Science (English Medium) कक्षा १२ - CBSE Question Bank Solutions for Mathematics

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.

The line  \[\vec{r} = \hat{i} + \lambda\left( 2 \hat{i} - m \hat{j}  - 3 \hat{k}  \right)\]  is parallel to the plane  \[\vec{r} \cdot \left( m \hat{i}  + 3 \hat{j}  + \hat{k}  \right) = 4 .\] Find m. 

Show that the line whose vector equation is \[\vec{r} = 2 \hat{i}  + 5 \hat{j} + 7 \hat{k}+ \lambda\left( \hat{i}  + 3 \hat{j}  + 4 \hat{k}  \right)\] is parallel to the plane whose vector  \[\vec{r} \cdot \left( \hat{i} + \hat{j}  - \hat{k}  \right) = 7 .\]  Also, find the distance between them.

Find the angle between the line \[\frac{x - 2}{3} = \frac{y + 1}{- 1} = \frac{z - 3}{2}\] and the plane

3x + 4y + z + 5 = 0.

State when the line \[\vec{r} = \vec{a} + \lambda \vec{b}\]  is parallel to the plane  \[\vec{r} \cdot \vec{n} = d .\]Show that the line  \[\vec{r} = \hat{i}  + \hat{j}  + \lambda\left( 3 \hat{i}  - \hat{j}  + 2 \hat{k}  \right)\]  is parallel to the plane  \[\vec{r} \cdot \left( 2 \hat{j} + \hat{k} \right) = 3 .\]   Also, find the distance between the line and the plane.

Show that the plane whose vector equation is \[\vec{r} \cdot \left( \hat{i}  + 2 \hat{j}  - \hat{k}  \right) = 1\] and the line whose vector equation is  \[\vec{r} = \left( - \hat{i}  + \hat{j} + \hat{k}  \right) + \lambda\left( 2 \hat{i}  + \hat{j}  + 4 \hat{k}  \right)\]   are parallel. Also, find the distance between them.