The differential equation which represents the family of curves given by tany = c(1 - e^x) is

• ${\mathrm{e}}^{\mathrm{x}}\tan \left(\mathrm{y}\right)\mathrm{dx} + \left(1 - {\mathrm{e}}^{\mathrm{x}}\right)\mathrm{dy} = 0$

• ${\mathrm{e}}^{\mathrm{x}}\tan \left(\mathrm{y}\right)\mathrm{dx} + \left(1 - {\mathrm{e}}^{\mathrm{x}}\right){\sec }^2\left(\mathrm{x}\right)\mathrm{dy} = 0$

• ${\mathrm{e}}^{\mathrm{x}}\left(1 - {\mathrm{e}}^{\mathrm{x}}\right)\mathrm{dx} + \tan \left(\mathrm{y}\right)\mathrm{dy} = 0$

• ${\mathrm{e}}^{\mathrm{x}}\tan \left(\mathrm{y}\right)\mathrm{dy} + \left(1 - {\mathrm{e}}^{\mathrm{x}}\right)\mathrm{dx} = 0$

42.

Which one of the following is the differential equation that represents the family of curves $\mathrm{y} = \frac{1}{2{\mathrm{x}}^2 - \mathrm{c}}$

where c is an arbitrary constant ?

• $\frac{\mathrm{dy}}{\mathrm{dx}} = 4{\mathrm{xy}}^2$

• $\frac{\mathrm{dy}}{\mathrm{dx}} = \frac{1}{\mathrm{y}}$

• $\frac{\mathrm{dy}}{\mathrm{dx}} = {\mathrm{x}}^{2\mathrm{y}}$

• $\frac{\mathrm{dy}}{\mathrm{dx}} = - 4{\mathrm{xy}}^2$

Which one of the following is the second degree polynomial function f(x) where f(0) = 5, f( - 1) = 10 and f(1) = 6 ?

• 5x² – 2x + 5

• 3x² – 2x – 5

• 3x² – 2x + 5

• 3x² – 10x + 5

What is the degree of the differential equation

$\frac{{\mathrm{d}}^3\mathrm{y}}{{\mathrm{dx}}^3} + {\left(\frac{\mathrm{dy}}{\mathrm{dx}}\right)}^2 - {\mathrm{x}}^2\left(\frac{{\mathrm{d}}^4\mathrm{y}}{{\mathrm{dy}}^4}\right) = 0$

• 1

• 2

• 3

• 4

Consider the following statements :

1) The function f(x) = ln(x) increases in the interval (0, ∞).

2) The function f(x) = tan(x) increases in the interval $\left( - \frac{\mathrm{\pi }}{2}, \frac{\mathrm{\pi }}{2}\right)$

Which of the above statements is/are correct ?

• 1 only

• 2 only

• Both 1 and 2

• Neither 1 nor 2

The function u(x, y) = c which satisfies the differential equation x(dx – dy) + y(dy – dx) = 0, is

• x² + y² = xy + c

• x² + y² = 2xy + c

• x² - y² = xy + c

• x² - y² = 2xy + c

If x^myⁿ = a^m+n, then what is $\frac{\mathrm{dy}}{\mathrm{dx}}$ equal to ?

• $\frac{\mathrm{my}}{\mathrm{nx}}$

• $- \frac{\mathrm{my}}{\mathrm{nx}}$

• $\frac{\mathrm{mx}}{\mathrm{ny}}$

• $- \frac{\mathrm{ny}}{\mathrm{mx}}$

The solution of the differential equation $\mathrm{dy} = \left(1 + {\mathrm{y}}^2\right)\mathrm{dx}$ is

• y = tan(x) + c

• y = tan(x + c)

• tan^-1(y + c) = x

• tan^-1(y + x) = 2x

The order and degree of the differential equation$\mathrm{k}\frac{\mathrm{dy}}{\mathrm{dx}} = \int {\left[1 + {\left(\frac{\mathrm{dy}}{\mathrm{dx}}\right)}^2\right]}^{\frac{2}{3}}$

• 1 and 1

• 2 and 3

• 2 and 4

• 1 and 4