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

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

\[\int\frac{x + 3}{\left( x + 4 \right)^2} e^x dx =\]

The two adjacent sides of a parallelogram are `2hati-4hatj-5hatk and 2 hati+2hatj+3hatj` . Find the two unit vectors parallel to its diagonals. Using the diagonal vectors, find the area of the parallelogram.

Find the values of x, y, z if the matrix `A = [(0,2y,z),(x,y,-z),(x , -y,z)]` satisfy the equation A'A = I.

If \[A = \begin{bmatrix}1 & 2 \\ 0 & 3\end{bmatrix}\] is written as B + C, where B is a symmetric matrix and C is a skew-symmetric matrix, then B is equal to.