The equation${e^x} - x - 1 = 0$ has

The number of solutions of $\sin ^2 \mathrm{x}+\left(2+2 \mathrm{x}-\mathrm{x}^2\right) \sin \mathrm{x}-3(\mathrm{x}-1)^2=0$, where $-\pi \leq \mathrm{x} \leq \pi$, is....................

Let $[t]$ denote the greatest integer $\leq t .$ Then the equation in $x ,[ x ]^{2}+2[ x +2]-7=0$ has

If the sum of the two roots of the equation $4{x^3} + 16{x^2} - 9x - 36 = 0$ is zero, then the roots are

If $x$ is real, then the value of ${x^2} - 6x + 13$ will not be less than

If $\alpha ,\beta,\gamma$ are the roots of equation $x^3 + 2x -5 = 0$ and if equation $x^3 + bx^2 + cx + d = 0$ has roots $2 \alpha + 1, 2 \beta + 1, 2 \gamma + 1$ , then value of $|b + c + d|$ is (where $b,c,d$ are coprime)-