JEE Main 2008 — Calculus

How many real solutions does the equation $x^7+14 x^5+16 x^3+30 x-560=0$ have?

Let $\mathrm{I}=\int_0^1 \frac{\sin \mathrm{x}}{\sqrt{\mathrm{x}}} \mathrm{dx}$ and $\mathrm{J}=\int_0^1 \frac{\cos \mathrm{x}}{\sqrt{\mathrm{x}}} \mathrm{dx}$. Then which one of the following is true?

Let $f(x)=\left\{\begin{array}{ll}(x-1) \sin \left(\frac{1}{x-1}\right), & \text { if } x \neq 1 \\ 0, & \text { if } x=1\end{array}\right.$. Then which one of the following is true?

Suppose the cube $x^3-p x+q$ has three distinct real roots where $p>0$ and $q>0$. Then which one of the following holds?

The area of the plane region bounded by the curves $x+2 y^2=0$ and $x+3 y^2=1$ is equal to

The solution of the differential equation $\frac{d y}{d x}=\frac{x+y}{x}$ satisfying the condition $y(1)=1$ is

The value of $\sqrt{2} \int \frac{\sin x d x}{\sin \left(x-\frac{\pi}{4}\right)}$ is