AIEEE 2006 Mock Test

The value of ${\int }_1^a[x]f^{\prime }(x)dx,a>1$, where $[x]$ denotes the greatest integer not exceeding $x$, is:

If the lines $3x-4y-7=0$ and $2x-3y-5=0$ are two diameters of a circle of area $49\pi$ square units, the equation of the circle is:

The differential equation whose solution is $A{x}^2+B{y}^2=1$, where $A$ and $B$ are arbitrary constants, is of:

Let $C$ be the circle with centre $(0,0)$ and radius $3$ units. The equation of the locus of the mid points of the chords of the circle $C$ that subtend an angle of $2\pi /3$ at its centre is:

If $(a,a^2)$ falls inside the angle made by the lines $y=x/2,x>0$ and $y=3x,x>0$, then $a$ belongs to: