Mathematics Mock Test - 10

The normal to the curve $5x^5-10x^3+x+2y+6=0$ at $A(0,-3)$ meets the curve again at two points $B$ and $C$. The length of the line segment $BC$ is:

The lines $x=py+q,z=ry+s$ and $x=p^{\prime }y+q^{\prime },z=r^{\prime }y+s^{\prime }$ are perpendicular to each other, if:

If $\int \frac{dx}{x-x^3}=A\ln \left|\frac{x^2}{1-x^2}\right|+c$, then the value of $A$ is:

Let $f(x)=x-[x]$ for every real number $x$, where $[x]$ is the greatest integer less than or equal to $x$. Then the value of ${\int }_{-1}^{1}f(x)dx$ is:

The curves $f(x)=\sin x$ and $g(x)=\cos x$ intersect infinitely many times, forming bounded regions of equal areas. The area of one such region is: