Determinant

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

Click on each question to see answer & solution.

Question No: #71

without expanding, prove that: $ \begin{vmatrix} 1 & ab & \frac{1}{a}+\frac{1}{b} \\ 1 & bc & \frac{1}{b}+\frac{1}{c} \\ 1 & ca & \frac{1}{c}+\frac{1}{a} \end{vmatrix} = \begin{vmatrix} b^2c^2 & bc & b+c \\ c^2a^2 & ca & c+a \\ a^2b^2 & ab & a+b \end{vmatrix} = 0 $
Ans: N.A.
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Question No: #72

Prove that $ \begin{vmatrix} 10! & 11! & 12! \\ 11! & 12! & 13! \\ 12! & 13! & 14! \end{vmatrix} = 2\times10!\times11!\times12! $

Question No: #73

Show that $ \begin{vmatrix} a^2 & 2a & 1 \\ 1 & a^2 & 2a \\ 2a & 1 & a^2 \end{vmatrix} $ is a perfect square quantity.

Question No: #74

Prove that $ \begin{vmatrix} 1 & a & a^2 \\ a^2 & 1 & a \\ a & a^2 & 1 \end{vmatrix} = (a^3-1)^2 $

Question No: #75

Prove that $ \begin{vmatrix} 2ab & a^2 & b^2 \\ a^2 & b^2 & 2ab \\ b^2 & 2ab & a^2 \end{vmatrix} = -(a^3+b^3)^2 $

Question No: #76

Prove $ \begin{vmatrix} a & b & 0 \\ 0 & a & b \\ b & 0 & a \end{vmatrix} = a^3+b^3 $. Hence find the value of $ \begin{vmatrix} 2ab & a^2 & b^2 \\ a^2 & b^2 & 2ab \\ b^2 & 2ab & a^2 \end{vmatrix} $

Question No: #77

Find the value (without expanding). $ \begin{vmatrix} a^{-1} & a^2 & bc \\ b^{-1} & b^2 & ca \\ c^{-1} & c^2 & ab \end{vmatrix} $

Question No: #78

If $ \Delta = \begin{vmatrix} a-x & c & b \\ c & b-x & a \\ b & a & c-x \end{vmatrix} =0 $ and `a+b+c=0`, then show that either `x=0` or `x=\pm\sqrt{3/2(a^2+b^2+c^2)}`

Question No: #79

Prove that $ \begin{vmatrix} a & b & c \\ b & c & a \\ c & a & b \end{vmatrix} = -(a+b+c)(a+b\omega+c\omega^2)(a+b\omega^2+c\omega) $ where `\omega` is the imaginary cube root of unity.

Question No: #80

Prove that `f(200)=0` where $ f(x) = \begin{vmatrix} 1 & x & x+1 \\ 2x & x(x-1) & x(x+1) \\ 3x(x-1) & x(x-1)(x-2) & x(x+1)(x-1) \end{vmatrix} $