If $f^{\text{'}}\left(x\right)=a\sin x+b\cos x, f^{\text{'}}\left(0\right)=4, f\left(0\right)=3$ and $f\left(\frac{\mathrm{\pi }}{2}\right)=5,$ then $f\left(x\right)=$

When an inductor of inductance \(\frac{6}{\pi} \mathrm{H}\), a capacitor of capacitance \(\frac{50}{\pi} \mu \mathrm{F}\) and resistor of resistance \(R\) are connected in series with an AC supply of rms voltage \(220 \mathrm{~V}\) and frequency \(50 \mathrm{~Hz}\), the rms current through the circuit is \(440 \mathrm{~mA}\). Match the inductive reactance, \(X_L\) the capacitive reactance, \(X_C\) the resistance \(R\) and the impedance \(Z\) of the circuit given in List-I with the corresponding values given in List-II.
\(\begin{array}{llll} \hline & \text { List- I } & & \text { List- II } \\ \hline \text {(A) } & X_L & \text { (i) } & 200 \Omega \\ \hline \text {(B) } & X_C & \text { (ii) } & 300 \Omega \\ \hline \text {(C) } & R & \text { (iii) } & 500 \Omega \\ \hline \text {(D) } & Z & \text { (iv) } & 600 \Omega \\ \hline \end{array}\)

In Young's double slit experiment, an interference pattern is obtained on a screen by a light of wavelength \(6000 Ã…\) coming from the coherent sources \(S_1\) and \(S_2\). At certain point \(P\) on the screen third dark fringe is formed. Then, the path difference \(S_1 P-S_2 P\) in microns is:

A force of \((4 i+2 j+k) \mathrm{N}\) is acting on a particle of mass 2 kg displaces the particle from a position of \((2 \bar{i}+2 \bar{j}+\bar{k}) \mathrm{m}\) to a position of \((4 \bar{i}+3 \bar{j}+2 \bar{k}) m\). The work done by the force on the particle in joules is

A car covers a distance at speed of $60 \mathrm{km} {\mathrm{h}}^{-1}$. It returns and comes back to the original point moving at a speed of $V$. If the average speed for the round trip is $48 \mathrm{km} {\mathrm{h}}^{-1}$, then the magnitude of $v$ is

At what temperature does the velocity of sound in air increased by 10% comparison with velocity at \(0^{\circ} \mathrm{C}\) ?

A short bar magnet has a magnetic moment of \(0.48 \mathrm{~J} \mathrm{~T}^{-1}\). The magnitude of magnetic field at a point at 10 cm distance from the centre of the magnet on its axis is

\(\bar{a}, \bar{b}\) are position vectors of the points \(A\) and \(B\) respectively, \(C\) and \(D\) are points on the line AB such that \(\overline{\mathrm{AB}}, \overline{\mathrm{AC}}\) and \(\overline{\mathrm{BD}}, \overline{\mathrm{BA}}\) are two pairs of like vectors. If \(\overline{\mathrm{AC}}=3 \overline{\mathrm{AB}}\) and \(\overline{\mathrm{BD}}=2 \overline{\mathrm{BA}}\), then \(\overline{\mathrm{CD}}=\)

If \(\int_0^{2 \pi}\left(\sin ^4 x+\cos ^4 x\right) d x=K \int_0^\pi \sin ^2 x d x+L \int_0^{\frac{\pi}{2}} \cos ^2 x d x\) and \(K, L \in N\), then the number of possible ordered pairs \((\mathrm{K}, \mathrm{L})\) is

A motor engine pumps \(1800 \mathrm{~L}\) of water per minute from a well of depth \(30 \mathrm{~m}\) and allows to pass through a pipe of cross-sectional area \(30 \mathrm{~cm}^2\). Then the power of the engine is (Acceleration due to gravity, \(g=10 \mathrm{~ms}^{-2}\) )

Speed of electron in its Ist Bohr's orbit is given by \(2.18 \times 10^6 \mathrm{~ms}^{-1}\). If the time period of electron in \(n\)th orbit is measured as 4.10 femto second, the value of \(n\) is

\(x=\cos ^{-1}\left(\frac{1}{\sqrt{1+t^2}}\right), y=\sin ^{-1}\left(\frac{t}{\sqrt{1+t^2}}\right) \Rightarrow \frac{d y}{d x}\)
is equal to

A person standing at the junction (crossing) of 2 straight paths represented by the equations 2x − 3y + 4 = 0 and 3x + 4 y − 5 = 0, wants to reach the path whose equation is 6 x − 7 y + 8 = 0 in the least time, then the equation of the path he should follow is