\(\smallint \left( {1 + x - \frac{1}{x}} \right){e^{x + \frac{1}{x}}}\;dx = \)

Five numbers are in \(A.P.\), whose sum is \(25\) and product is \(2520 .\) If one of these five numbers is \(-\frac{1}{2},\) then the greatest number amongst them is

If \(\int_0^1 \frac{1}{\sqrt{3+x}+\sqrt{1+x}} d x=a+b \sqrt{2}+c \sqrt{3}\), where \(a, b, c\) are rational numbers, then \(2 a+3 b-4 c\) is equal to :

Let \(\overrightarrow{ a }=2 \hat{ i }+\hat{ j }+\hat{ k }\), and \(\overrightarrow{ b }\) and \(\overrightarrow{ c }\) be two nonzero vectors such that \(|\vec{a}+\vec{b}+\vec{c}|=|\vec{a}+\vec{b}-\vec{c}| \quad\) and \(\vec{b} \cdot \vec{c}=0\). Consider the following two statement: \((A)\) \(|\overrightarrow{ a }+\lambda \overrightarrow{ c }| \geq|\overrightarrow{ a }|\) for all \(\lambda \in R\). \((B)\) \(\overrightarrow{ a }\) and \(\overrightarrow{ c }\) are always parallel

Let \(f: R \rightarrow R\) be a function defined \(f(x)=\frac{2 e^{2 x}}{e^{2 x}+\varepsilon}\). Then \(f\left(\frac{1}{100}\right)+f\left(\frac{2}{100}\right)+f\left(\frac{3}{100}\right)+\ldots .+f\left(\frac{99}{100}\right)\) is equal to

If \(24 \int_0^{\frac{\pi}{4}}\left(\sin \left|4 x-\frac{\pi}{12}\right|+[2 \sin x]\right) \mathrm{d} x=2 \pi+\alpha\), where \([\cdot]\) denotes the greatest integer function, then \(\alpha\) is equal to _______.

If the real part of the complex number \((1-\cos \theta+2 i \sin \theta)^{-1}\) is \(\frac{1}{5}\) for \(\theta \in(0, \pi)\), then the value of the integral \(\int_{0}^{\theta} \sin x \,d x\) is equal to:

A candidate has to go to the examination centre to appear in an examination. The candidate uses only one means of transportation for the entire distance out of bus, scooter and car. The probabilities of the candidate going by bus, scooter and car, respectively, are \(\dfrac{2}{5}\), \(\dfrac{1}{5}\) and \(\dfrac{2}{5}\). The probabilities that the candidate reaches late at the examination centre are \(\dfrac{1}{5}\), \(\dfrac{1}{3}\) and \(\dfrac{1}{4}\) if the candidate uses bus, scooter and car, respectively. Given that the candidate reached late at the examination centre, the probability that the candidate travelled by bus is:

Let \(\mathrm{A}\) be a \(3 \times 3\) real matrix. If \(\operatorname{det}(2 \operatorname{Adj}(2 \operatorname{Adj}(\operatorname{Adj}(2 \mathrm{~A}))))=2^{41}\), then the value of \(\operatorname{det}\left(A^{2}\right)\) equal ..... .

Consider a triangle having vertices \(A(-2,3), B(1,9)\) and \(C(3,8)\). If a line \(L\) passing through the circum-center of triangle \(\mathrm{ABC}\), bisects line \(\mathrm{BC}\), and intersects \(\mathrm{y}\)-axis at point \(\left(0, \frac{\alpha}{2}\right)\), then the value of real number \(\alpha\) is \(.....\)

If \(f(x)=x^3-x^2 f^{\prime}(1)+x f^{\prime \prime}(2)-f^{\prime \prime \prime}(3), x \in R\), then

The value of  \(\operatorname{Lim}_{n \rightarrow \infty} \frac{1+2-3+4+5-6+\ldots+(3 n-2)+(3 n-1)-3 n}{\sqrt{2 n^4+4 n+3-} \sqrt{n^4+5 n+4}}\) is :

The maximum volume (in \(cu.m\)) of the right circular cone having slant height \(3\, m\) is