Science (English Medium) Class 12 - CBSE Question Bank Solutions for Mathematics

Find the equation of the plane passing through the line of intersection of the planes 2x − 7y + 4z − 3 = 0, 3x − 5y + 4z + 11 = 0 and the point (−2, 1, 3).

Find the equation of the plane passing through the points (3, 4, 1) and (0, 1, 0) and parallel to the line 

Show that the lines \[\vec{r} = \left( 2 \hat{j}  - 3 \hat{k} \right) + \lambda\left( \hat{i}  + 2 \hat{j}  + 3 \hat{k} \right) \text{ and } \vec{r} = \left( 2 \hat{i}  + 6 \hat{j} + 3 \hat{k} \right) + \mu\left( 2 \hat{i}  + 3 \hat{j} + 4 \hat{k}  \right)\]  are coplanar. Also, find the equation of the plane containing them.

Show that the lines \[\frac{x + 1}{- 3} = \frac{y - 3}{2} = \frac{z + 2}{1} \text{ and }\frac{x}{1} = \frac{y - 7}{- 3} = \frac{z + 7}{2}\]  are coplanar. Also, find the equation of the plane containing them. 

Show that the lines  \[\frac{5 - x}{- 4} = \frac{y - 7}{4} = \frac{z + 3}{- 5}\] and  \[\frac{x - 8}{7} = \frac{2y - 8}{2} = \frac{z - 5}{3}\] are coplanar.

Show that the lines  \[\frac{x + 3}{- 3} = \frac{y - 1}{1} = \frac{z - 5}{5}\] and  \[\frac{x + 1}{- 1} = \frac{y - 2}{2} = \frac{z - 5}{5}\]  are coplanar. Hence, find the equation of the plane containing these lines.

Find the values of  \[\lambda\] for which the lines

If the lines  \[x =\]  5 ,  \[\frac{y}{3 - \alpha} = \frac{z}{- 2}\] and   \[x = \alpha\] \[\frac{y}{- 1} = \frac{z}{2 - \alpha}\] are coplanar, find the values of  \[\alpha\].

If the straight lines  \[\frac{x - 1}{2} = \frac{y + 1}{k} = \frac{z}{2}\] and \[\frac{x + 1}{2} = \frac{y + 1}{2} = \frac{z}{k}\] are coplanar, find the equations of the planes containing them.

For the function y = x², if x = 10 and ∆x = 0.1. Find ∆y.

Verify that xy = a e^x + b e^−x + x² is a solution of the differential equation \[x\frac{d^2 y}{d x^2} + 2\frac{dy}{dx} - xy + x^2 - 2 = 0.\]

Show that y = C x + 2C² is a solution of the differential equation \[2 \left( \frac{dy}{dx} \right)^2 + x\frac{dy}{dx} - y = 0.\]

Show that y² − x² − xy = a is a solution of the differential equation \[\left( x - 2y \right)\frac{dy}{dx} + 2x + y = 0.\]

Verify that y = A cos x + sin x satisfies the differential equation \[\cos x\frac{dy}{dx} + \left( \sin x \right)y=1.\]

Find the differential equation corresponding to y = ae^2x + be^−3x + ce^x where a, b, c are arbitrary constants.

Show that the differential equation of all parabolas which have their axes parallel to y-axis is \[\frac{d^3 y}{d x^3} = 0.\]

From x² + y² + 2ax + 2by + c = 0, derive a differential equation not containing a, b and c.

\[\frac{dy}{dx} = \sin^3 x \cos^4 x + x\sqrt{x + 1}\]

\[\frac{dy}{dx} = \frac{1}{x^2 + 4x + 5}\]

\[\frac{dy}{dx} = y^2 + 2y + 2\]