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

If w is a complex cube root of unity, show that

`([[1         w          w^2],[w            w^2             1],[w^2           1             w]]+[[w          w^2          1],[w^2             1               w],[w            w^2              1]])[[1],[w],[w^2]]=[[0],[0],[0]]`

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

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

If A = [a_ij] is a skew-symmetric matrix, then write the value of  \[\sum_i \sum_j\]  a_ij.

Find the values of x and y, if \[2\begin{bmatrix}1 & 3 \\ 0 & x\end{bmatrix} + \begin{bmatrix}y & 0 \\ 1 & 2\end{bmatrix} = \begin{bmatrix}5 & 6 \\ 1 & 8\end{bmatrix}\]

If  \[x\binom{2}{3} + y\binom{ - 1}{1} = \binom{10}{5}\] , find the value of x.

If  \[2\begin{bmatrix}3 & 4 \\ 5 & x\end{bmatrix} + \begin{bmatrix}1 & y \\ 0 & 1\end{bmatrix} = \begin{bmatrix}7 & 0 \\ 10 & 5\end{bmatrix}\] , find x − y.

If  \[\begin{bmatrix}xy & 4 \\ z + 6 & x + y\end{bmatrix} = \begin{bmatrix}8 & w \\ 0 & 6\end{bmatrix}\] , write the value of (x + y + z).

If  \[\binom{x + y}{x - y} = \begin{bmatrix}2 & 1 \\ 4 & 3\end{bmatrix}\binom{1}{ - 2}\] , then write the value of (x, y).

If \[I = \begin{bmatrix}1 & 0 \\ 0 & 1\end{bmatrix}, J = \begin{bmatrix}0 & 1 \\ - 1 & 0\end{bmatrix} and B = \begin{bmatrix}\cos \theta & \sin \theta \\ - \sin \theta & \cos \theta\end{bmatrix}\] then B equals ) 

The trace of the matrix \[A = \begin{bmatrix}1 & - 5 & 7 \\ 0 & 7 & 9 \\ 11 & 8 & 9\end{bmatrix}\], is

If \[\vec{a,} \vec{b}\] are two vectors, then write the truth value of the following statement: 
 \[\vec{a} = - \vec{b} \Rightarrow \left| \vec{a} \right| = \left| \vec{b} \right|\]

If \[\vec{a,} \vec{b}\] are two vectors, then write the truth value of the following statement: 
\[|\vec{a}| =  |\vec{b}| \Rightarrow \vec{a}  = ± \vec{b} \]

If \[\vec{a,} \vec{b}\] are two vectors, then write the truth value of the following statement: 
\[\left| \vec{a} \right| = \left| \vec{b} \right| \Rightarrow \vec{a} = \vec{b}\]

The two vectors \[\hat{j} + \hat{k}\] and \[3 \hat{i} - \hat{j} + 4 \hat{k}\] represents the sides \[\overrightarrow{AB}\] and \[\overrightarrow{AC}\] respectively of a triangle ABC. Find the length of the median through A.