If $$\begin{gathered} \mathrm{f}\left(\mathrm{x}\right) \left\{= \mathrm{x} \mathrm{for} \mathrm{x} \le 0 \\ = 0 \mathrm{for} \mathrm{x} > 0\right\} \end{gathered}$$, then f(x) at x = 0 is

• continuous but not differentiable

• not continuous but differentiable

• continuous and differentiable

• not continuous and not differentiable

The value of ${\cos }^{-1}\left(\cot \left(\frac{\mathrm{\pi }}{2}\right)\right) + {\cos }^{-1}\left(\sin \left(\frac{2\mathrm{\pi }}{3}\right)\right)$ is

• $\frac{2\mathrm{\pi }}{3}$

• $\frac{\mathrm{\pi }}{3}$

• $\frac{\mathrm{\pi }}{2}$

• $\mathrm{\pi }$

The objective function z = 4x₁ + 5x₂, subject to 2x₁ + x₂ $\ge$ 7, 2x₁ + 3x₂ $\le$ 15, x₂ $\le$ 3, x₁x₂ $\ge$ 0 has minimum value at the point

• on X-axis

• on Y-axis

• at the origin

• on the line parallel to X-axis

${\int }_0^1{\mathrm{xtan}}^{-1}\left(\mathrm{x}\right)\mathrm{dx}$ =

• $\frac{\mathrm{\pi }}{4} + \frac{1}{2}$

• $\frac{\mathrm{\pi }}{4} - \frac{1}{2}$

• $\frac{1}{2} - \frac{\mathrm{\pi }}{4}$

• $- \frac{\mathrm{\pi }}{4} - \frac{1}{2}$

If $\int \frac{1}{\sqrt{9 - 16{\mathrm{x}}^2}}\mathrm{dx} = {\mathrm{\alpha sin}}^{-1}\left(\mathrm{\beta x}\right) + \mathrm{c}$, then $\mathrm{\alpha } + \frac{1}{\mathrm{\beta }}$ =

• 1

• $\frac{7}{12}$

• $\frac{19}{12}$

• $\frac{9}{12}$

The solution of the differential equation

$\frac{\mathrm{dy}}{\mathrm{dx}} = \tan \left(\frac{\mathrm{y}}{\mathrm{x}}\right) + \frac{{\displaystyle \mathrm{y}}}{{\displaystyle \mathrm{x}}}$ is

• $\cos \left(\frac{\mathrm{y}}{\mathrm{x}}\right) = \mathrm{cx}$

• $\sin \left(\frac{\mathrm{y}}{\mathrm{x}}\right) = \mathrm{cx}$

• $\cos \left(\frac{\mathrm{y}}{\mathrm{x}}\right) = \mathrm{cy}$

• $\sin \left(\frac{\mathrm{y}}{\mathrm{x}}\right) = \mathrm{cy}$

If ${\int }_0^{\frac{\mathrm{\pi }}{2}}\log \left(\cos \left(\mathrm{x}\right)\right)\mathrm{dx} = \frac{\mathrm{\pi }}{2}\log \left(\frac{1}{2}\right)$, then ${\int }_0^{\frac{\mathrm{\pi }}{2}}\log \left(\sec \left(\mathrm{x}\right)\right)\mathrm{dx}$ =

• $\frac{\mathrm{\pi }}{2}\log \left(\frac{1}{2}\right)$

• $1 - \frac{\mathrm{\pi }}{2}\log \left(\frac{1}{2}\right)$

• $1 + \frac{\mathrm{\pi }}{2}\log \left(\frac{1}{2}\right)$

• $\frac{\mathrm{\pi }}{2}\log \left(2\right)$

18.

If the angle between the planes $\mathrm{r} . \left(\mathrm{m}\hat{\mathrm{i}} - \hat{\mathrm{j}} + 2\hat{\mathrm{k}}\right)$ + 3 = 0 and $\mathrm{r} . \left(2\hat{\mathrm{i}} - \mathrm{m}\hat{\mathrm{j}} - \hat{\mathrm{k}}\right) - 5$ = 0 is $\frac{\mathrm{\pi }}{3}$, then m =

• 2

• $\pm 3$

• 3

• - 2

If the origin and the points P(2, 3, 4 ), Q(1, 2, 3) and R(x, y, z) are coplanar, then

• x - 2y - z = 0

• x + 2y + z = 0

• x - 2y + z = 0

• 2x - 2y + z = 0

If lines represented by equation px² - qy² = 0 are distinct, then

• pq > 0

• pq < 0

• pq = 0

• p + q = 0