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

Integration of \[\frac{1}{1 + \left( \log_e x \right)^2}\] with respect to log_e x is

\[\int\frac{8x + 13}{\sqrt{4x + 7}} \text{ dx }\]

\[\int\frac{1 + x + x^2}{x^2 \left( 1 + x \right)} \text{ dx}\]

Find the equation of the plane passing through the following points.

 (2, 1, 0), (3, −2, −2) and (3, 1, 7)

Find the equation of the plane passing through the following points.

 (−5, 0, −6), (−3, 10, −9) and (−2, 6, −6)

Find the equation of the plane passing through the following point

 (1, 1, 1), (1, −1, 2) and (−2, −2, 2)

Find the equation of the plane passing through the following points. 

(2, 3, 4), (−3, 5, 1) and (4, −1, 2) 

Find the equation of the plane passing through the following point

(0, −1, 0), (3, 3, 0) and (1, 1, 1)

Show that the four points (0, −1, −1), (4, 5, 1), (3, 9, 4) and (−4, 4, 4) are coplanar and find the equation of the common plane.

Show that the following points are coplanar.
 (0, −1, 0), (2, 1, −1), (1, 1, 1) and (3, 3, 0) 

Show that the following points are coplanar. 

Find the coordinates of the point P where the line through A (3, -4 , -5 ) and B (2, -3 , 1) crosses the plane passing through three points L(2,2,1), M(3,0,1) and N(4, -1,0 ) . Also, find the ratio in which P diveides the line segment AB.

Find : \[\int\left( 2x + 5 \right)\sqrt{10 - 4x - 3 x^2}dx\] .

Find the vector equations of the following planes in scalar product form  \[\left( \vec{r} \cdot \vec{n} = d \right):\] \[\vec{r} = \left( 2 \hat{i} - \hat{k} \right) + \lambda \hat{i} + \mu\left( \hat{i} - 2 \hat{j} - \hat{k}
\right)\]

Find the vector equations of the following planes in scalar product form  \[\left( \vec{r} \cdot \vec{n} = d \right):\] \[\vec{r} = \left( 1 + s - t \right) \hat{t}  + \left( 2 - s \right) \hat{j}  + \left( 3 - 2s + 2t \right) \hat{k} \]

Find the vector equations of the following planes in scalar product form  \[\left( \vec{r} \cdot \vec{n} = d \right):\]\[\vec{r} = \left( \hat{i}  + \hat{j}  \right) + \lambda\left( \hat{i}  + 2 \hat{j}  - \hat{k}  \right) + \mu\left( - \hat{i}  + \hat{j} - 2 \hat{k} \right)\]

Find the vector equations of the following planes in scalar product form  \[\left( \vec{r} \cdot \vec{n} = d \right):\]\[\vec{r} = \hat{i} - \hat{j} + \lambda\left( \hat{i}  + \hat{j}  + \hat{k}  \right) + \mu\left( 4 \hat{i}  - 2 \hat{j}  + 3 \hat{k} \right)\]

Find the Cartesian forms of the equations of the following planes. \[\vec{r} = \left( \hat{i}  - \hat{j}  \right) + s\left( - \hat{i}  + \hat{j}  + 2 \hat{k} \right) + t\left( \hat{i} + 2 \hat{j} + \hat{k}  \right)\]

Find the Cartesian forms of the equations of the following planes.