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

Using vector method, prove that the following points are collinear:
A (6, −7, −1), B (2, −3, 1) and C (4, −5, 0)

Using vector method, prove that the following points are collinear:
A (2, −1, 3), B (4, 3, 1) and C (3, 1, 2)

Using vector method, prove that the following points are collinear:
A (1, 2, 7), B (2, 6, 3) and C (3, 10, −1)

Using vector method, prove that the following points are collinear:
A (−3, −2, −5), B (1, 2, 3) and C (3, 4, 7)

Evaluate : \[\int\frac{\cos 2x + 2 \sin^2 x}{\cos^2 x}dx\] .

Find : \[\int\frac{dx}{\sqrt{3 - 2x - x^2}}\] .

Find :  \[\int\frac{e^x}{\left( 2 + e^x \right)\left( 4 + e^{2x} \right)}dx.\] 

Find the angle between the given planes. \[\vec{r} \cdot \left( 2 \hat{i} - 3 \hat{j} + 4 \hat{k} \right) = 1 \text{ and } \vec{r} \cdot \left( - \hat{i}  + \hat{j}  \right) = 4\]

Find the angle between the given planes. \[\vec{r} \cdot \left( 2 \hat{i} - \hat{j}  + 2 \hat{k}  \right) = 6 \text{ and } \vec{r} \cdot \left( 3 \hat{i}  + 6 \hat{j}  - 2 \hat{k}  \right) = 9\]

Find the angle between the planes.

2x − y + z = 4 and x + y + 2z = 3

Find the angle between the planes.

x + y − 2z = 3 and 2x − 2y + z = 5

Find the angle between the planes.

 x − y + z = 5 and x + 2y + z = 9

Find the angle between the planes.
 2x − 3y + 4z = 1 and − x + y = 4

Find the angle between the planes.

Show that the following planes are at right angles.

\[\vec{r} \cdot \left( 2 \hat{i} - \hat{j} + \hat{k}  \right) = 5 \text{ and }  \vec{r} \cdot \left( - \hat{i}  - \hat{j} + \hat{k}  \right) = 3\]

Show that the following planes are at right angles.

The acute angle between the planes 2x − y + z = 6 and x + y + 2z = 3 is

\[\int\frac{5 x^4 + 12 x^3 + 7 x^2}{x^2 + x} dx\]

\[\int \left( e^x + 1 \right)^2 e^x dx\]