Suppose A and B are the coefficients of \(30^{\text {th }}\) and \(12^{\text {th }}\) terms respectively in the binomial expansion of \((1+x)^{2 \mathrm{n}-1}\). If \(2 \mathrm{~A}=5 \mathrm{~B}\), then n is equal to :

Let \(f: \mathbb{R} \rightarrow \mathbb{R}\) be such that \(f(xy) = f(x)f(y)\), for all \(x, y \in \mathbb{R}\) and \(f(0) \neq 0\). Let \(g: [1, \infty) \rightarrow \mathbb{R}\) be a differentiable function such that
\(x^2 g(x) = \int\limits_1^x (t^2 f(t) - tg(t))\,dt\).
Then \(g(2)\) is equal to :

If the system of equations \( 11 x+y+\lambda z=-5 \) \( 2 x+3 y+5 z=3 \) \( 8 x-19 y-39 z=\mu\) has infinitely many solutions, then \(\lambda^4-\mu\) is equal to :

The shortest distance between the lines \(\dfrac{x-4}{1} = \dfrac{y-3}{2} = \dfrac{z-2}{-3}\) and \(\dfrac{x+2}{2} = \dfrac{y-6}{4} = \dfrac{z-5}{-5}\) is :

If \(z\) is a complex number such that \(\frac{z-i}{z-1}\) is purely imaginary, then the minimum value of \(\mid \mathrm{z}-(3+3 \mathrm{i}) \mid\) is :

The following system of linear equations  \(7 x+6 y-2 z=0\) ; \(3 x+4 y+2 z=0\) ; \({x}-2{y}-6{z}=0,\) has

Let \(9\) distinct balls be distributed among \(4\) boxes, \(B_{1}, B_{2}, B_{3}\) and \(B_{4}\). If the probability that \(B_{3}\) contains exactly \(3\) balls is \(k\left(\frac{3}{4}\right)^{9}\) then \(\mathrm{k}\) lies in the set:

The set of all values of \(k\) for which \(\left(\tan ^{-1} x \right)^{3}+\left(\cot ^{-1} x \right)^{3}= k \pi^{3}, x \in R\), is the interval

The mean age of \(25\) teachers in a school is \(40\) years. A teacher retires at the age of \(60\) years and a new teacher is appointed in his place. If the mean age of the teachers in this school now is \(39\) years, then the age (in years) of the newly appointed teacher is..........

The mean and the median of the following ten numbers in increasing order \(10, 22, 26, 29, 34, x, 42, 67, 70, y\) are \(42\) and \(35\) respectively, then \(\frac{y}{x}\) is equal to 

The range of \(f(x)=4 \sin ^{-1}\left(\frac{x^2}{x^2+1}\right)\) is

Let \(\vec{a}=\hat{\mathrm{i}}+\hat{\mathrm{j}}+\hat{\mathrm{k}}, \overrightarrow{\mathrm{b}}=2 \hat{\mathrm{i}}+2 \hat{\mathrm{j}}+\hat{\mathrm{k}}\) and \(\overrightarrow{\mathrm{d}}=\vec{a} \times \overrightarrow{\mathrm{b}}\). If \(\overrightarrow{\mathrm{c}}\) is a vector such that \(\vec{a} \cdot \overrightarrow{\mathrm{c}}=|\overrightarrow{\mathrm{c}}|\), \(|\overrightarrow{\mathrm{c}}-2 \vec{a}|^2=8\) and the angle between \(\overrightarrow{\mathrm{d}}\) and \(\overrightarrow{\mathrm{c}}\) is \(\frac{\pi}{4}\), then \(|10-3 \overrightarrow{\mathrm{~b}} \cdot \overrightarrow{\mathrm{c}}|+|\overrightarrow{\mathrm{d}} \times \overrightarrow{\mathrm{c}}|^2\) is equal to

\(\lim _{x \rightarrow \frac{\pi}{4}} \frac{8 \sqrt{2}-(\cos x+\sin x)^{7}}{\sqrt{2}-\sqrt{2} \sin 2 x}\) is equal to