A vertical pole subtends an angle \(\tan ^{-1}\left(\frac{1}{2}\right)\) at a point \(P\) on the ground. If the angles substended by the upper half and the lower half of the pole at \(P\) are respectively \(\alpha\) and \(\beta\), then \((\tan \alpha, \tan \beta)\) is equal to

If \(f_n(x)=\log \log \log \ldots \log x \quad(\log\) is repeated \(n\)-times), then \(\int\left(x f_1(x) f_2(x) \ldots f_n(x)\right)^{-1} d x\) is equal to

\(\int e^{-2 x}\left(\tan 2 x-2 \sec ^2 2 x \tan 2 x\right) d x=\)

If \(\quad f:[2, \infty) \rightarrow B \quad\) defined by \(f(x)=x^2-4 x+5\) is a bijection, then \(B\) is equal to

If $y+c=0$ is a tangent to the circle $x^2+y^2-6x-2y+1=0$ at $\left(a,4\right)$, then

In a \(\triangle A B C, \cos \left(\frac{B+2 C+3 A}{2}\right)+\cos \left(\frac{A-B}{2}\right)\) is equal to

The number of different permutations of letters that can be formed by taking 4 letters at a time from the letters of the word 'REPETITION' is

If $f\left(x\right)$ is a polynomial of degree $n$ with rational coefficients and $1+2i, 2-\sqrt{3}$ and $5$ are three roots of $f\left(x\right)=0,$ then the least value of $n$ is

Which of the following hormones is an example of polypeptide?

A cell of emf \(10 \mathrm{~V}\) and internal resistance \(3 \Omega\) is connected in parallel with another cell of emf \(7 \mathrm{~V}\) and internal resistance \(\frac{3}{5} \Omega\), such that their positive terminals are joined together and so their are negative terminals. Their positive terminals are joined with the negative terminal and their negative terminal is joined with the positive terminal of a third cell of emf \(20 \mathrm{~V}\) with internal resistance \(2 \Omega\). The combination can be replaced by a battery of emf \(E\) and internal resistance \(r\), then the values of \(E\) and \(R\) are respectively

In a \(p-n\) junction diode, an electric field of magnitude \(2 \times 10^5 \mathrm{~V} /\) mexists in the depletion region. A particle with charge \(-3 e\) can diffuse from \(n\)-side to \(p\)-side, if it has minimum kinetic energy \(0.6 \mathrm{eV}\). The width of the depletion region of the \(p-n\) junction is

Two blocks of masses \(2 \mathrm{~kg}\) and \(1 \mathrm{~kg}\) are tied to the ends of a string which passes over a light frictionless pulley. The blocks are held at rest at the same horizontal level and then released suddenly. The distance traversed by their centre of mass in 2 seconds is (acceleration due to gravity \(=10 \mathrm{~ms}^{-2}\) )

A solid metal sphere released in a vertical liquid column has attained terminal velocity in the downward direction. The magnitudes of viscous force, buoyant force and gravitational force acting on it are \(\mathrm{F}_{\mathrm{v}}, \mathrm{F}_{\mathrm{B}}\) and \(\mathrm{F}_{\mathrm{W}}\) respectively. Then the correct relation between them is