The function f(x) = $\log \left(1 + \mathrm{x}\right) - \frac{2\mathrm{x}}{2 +\hspace{0.17em}\mathrm{x}}$ is increasing on

• $\left(0, \infty \right)$

• $\left(- \infty , 0\right)$

• $\left(- \infty , \infty \right)$

• None of the above

If f(x) = $\mathrm{kx} - \sin \left(\mathrm{x}\right)$, then sin(x)is monotonically increasing, then

• k > 1

• k > - 1

• k < 1

• k < - 1

The positive root of x² - 78.8 = 0 after first approximation by Newton Raphson method assuming initial approximation to the root is 14, is

• 9.821

• 9.814

• 9.715

• 9.915

In the usual notation the value of $∆\nabla$ is equal to

• $∆ -\hspace{0.17em}\nabla$

• $∆ + \nabla$

• $\nabla - ∆$

• None of the above

15.

The maximum area of the rectangle that can be inscribed in a circle of radius r, is

• ${\mathrm{\pi r}}^2$

• r²

• $\raisebox{1ex}{${\mathrm{\pi r}}^2$}\!\left/ \!\raisebox{-1ex}{$4$}\right.$

• 2r²

The set of points where the function $\mathrm{f}\left(\mathrm{x}\right) = \left|\mathrm{x} - 1\right|{\mathrm{e}}^{\mathrm{x}}$ is differentiable, is

• R

• R - {1}

• R - {- 1}

• R - {0}

If $\mathrm{x} = \mathrm{\varphi }\left(\mathrm{t}\right), \mathrm{y} = \mathrm{\psi }\left(\mathrm{t}\right), \mathrm{then} \frac{{\mathrm{d}}^2\mathrm{y}}{{\mathrm{dx}}^2}$ is equal to

• $\frac{\mathrm{\varphi }\text{'}\mathrm{\psi }\text{'}\text{'} - \mathrm{\psi }\text{'}\mathrm{\varphi }\text{'}\text{'}}{{\left(\mathrm{\varphi }\text{'}\right)}^2}$

• $\frac{\mathrm{\varphi }\text{'}\mathrm{\psi }\text{'}\text{'} - \mathrm{\psi }\text{'}\mathrm{\varphi }\text{'}\text{'}}{{\left(\mathrm{\varphi }\text{'}\right)}^3}$

• $\frac{\mathrm{\varphi }\text{'}\text{'}}{\mathrm{\psi }\text{'}\text{'}}$

• $\frac{\mathrm{\psi }\text{'}\text{'}}{\mathrm{\varphi }\text{'}\text{'}}$

$\int \frac{1 + \sin \left(\mathrm{x}\right)}{1 + \cos \left(\mathrm{x}\right)}\mathrm{dx}$ is equal to

• $\mathrm{xtan}\left(\frac{\mathrm{x}}{2}\right) + \mathrm{c}$

• $\log \left(1 + \cos \left(\mathrm{x}\right)\right) +\hspace{0.17em}\mathrm{c}$

• $\cot \left(\frac{\mathrm{x}}{2}\right) +\hspace{0.17em}\mathrm{c}$

• $\log \left(\mathrm{x} + \sin \left(\mathrm{x}\right)\right) +\hspace{0.17em}\mathrm{c}$

The order of the differential equation whose solution is y = a cos(x) + b sin(x) + ce^-x , is

• 3

• 1

• 2

• 4

The differential equation of all parabolas with axis parallel to the axis of y is

• y₂ = 2y₁ + x

• y₃ = 2y₁

• ${\mathrm{y}}_2^3$ = y₁

• None of these