For the following questions answer them individually
Let be the origin, the point A be $$z=\sqrt{3}+2\sqrt{2}i$$, the point $$B(z_{2})$$ be such that $$\sqrt{3}|z_{2}|=|z_{1}|$$ and $$arg(z_{2})=arg(z_{1})+\frac{\pi}{6}$$. Then
Let $$f=\mathbb{R}\rightarrow \mathbb{R}$$ be a function defined by $$f(x)=(2+3a)x^{2}+(\frac{a+2}{a-1})x+b,a\neq 1$$ If $$f(x+y)=f(x)+f(y)+1-\frac{2}{7}xy$$, then the value of $$28\sum_{i=1}^{5}|f(i)|$$ is
Let ABCD be a trapezium whose vertices lie on the parabola $$y^{2}=4x$$. Let the sides AD and BC of the trapezium be parallel to y -axis. If the diagonal AC is of length $$\frac{25}{4}$$ and it passes through the point (1,0), then the area of ABCD is
The sum of all local minimum values of the function
$$f(x) = \left\{\begin{array}{l l}1-2x, & \quad {x<-1}\\ \frac{1}{3}(7+2|x|), & \quad {-1\leq x\leq 2}\\\frac{11}{8}(x-4)(x-5), & \quad {x>2}\\ \end{array}\right.$$ is
Let $$^{n}C_{r-1}=28,^{n}C_{r}=56$$ and $$^{n}C_{r+1}=70$$. Let $$A(4\cos t,4\sin t),B(2\sin t,-2\cos t)$$ and $$C(3r-n,r^{2}-n-1)$$ be the vertices of a triangle ABC, where $$t$$is a parameter. If $$(3x-1)^{2}+(3y)^{2}=\alpha$$, is the locus of the centroid of triangle ABC, then $$\alpha$$ equals
Let the equation of the circle, which touches x-axis at the point (a,0),a > 0 and cuts off an intercept of length b on y-axis be $$x^{2}+y^{2}-\alpha x +\beta y+\gamma =0$$. If the circle lies below x-axis, then the ordered pair $$(2a,b^{2})$$ is equal to
If $$f(x)=\frac{2^{x}}{2^{x}+\sqrt{2}},x \in \mathbb{R}$$, then $$\sum_{k=1}^{81}f(\frac{k}{82})$$ is equals to
Two number $$k_{1}$$ and $$k_{2}$$ are randomly chosen from the set of natural numbers. Then, the probability that the value of $$i^{k_{1}}+i^{k_{2}},(i=\sqrt{-1})$$ is non-zero, equals
If the image of the point (4,4,3)in the line $$\frac{x-1}{2}=\frac{y-2}{1}=\frac{z-1}{3}$$ is $$(\alpha ,\beta ,\gamma)$$, then $$\alpha +\beta +\gamma$$ is equal to
$$\cos \left(\sin^{-1}\frac{3}{5}+\sin^{-1}\frac{5}{13}+\sin^{-1}\frac{33}{65}\right)$$ is equal to:
