For the following questions answer them individually
The value of the integral $$\int_{\frac{\pi}{24}}^{\frac{5\pi}{24}} \frac{dx}{1+\sqrt[3]{\tan2x}}$$ is:
The sum of all possible values of $$n\epsilon N$$, so that the coefficients of $$x,x^{2}\text{ and }x^{3}$$ in the expansion of $$(1+x^{2})^{2}(1+x)^{n}$$, are in arithmetic progression is:
Number of solutions of $$\sqrt{3}\cos2\theta+8\cos\theta+3\sqrt{3}=0,\theta\epsilon[-3\pi,2\pi]$$ is:
Let $$S=\left\{z:3\leq|2z-3(1+i)|\leq7\right\}$$ be a set of complex nwnbers. Then $$\min_{Z \epsilon S}\left|\left(z+\frac{1}{2}(5+3i)\right)\right|$$ is equal to :
Let $$\overrightarrow{a}=-\widehat{i}+\widehat{j}+2\widehat{k},\overrightarrow{b}=\widehat{i}-\widehat{j}-3\widehat{k},\overrightarrow{c}=\overrightarrow{a} \times \overrightarrow{b}\text{ and }\overrightarrow{d}=\overrightarrow{c}\times\overrightarrow{a}$$. Then $$\large (\overrightarrow{a}-\overrightarrow{b}).\overrightarrow{d}$$ is equal to:
A building construction work can be completed by two masons A and B together in 22.5 days. Mason A alone can complete the construction work in 24 days less than mason B alone. Then mason A alone will complete the construction work in :
Let the domain of the function $$f(x)=\log_{3}\log_{5}\log_{7}(9x-x^{2}-13)$$ be the interval (m, n). Let the hyperbola $$\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1$$ have eccentricity $$\frac{n}{3}$$ and the length of the latus rectum $$\frac{8m}{3}$$. Then $$b^{2}-a^{2}$$ is equal to:
The value of $$\frac{100_{C_{50}}}{51}+\frac{100_{C_{51}}}{52}+...+\frac{100_{C_{100}}}{101}$$ is:
The vertices B and C of a triangle ABC lie on the line $$\frac{x}{1}=\frac{1-y}{-2}=\frac{z-2}{3}$$ The coordinates of A and B are (1, 6, 3) and (4, 9, $$\alpha$$) respectively and C is at a distance of 10 units from B. The area (in sq. units) of $$\triangle$$ABC is :
Let $$f(x) = \left\{\begin{array}{l l}\frac{ax^{2}2ax+3}{4x^{2}+4x-3} ,& x\neq\quad -\frac{3}{2},\frac{1}{2}\\b, & \quad x=-\frac{3}{2},\frac{1}{2}\\\end{array}\right.$$
be continuous at $$x=-\frac{3}{2}$$. If $$f_{0}f(x)=-\frac{7}{5}$$ then x is equal to:
