The value of \(\int_0^1\left(2 x^3-3 x^2-x+1\right)^{\frac{1}{3}} d x\) is equal to :

If \(2 \tan ^2 \theta-5 \sec \theta=1\) has exactly \(7\) solutions in the interval \(\left[0, \frac{n \pi}{2}\right]\), for the least value of \(n \in N\) then \(\sum_{\mathrm{k}=1}^{\mathrm{n}} \frac{\mathrm{k}}{2^{\mathrm{k}}}\) is equal to :

Let a line \(\mathrm{y}=\mathrm{mx}(\mathrm{m}>0)\) intersect the parabola, \(\mathrm{y}^{2}=\mathrm{x}\) at a point \(\mathrm{P},\) other than the origin. Let the tangent to it at \(P\) meet the \(x\) -axis at the point \(Q\). If area \((\Delta \mathrm{OPQ})=4\) sq. units, then \(\mathrm{m}\) is equal to

Let three vectors \(\overrightarrow{ a }, \overrightarrow{ b }\) and \(\overrightarrow{ c }\) be such that \(\overrightarrow{ c }\) is coplanar with \(\overrightarrow{ a }\) and \(\overrightarrow{ b }, \overrightarrow{ a } \cdot \overrightarrow{ c }=7\) and \(\overrightarrow{ b }\) is perpendicular to \(\overrightarrow{ c },\) where \(\overrightarrow{ a }=-\hat{ i }+\hat{ j }+\hat{ k }\) and \(\overrightarrow{ b }=2 \hat{ i }+\hat{ k },\) then the value of \(2|\overrightarrow{ a }+\overrightarrow{ b }+\overrightarrow{ c }|^{2}\) is .........

Let \(I_{n}=\int_{1}^{e} x^{19}(\log |x|)^{n} d x,\) where \(n \in N\). If \((20) I _{10}=\alpha I _{9}+\beta I _{8},\) for natural numbers \(\alpha\) and \(\beta\), then \(\alpha-\beta\) equal to ..... .

Let M denote the set of all real matrices of order \(3 \times 3\) and let \(\mathrm{S}=\{-3,-2,-1,1,2\}\). Let
\(\mathrm{S}_1=\left\{\mathrm{A}=\left[a_{\mathrm{ij}}\right] \in \mathrm{M}: \mathrm{A}=\mathrm{A}^{\mathrm{T}} \text { and } a_{\mathrm{ij}} \in \mathrm{~S}, \forall \mathrm{i}, \mathrm{j}\right\}, \)
\( \mathrm{S}_2=\left\{\mathrm{A}=\left[a_{\mathrm{ij}}\right] \in \mathrm{M}: \mathrm{A}=-\mathrm{A}^{\mathrm{T}} \text { and } a_{\mathrm{ij}} \in \mathrm{~S}, \forall \mathrm{i}, \mathrm{j}\right\}, \)
\( \mathrm{S}_3=\{\mathrm{A}=\left[a_{\mathrm{ij}}\right] \in \mathrm{M}: a_{11}+a_{22}+a_{33}=0\) and \(a_{\mathrm{ij}} \in \mathrm{~S}, \forall \mathrm{i}, \mathrm{j}\}\)
If \(n\left(\mathrm{~S}_1 \cup_2 \mathrm{US}_3\right)=125 \alpha\), then \(\alpha\) equals _______

If \(\lim _{x \rightarrow 1} \frac{(5 x+1)^{1 / 3}-(x+5)^{1 / 3}}{(2 x+3)^{1 / 2}-(x+4)^{1 / 2}}=\frac{m \sqrt{5}}{n(2 n)^{2 / 3}}\), where \(\operatorname{gcd}(m, n)=1\), then \(8 m+12 n\) is equal to ...........

The number of 6-letter words, with or without meaning, that can be formed using the letters of the word MATHS such that any letter that appears in the word must appear at least twice, is _______.

The integral \(80 \int_0^{\frac{\pi}{4}}\left(\frac{\sin \theta+\cos \theta}{9+16 \sin 2 \theta}\right) d \theta\) is equal to :

Let \(\alpha = 3+4+8+9+13+14+\ldots\) upto 40 terms. If \((\tan\beta)^{\frac{\alpha}{1020}}\) is a root of the equation \(x^2+x-2=0\), \(\beta \in \left(0, \dfrac{\pi}{2}\right)\), then \(\sin^2\beta + 3\cos^2\beta\) is equal to:

Let \([ x ]\) denote the greatest integer \(\leq x\). Consider the function \(f(x)=\max \left\{x^2, 1+[x]\right\}\). Then the value of the integral \(\int \limits_0^2 f ( x ) dx\) is :

If \(0 < a , b < 1,\) and \(\tan ^{-1} a +\tan ^{-1} b =\frac{\pi}{4},\) then the value of \((a+b)-\left(\frac{a^{2}+b^{2}}{2}\right)+\left(\frac{a^{3}+b^{3}}{3}\right)-\left(\frac{a^{4}+b^{4}}{4}\right)+\ldots\) is ..... .

Let \(A =\left\{\theta \in(0,2 \pi): \frac{1+2 i \sin \theta}{1- i \sin \theta}\right.\) is purely imaginary \(\}\). Then the sum of the elements in \(A\) is