If \(\int \cos x \log \left(\tan \frac{x}{2}\right) d x$$=\sin x \log \left(\tan \frac{x}{2}\right)+f(x),\) then \(f(x)\) is equal to (assuming \(c\) is a arbitrary real constant)

\(7^{2 n}+16 n-1(n \in N)\) is divisible by

Lines \(x+y=1\) and \(3 y=x+3\) intersect the ellipse \(x^{2}+9 y^{2}=9\) at the points \(P, Q\) and \(R\). The area of the \(\triangle P Q R\) is

If \(M=\int_{0}^{n / 2} \frac{\cos x}{x+2} d x, N=\int_{0}^{\frac{\pi}{4}} \frac{\sin x \cos x}{(x+1)^{2}} d x,\) then
the value of \(M-N\) is

Let \(f(x)\) be a second degree polynomial. If \(f(1)=f(-1)\) and \(p, q, r\) are in A.P., then \(f^{\prime}(p), f^{\prime}(q), f^{\prime}(r)\) are

The value of \(\int_{-2}^2(x \cos x+\sin x+1) d x\) is

Time period \(T\) of a simple pendulum of length \(l\) is given by \(T=2 \pi \sqrt{\frac{l}{g}}\). If the length is increased by \(2 \%\), then an approximate change in the time period is

Which of the following has the strongest H-bond?

A body of mass \(2 \mathrm{~kg}\) moves in a horizontal circular path of radius \(5 \mathrm{~m}\). At an instant, its speed is \(2 \sqrt{5} \mathrm{~m} / \mathrm{s}\) and is increasing at the rate of \(3 \mathrm{~m} / \mathrm{s}^2\). The magnitude of force acting on the body at the instant is,

The ratio of magnetic field and magnetic moment at the centre of a current carrying circular loop is \(x\). When both the current and radius is doubled the ratio will be

During the preparation of \(\mathrm{NH}_3\) in Haber's process, the promoter(s) used is / are -

A glass slab consists of thin uniform layers of progressively decreasing refractive indices RI (see figure) such that the RI of any layer is \(\mu-m \Delta \mu\). Here, \(\mu\) and \(\Delta \mu\) denote the \(\mathrm{RI}\) of 0 th layer and the difference in RI between any two consecutive layers, respectively. The integer \(m=0,1,2,3, \ldots\) denotes the numbers of the successive layers. A ray of light from the oth layer enters the 1st layer at an angle of incidence of \(30^{\circ}\). After undergoing the mth refraction, the ray emerges parallel to the interface. If \(\mu=1.5\) and \(\Delta \mu=0.015,\) the value of \(m\) is

Consider the circuit shown. If all the cells have negligible internal resistance, what will be the current through the \(2 \Omega\) resistor when steady state is reached?