A thin glass plate of thickness is \(\frac{{2500}}{3}\,\lambda \) (\(\lambda \) is wavelength of light used) and refractive index \(\mu  = 1.5\) is inserted between one of the slits and the screen in Young's double slit experiment. At a point on the screen equidistant from the slits, the ratio of the intensities before and after the introduction of the glass plate is

A very long wire \(ABDMNDC\) is shown in figure carrying current \(I. AB\) and \(BC\) parts are straight, long and at right angle. At \(D\) wire forms a circular turn \(DMND\) of radius \(R. AB.\) \(\mathrm{BC}\) parts are tangential to circular turn at \(\mathrm{N}\) and \(D\). Magnetic field at the centre of circle is

Let \(f : R \rightarrow R\) be defined as \(f ( x )= x ^{3}+ x -5\). If \(g ( x )\) is a function such that \(f ( g ( x ))= x\), \(\forall x \in R\), then \(g ^{\prime}(63)\) is equal to

Let  \(a,b,c\; \in R.\) If \(f\left( x \right) = a{x^2} + bx + c\) is such that \(a + b + c = 3\) and \(f\left( {x + y} \right) = f\left( x \right) + f\left( y \right) + xy,\) \(\forall x,y \in R,\) then \(\mathop \sum \limits_{n = 1}^{10} f\left( n \right)\) is equal to : 

The locus of the centroid of the triangle formed by any point \(\mathrm{P}\) on the hyperbola \(16 \mathrm{x}^{2}-9 \mathrm{y}^{2}+\) \(32 x+36 y-164=0\), and its foci is:

Let \(f: \mathbf{R} \rightarrow \mathbf{R}\) be a twice differentiable function such that \(f(x+y)=f(x) f(y)\) for all \(x, y \in \mathbf{R}\). If \(f^{\prime}(0)=4 \mathrm{a}\) and \(f\) satisfies \(f^{\prime \prime}(x)-3 \mathrm{a} f^{\prime}(x)-f(x)=0\), \(\mathrm{a}\gt0\), then the area of the region \(\mathrm{R}=\{(x, y) \mid 0 \leq y \leq f(\mathrm{a} x), 0 \leq x \leq 2\}\) is:

In Bohr's atomic model, the electron is assumed to revolve in a circular orbit of radius \(0.5 \times 10^{-10}\,m\). If the speed of electron is \(2.2 \times 10^{6} \;{m} / {s}\), then the current associated with the electron will be \(....\,\times 10^{-2} \;{mA}\). [Take \(\pi\) as \(\left.\frac{22}{7}\right]\)