AIEEE 2006 Mock Test

The two lines $x=ay+b,z=cy+d$; and $x=a^{\prime }y+b^{\prime },z=c^{\prime }y+d^{\prime }$ are perpendicular to each other if:

The locus of the vertices of the family of parabolas $y=\frac{a^3{x}^2}{3}+\frac{a^{2}x}{2}-2a$ is:

The values of $a$ for which the points $A,B,C$ with position vectors $2\hat{i}-\hat{j}+\hat{k}$, $\hat{i}-3\hat{j}-5\hat{k}$ and $a\hat{i}-3\hat{j}+\hat{k}$ respectively are the vertices of a right-angled triangle with $\angle C=\pi /2$ are:

${\int }_{-\pi /2}^{\pi /2}\left[[x+\pi ]+\cos (x+3\pi )\right]dx$ is equal to (where $[\cdot ]$ denotes the greatest integer function):

If $x$ is real, the maximum value of $\frac{3x^2+9x+17}{3x^2+9x+7}$ is: