\(\int \limits_{\pi / 6}^{\pi / 3} \tan ^{3} x \cdot \sin ^{2} 3 x\left(2 \sec ^{2} x \cdot \sin ^{2} 3 x+3 \tan x \cdot \sin 6 x\right) d x\) is equal to

Let \(f(x)=2 \cos ^{-1} x+4 \cot ^{-1} x-3 x^{2}-2 x+10, x \in[-\) \(1,1]\). If \([ a , b ]\) is the range of the function then \(4 a -\) \(b\) is equal to

The distance of the point \((1,-2,3)\) from the plane \(x-y+z=5\) measured parallel to a line, whose direction ratios are \(2,3,-6\) is :

Let \(f, \mathrm{~g}:(1, \infty) \rightarrow \mathbb{R}\) be defined as \(f(\mathrm{x})=\frac{2 x+3}{5 x+2}\) and \(g(x)=\frac{2-3 x}{1-x}\). If the range of the function \(f \circ g:[2,4] \rightarrow \mathbb{R}\) is \([\alpha, \beta]\), then \(\frac{1}{\beta-\alpha}\) is equal to

The number of ways of giving \(20\) distinct oranges to \(3\) children such that each child gets atleast one orange is \(............\).

If the system of equations:
\(x+y+z=5\)
\(x+2y+3z=9\)
\(x+3y+\lambda z=\mu\)
has infinitely many solutions, then the value of \(\lambda+\mu\) is:

Let \(\vec{a}, \vec{b}\) and \(\vec{c}\) be three units vectors such that \(\overrightarrow{\mathrm{a}}+\overrightarrow{\mathrm{b}}+\overrightarrow{\mathrm{c}}=\overrightarrow{0} .\) If \(\lambda=\overrightarrow{\mathrm{a}} \cdot \overrightarrow{\mathrm{b}}+\overrightarrow{\mathrm{b}} \cdot \overrightarrow{\mathrm{c}}+\overrightarrow{\mathrm{c}} \cdot \overrightarrow{\mathrm{a}} \) and \(\overrightarrow{\mathrm{d}}=\overrightarrow{\mathrm{a}} \times \overrightarrow{\mathrm{b}}+\overrightarrow{\mathrm{b}} \times \overrightarrow{\mathrm{c}}+\overrightarrow{\mathrm{c}} \times \overrightarrow{\mathrm{a}},\) then the ordered pair \((\lambda, {\mathrm{\vec d}})\) is equal to 

The sum of the solutions \(x \in \mathbb{R}\) of the equation \(\frac{3 \cos 2 x+\cos ^3 2 x}{\cos ^6 x-\sin ^6 x}=x^3-x^2+6\) is

If the set \(\left\{\operatorname{Re}\left(\frac{z-\bar{z}+z \bar{z}}{2-3 z+5 \bar{z}}\right): z \in C , \operatorname{Re}(z)=3\right\}\) is equal to the interval \((\alpha, \beta]\), then \(24(\beta-\alpha)\) is equal to

Let f be a twice differentiable non-negative function such that \( (f(x))^{2}=25+\int_{0}^{x}((f(t))^{2}+(f'(t))^{2})dt \). Then the mean of \(f\left(\log _e(1)\right), f\left(\log _e(2)\right), \ldots \ldots, f\left(\log _e(625)\right)\) is equal to:

Let 729, 81, 9, 1, .... be a sequence and \( P_{n} \) denote the product of the first n terms of this sequence. If \( 2\sum_{n=1}^{40}(P_{n})^{\frac{1}{n}}=\frac{3^{\alpha}-1}{3^{\beta}} \) and \( \gcd(\alpha,\beta)=1 \), then \( \alpha+\beta \) is equal to

Let \(f, g: N \rightarrow N\) such that \(f(n+1)=f(n)+f(1)\) \(\forall \, n \in N\) and \(g\) be any arbitrary function. Which of the following statements is \(NOT\) true ?

Let \(\lambda \in R , \vec{a}=\lambda \hat{i}+2 \hat{j}-3 \hat{k}, \vec{b}=\hat{i}-\lambda \hat{j}+2 \hat{k}\) If \(((\vec{a}+\vec{b}) \times(\vec{a} \times \vec{b})) \times(\vec{a}-\vec{b})=8 \hat{i}-40 \hat{j}-24 \hat{k}\), then \(|\lambda(\vec{a}+\vec{b}) \times(\vec{a}-\vec{b})|^2\) is equal to