\(\lim _{n \rightarrow \infty}\left[\left(1+\frac{1}{n^2}\right)\left(1+\frac{4}{n^2}\right)\left(1+\frac{9}{n^2}\right) \ldots(2)\right]^{1 / n}=\)

If the product of the lengths of the perpendiculars drawn from the foci to the tangent \(y=\frac{-3}{4} x+3 \sqrt{2}\) of the ellipse \(\frac{x^2}{a^2}+\frac{y^2}{b^2}=1\) is 9 , then the eccentricity of that ellipse is

The probability of securing a success in a trial is three times that of a failure. The probability of getting atleast 4 successes in 5 trials is

If \(\frac{3 x+1}{(x-1)^2\left(x^2+1\right)}=\frac{\mathrm{A}}{x-1}+\frac{\mathrm{B}}{(x-1)^2}+\frac{\mathrm{C} x+\mathrm{D}}{x^2+1}\), then \(2(\mathrm{~A}-\mathrm{C}+\mathrm{B}+\mathrm{D})=\)

If \(f:[2,3] \rightarrow R\) is defined by \(f(x)=x^3+3 x-2\), then the range \(f(x)\) is contained in the interval

If \(\ell, \mathrm{m}, \mathrm{n}\) and \(\mathrm{a}, \mathrm{b}, \mathrm{c}\) are direction cosines of two lines then

\(\mathrm{A}, \mathrm{B}\) are the events in a random experiment. If \(\mathrm{P}(\mathrm{A})=\frac{1}{2}, \mathrm{P}(\mathrm{B})=\frac{1}{3}, \mathrm{P}(\mathrm{A} \cap \mathrm{B})=\frac{1}{4}\), then \(\mathrm{P}\left(\frac{\mathrm{A}^{\mathrm{c}}}{\mathrm{B}^{\mathrm{c}}}\right)+\mathrm{P}\left(\frac{\mathrm{A}}{\mathrm{B}}\right)=\)

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 functional groups involved in the conversion of glucose to gluconic acid and gluconic acid to saccharic acid respectively are

In a triangle ABC , if \(a=5, b=3, c=7\), then \(\sqrt{\frac{\sin (A-B)}{\sin (A+B)}}=\)

The displacement of a particle executing SHM is given by \(y=5 \sin \left(4 t+\frac{\pi}{3}\right) .\) If \(T\) is the time period and the mass of the particle is \(2 \mathrm{~g}\), the kinetic energy of the particle when \(t=\frac{T}{4}\) is given by

If \(\overrightarrow{\mathbf{a}}+\overrightarrow{\mathbf{b}}+\overrightarrow{\mathbf{c}}=\overrightarrow{\mathbf{0}}\) and \(|\overrightarrow{\mathbf{a}}|=3,|\overrightarrow{\mathbf{b}}|=4\) and \(|\overrightarrow{\mathbf{c}}|=\sqrt{37}\), then the angle between \(\overrightarrow{\mathbf{a}}\) and \(\overrightarrow{\mathbf{b}}\) is :

If \(\mathrm{e}^{\mathrm{x}}=\mathrm{y}+\sqrt{y^2-1}\), then \(\frac{d y}{d x}=\)