\(\lim _{n \rightarrow \infty}\left[\left(1+\frac{1}{n^2}\right)\left(1+\frac{2^2}{n^2}\right) \ldots . .\left(1+\frac{n^2}{n^2}\right)\right]^{1 / n}=\)

A quadrilateral $ABCD$ is divided by the diagonal $AC$ into two triangles of equal areas. If $A, B, C$ are respectively $\left(3,4\right), \left(-3,6\right), \left(-5,1\right)$, then the locus of $D$ is

\(x\) and \(y\) are two positive integers such that \(2 x+3 y=50\). If \(x^2 y^3\) is maximum for \(x=\alpha\) and \(y=\beta\), then \(\frac{\alpha}{2}+\frac{\beta}{5}=\)

If $\Delta =\left|\begin{array}{lll}1& a& a^2\\ 1& b& b^2\\ 1& c& c^2\end{array}\right|=K\left(a-b\right)\left(b-c\right)\left(c-a\right),$ then $K=$

A person who tosses an unbiased coin gains two points for turning up a head and loses one point for a tail. If three coins are tossed and the total score \(X\) is observed, then the range of \(x\) is

The domain of the function \(f(x)=\sin ^{-1}\left[\log _4\left(\frac{x}{4}\right)\right]+\sqrt{17 x-x^2-16}\) is

Let \(\overrightarrow{\mathbf{a}}=a_1 \hat{\mathbf{i}}+a_2 \hat{\mathbf{j}}+a_3 \hat{\mathbf{k}}\) Assertion (A) : The identity \(|\overrightarrow{\mathbf{a}} \times \hat{\mathbf{i}}|^2+|\overrightarrow{\mathbf{a}} \times \hat{\mathbf{j}}|^2+|\overrightarrow{\mathbf{a}} \times \hat{\mathbf{k}}|^2=2|\overrightarrow{\mathbf{a}}|^2\) holds for \(\vec{a}\). Reason (R) : \(\overrightarrow{\mathbf{a}} \times \hat{\mathbf{i}}=a_3 \hat{\mathbf{j}}-a_2 \hat{\mathbf{k}}\), \(\overrightarrow{\mathbf{a}} \times \hat{\mathbf{j}}=a_1 \hat{\mathbf{k}}-a_3 \hat{\mathbf{i}}, \overrightarrow{\mathbf{a}} \times \hat{\mathbf{k}}=a_2 \hat{\mathbf{i}}-a_1 \hat{\mathbf{j}}\) Which of the following is correct?

The quadratic equation whose roots are \(\ell=\lim _{\theta \rightarrow 0}\left(\frac{3 \sin \theta-4 \sin ^3 \theta}{\theta}-\right)\) and \(\mathrm{m}=\lim _{\theta \rightarrow 0}\left(\frac{2 \tan \theta}{\theta\left(1-\tan ^2 \theta\right)}\right)\) is

In Young's double slit experiment, intensity of light at a point on the screen where the path difference becomes \(\lambda\) is \(I\). The intensity at a point on the screen where the path difference becomes \(\frac{\lambda}{3}\) is,

In a Young's double slit experiment, \(m\) th order and \(n\)th order of bright fringes are formed at point \(P\) on a distant screen, if monochromatic source of wavelength \(400 \mathrm{~nm}\) and \(600 \mathrm{~nm}\) are used respectively. The minimum value of \(m\) and \(n\) are respectively,

A clock dial has point charges \(-\mathrm{q},-2 \mathrm{q},-3 \mathrm{q}, \ldots \ldots \ldots .-12 \mathrm{q}\) at the positions of the corresponding numbers on the dial respectively. The time at which the hour's hand points the direction of the net electric field at the centre of the dial is (Assume clock hands do not influence the net electric (field)

The sum of all distinct roots of the equation \(x^5-3 x^4+5 x^3-5 x^2+3 x-1=0\) is

In Fischer-Ringe's method of separation of nobles gas mixture from air ............. is used