Determinant

Study Material (PDF)
Self Assesment
Study Material (PDF)
Self Assesment

Click on each question to see answer & solution.

Question No: #51

Find the value of: $ \begin{vmatrix} a-b & b-c & c-a \\ b-c & c-a & a-b \\ c-a & a-b & b-c \end{vmatrix} $

Question No: #52

Without expanding, prove that $ \begin{vmatrix} 3 & 4 & 5 \\ 4 & 3 & 7 \\ 5 & 7 & 5 \end{vmatrix} $ is divisible by `23`.

Question No: #53

Find the value (without expanding) $ \begin{vmatrix} 49 & 1 & 6 \\ 39 & 7 & 4 \\ 26 & 2 & 3 \end{vmatrix} $

Question No: #54

Find the value (without expanding): $ \begin{vmatrix} 41 & 1 & 5 \\ 79 & 7 & 9 \\ 29 & 5 & 3 \end{vmatrix} $

Question No: #55

If `x=-9` is one root of the equation $ \begin{vmatrix} x & 3 & 7 \\ 2 & x & 2 \\ 7 & 6 & x \end{vmatrix} = 0 $, then find other two roots.

Question No: #56

If $ \begin{vmatrix} a & b & c \\ x & y & z \\ m & n & p \end{vmatrix} = \lambda $, then find the value of $ \begin{vmatrix} 6a & 2b & 2c \\ 3x & y & z \\ 3m & n & p \end{vmatrix} $ in terms of `\lambda`

Question No: #57

If $ \begin{vmatrix} x & z & y \\ a & c & b \\ 1 & 1 & 1 \end{vmatrix} = 7 $, then show that $ \begin{vmatrix} c-b & y-z & zb-yc \\ b-a & x-y & ay-xb \\ a-c & z-x & cx-az \end{vmatrix}=49 $

Question No: #58

Find the value of $ \begin{vmatrix} 1 & \omega^3 & \omega^2 \\ \omega^3 & 1 & \omega \\ \omega^2 & \omega & 1 \end{vmatrix} $ where `\omega` is the imaginary cube root of unity.

Question No: #59

Find the value of $ \begin{vmatrix} 1 & \omega^n & \omega^{2n} \\ \omega^{2n} & 1 & \omega^n \\ \omega^n & \omega^{2n} & 1 \end{vmatrix} $

Question No: #60

Find the value of $ \begin{vmatrix} 1+2\omega^{100}+\omega^{200} & \omega^2 & 1 \\ 1 & 1+\omega^{100}+2\omega^{200} & \omega \\ \omega & \omega^2 & 2+\omega^{100}+\omega^{200} \end{vmatrix} $ where `omega` is the imaginary cube root of unity.