PUC Science 2nd PUC Class 12 - Karnataka Board PUC Question Bank Solutions for Mathematics

The solution of the differential equation \[\frac{dy}{dx} + 1 = e^{x + y}\], is

The solution of x² + y²  \[\frac{dy}{dx}\]= 4, is

The solution of the differential equation x dx + y dy = x² y dy − y² x dx, is

The solution of the differential equation (x² + 1) \[\frac{dy}{dx}\] + (y² + 1) = 0, is

The solution of the differential equation \[\frac{dy}{dx} - ky = 0, y\left( 0 \right) = 1\] approaches to zero when x → ∞, if

The solution of the differential equation \[\left( 1 + x^2 \right)\frac{dy}{dx} + 1 + y^2 = 0\], is

The solution of the differential equation \[\frac{dy}{dx} = \frac{x^2 + xy + y^2}{x^2}\], is

The number of arbitrary constants in the general solution of differential equation of fourth order is

The number of arbitrary constants in the particular solution of a differential equation of third order is

Which of the following differential equations has y = x as one of its particular solution?

The general solution of the differential equation \[\frac{dy}{dx} = e^{x + y}\], is

The general solution of the differential equation \[\frac{y dx - x dy}{y} = 0\], is

The general solution of a differential equation of the type \[\frac{dx}{dy} + P_1 x = Q_1\] is

The general solution of the differential equation e^x dy + (y e^x + 2x) dx = 0 is

Show that the cone of the greatest volume which can be inscribed in a given sphere has an altitude equal to \[ \frac{2}{3} \] of the diameter of the sphere.

Prove that the semi-vertical angle of the right circular cone of given volume and least curved surface is \[\cot^{- 1} \left( \sqrt{2} \right)\] .

A given quantity of metal is to be cast into a half cylinder with a rectangular base and semicircular ends. Show that in order that the total surface area may be minimum the ratio of the length of the cylinder to the diameter of its semi-circular ends is \[\pi : (\pi + 2)\].

Find the angle between the line \[\vec{r} = \left( 2 \hat{i}+ 3 \hat {j}  + 9 \hat{k}  \right) + \lambda\left( 2 \hat{i} + 3 \hat{j}  + 4 \hat{k}  \right)\]  and the plane  \[\vec{r} \cdot \left( \hat{i}  + \hat{j}  + \hat{k}  \right) = 5 .\]

Find the angle between the line \[\frac{x - 1}{1} = \frac{y - 2}{- 1} = \frac{z + 1}{1}\]  and the plane 2x + y − z = 4.