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

​Solve the following determinant equation:

Express the matrix \[A = \begin{bmatrix}3 & - 4 \\ 1 & - 1\end{bmatrix}\]  as the sum of a symmetric and a skew-symmetric matrix.

If a, b, c are real numbers such that
\[\begin{vmatrix}b + c & c + a & a + b \\ c + a & a + b & b + c \\ a + b & b + c & c + a\end{vmatrix} = 0\] , then show that either
\[a + b + c = 0 \text{ or, } a = b = c\]

If \[A = \begin{bmatrix}\cos x & \sin x \\ - \sin x & \cos x\end{bmatrix}\] , find x satisfying 0 < x < \[\frac{\pi}{2}\] when A + A^T = I

If \[a, b\] and c  are all non-zero and 

If \[\begin{vmatrix}a & b - y & c - z \\ a - x & b & c - z \\ a - x & b - y & c\end{vmatrix} =\] 0, then using properties of determinants, find the value of  \[\frac{a}{x} + \frac{b}{y} + \frac{c}{z}\]  , where \[x, y, z \neq\] 0

Find the area of the triangle with vertice at the point:

(3, 8), (−4, 2) and (5, −1)

Find the area of the triangle with vertice at the point:

(2, 7), (1, 1) and (10, 8)

Find the area of the triangle with vertice at the point:

 (−1, −8), (−2, −3) and (3, 2)

Find the area of the triangle with vertice at the point:

 (0, 0), (6, 0) and (4, 3)

Using determinants show that the following points are collinear:

(5, 5), (−5, 1) and (10, 7)

Using determinants show that the following points are collinear:

(1, −1), (2, 1) and (4, 5)

Using determinants show that the following points are collinear:

(3, −2), (8, 8) and (5, 2)

Using determinants show that the following points are collinear:

(2, 3), (−1, −2) and (5, 8)

Find the value of \[\lambda\]  so that the points (1, −5), (−4, 5) and \[\lambda\]  are collinear.