Arts (English Medium) इयत्ता १२ - CBSE Question Bank Solutions

If A is a square matrix, then AA is a

If A and B are symmetric matrices, then ABA is

If A = [a_ij] is a square matrix of even order such that a_ij = i² − j², then 

If \[A = \begin{bmatrix}2 & 0 & - 3 \\ 4 & 3 & 1 \\ - 5 & 7 & 2\end{bmatrix}\]  is expressed as the sum of a symmetric and skew-symmetric matrix, then the symmetric matrix is  

If A and B are two matrices of order 3 × m and 3 × n respectively and m = n, then the order of 5A − 2B is 

If A and B are matrices of the same order, then AB^T − BA^T is a 

The matrix  \[A = \begin{bmatrix}0 & - 5 & 8 \\ 5 & 0 & 12 \\ - 8 & - 12 & 0\end{bmatrix}\] is a 

The matrix   \[A = \begin{bmatrix}1 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 4\end{bmatrix}\] is

Find the vector and Cartesian equations of the line passing through (1, 2, 3) and parallel to the planes \[\vec{r} \cdot \left( \hat{i}  - \hat{j} + 2 \hat{k}  \right) = 5 \text{ and } \vec{r} \cdot \left( 3 \hat{i} + \hat{j}  + 2 \hat{k} \right) = 6\]

Find the vector and Cartesian forms of the equation of the plane passing through the point (1, 2, −4) and parallel to the lines \[\vec{r} = \left( \hat{i} + 2 \hat{j}  - 4 \hat{k}  \right) + \lambda\left( 2 \hat{i}  + 3 \hat{j}  + 6 \hat{k}  \right)\] and \[\vec{r} = \left( \hat{i}  - 3 \hat{j}  + 5 \hat{k}  \right) + \mu\left( \hat{i}  + \hat{j}  - \hat{k} \right)\] Also, find the distance of the point (9, −8, −10) from the plane thus obtained.  

Find the vector equation of the line passing through (1, 2, 3) and parallel to the planes  \[\vec{r} \cdot \left( \hat{i}  - \hat{j}  + 2 \hat{k}  \right) = 5 \text{ and } \vec{r} \cdot \left( 3 \hat{i}  + \hat{j}  + \hat{k}  \right) = 6 .\]

Find the vector and cartesian equations of the plane passing throuh the points (2,5,- 3), (-2, - 3,5) and (5,3,-3). Also, find the point of intersection of this plane with the line passing through points (3, 1, 5) and (–1, –3, –1).

Find the equation of the plane passing through the intersection of the planes `vec(r) .(hat(i) + hat(j) + hat(k)) = 1"and" vec(r) . (2 hat(i) + 3hat(j) - hat(k)) +4 = 0 `and parallel to x-axis. Hence, find the distance of the plane from x-axis.

Find the vector and cartesian equation of the plane passing through the point (2, 5, - 3), (-2, -3, 5) and (5, 3, -3). Also, find the point of intersection of this plane with the line passing through points (3, 1, 5) and (-1, -3, -1).

Vector equation of a line which passes through a point (3, 4, 5) and parallels to the vector `2hati + 2hatj - 3hatk`.

Find the vector and Cartesian equations of the plane passing through the points (2, 2 –1), (3, 4, 2) and (7, 0, 6). Also find the vector equation of a plane passing through (4, 3, 1) and parallel to the plane obtained above.

Find the vector equation of the plane that contains the lines `vecr = (hat"i" + hat"j") + λ (hat"i" + 2hat"j" - hat"k")` and the point (–1, 3, –4). Also, find the length of the perpendicular drawn from the point (2, 1, 4) to the plane thus obtained.

Show that a matrix which is both symmetric and skew symmetric is a zero matrix.

Express the matrix A as the sum of a symmetric and a skew-symmetric matrix, where A = `[(2, 4, -6),(7, 3, 5),(1, -2, 4)]`

Let A = `[(2, 3),(-1, 2)]`. Then show that A² – 4A + 7I = O. Using this result calculate A⁵ also.