The locus of \(z=x+i y\), such that \(\operatorname{Im}\left(\frac{z-3 i}{i z+4}\right)\) \(=0\) is

If \(y=f(x)^{g(x)}\) and \(\frac{d y}{d x}=y\left[\mathrm{H}(x) f^{\prime}(x)+\mathrm{G}(x) g^{\prime}(x)\right]\), then \(\int \frac{\mathrm{G}(x) \mathrm{H}(x) f^{\prime}(x)}{g(x)} d x=\)

If \(\left|Z_1-3-4 i\right|=5\) and \(\left|Z_2\right|=15\) then the sum of the maximum and minimum values of \(\left|Z_1-Z_2\right|\) is

If \(y=\cos ^{-1}\left(\frac{6 x-2 x^2-4}{2 x^2-6 x+5}\right)\) then \(\frac{d y}{d x}=\)

If \(f(t)=\int_{-t}^t \frac{e^{-|x|}}{2} d x\), then \(\lim _{t \rightarrow \infty} f(t)\) is equal

If \(y=e^{a x}(\cos b x+\sin b x)\) satisfies the equation \(\frac{d^2 y}{d x^2}-K \frac{d y}{d x}+L y=0\), then \(L+b K=\)

If \(\mathrm{M}\) is the foot of the perpendicular drawn from \(\mathrm{P}(1,2,-1)\) to the plane passing through the point \(A(3,-2,1)\) and perpendicular to the vector \(4 \hat{i}+7 \hat{j}-4 \hat{k}\), then the length of PM is

Two identical bar magnets of magnetic moment \(M\) each, are placed along \(X\) and \(Y\)-axes, respectively at a distance \(d\) from the origin (as shown in the figure). The origin lies on perpendicular bisector of magnet placed on \(X\)-axis and on the magnetic axis of magnet placed on \(Y\)-axis. If the magnitude of total magnetic field at the origin is \(B=\alpha\left[\frac{\mu_0}{4 \pi} \frac{M}{d^3}\right]\), then the value of constant \(\alpha\) will be \(\langle d>>l\), where \(l\) is the length of the bar magnets and direction of \(\mathrm{N}\) to \(\mathrm{S}\) in magnets is opposite with respect to each other)

If \(2 x+3 y+12=0\) and \(x-y+4 \lambda=0\) are conjugate with respect to the parabola \(y^2=8 x\), then \(\lambda\) is equal to

The height of transmitting antenna if TV programs have to cover the population in area of radius $64 \mathrm{km}$ is
[Use Radius of earth $=6.4\times {10}^6 \mathrm{m}$]

A radioactive nucleus can decay by two different processes. The half lives of the first and second decay processes are \(5 \times 10^3\) and \(10^5\) years respectively. Then, the effective half-life of the nucleus is

Two gases \(A\) and \(B\) are contained in two separate, but otherwise identical containers. Gas \(A\) consists of monatomic molecules, each with atomic mass of \(4 u\) whereas Gas \(B\) consists of rigid diatomic molecules, each with atomic mass of \(20 u\). If gas \(A\) is kept at \(27^{\circ} \mathrm{C}\), at what temperature should gas \(B\) be kept so that both have the same rms speed?