Functions $f(x)$ and $g(x)$ are such that $f(x) + \int\limits_0^x {g(t)dt = 2\sin x - \frac{\pi}{2}}$ and $f'(x)g(x) = \cos^2 x$. The number of solutions of the equation $f(x) + g(x) = 0$ in the interval $(0, 3\pi)$ is:

Which one of the following statements is $NOT \text{ } CORRECT$?

Which of the following statements is true for the function $f(x) = \begin{cases} \sqrt{x} & x \ge 1 \\ x^3 & 0 \le x < 1 \\ \frac{x^3}{3} - 4x & x < 0 \end{cases}$

If $f$ is defined by $f(x) = \begin{cases} x & \text{for } 0 \leq x < 1 \\ 2-x & \text{for } x \geq 1 \end{cases}$,then at $x=1$,$f(x)$ is

Let $y(x) = (1+x)(1+x^2)(1+x^4)(1+x^8)(1+x^{16})$. Then $y'(x) - y''(x)$ at $x = -1$ is equal to:

If $f(x) = \begin{cases} ax^2 - bx + 2, & x < 3 \\ bx^2 - 3, & x \geq 3 \end{cases}$ is differentiable at every $x \in R$,then the area (in sq units) of the triangle formed by the line $\frac{x}{a} + \frac{y}{b} = 1$ with the coordinate axes is