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Determinant
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Determinant
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All Questions (PDF)
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Study Material (PDF)
All Questions (PDF)
English Version
Self Assesment
Sort As
Serial Number
Newly Added
Language
English
Click on each question to see answer & solution.
Question No: #1
Prove that: $ \begin{vmatrix} 1 & b+c & c^2 \\ 1 & c+a & a^2 \\ 1 & a+b & b^2 \end{vmatrix} = (a-b)(b-c)(c-a) $
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Question No: #2
Without expanding prove that: $ \begin{vmatrix} a+b & 2a+b & 3a+b \\ 2a+b & 3a+b & 4a+b \\ 4a+b & 5a+b & 6a+b \end{vmatrix} = 0 $
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Question No: #3
Show that $ \begin{vmatrix} -a(b^2+c^2-a^2) & 2b^3 & 2c^3 \\ 2a^3 & -b(c^2+a^2-b^2) & 2c^3 \\ 2a^3 & 2b^3 & -c(a^2+b^2-c^2) \end{vmatrix} = abc(a^2+b^2+c^2)^3 $
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Question No: #4
Show that $ \begin{vmatrix} -a^2 & ab & ac \\ ab & -b^2 & bc \\ ca & bc & -c^2 \end{vmatrix} = 4a^2b^2c^2 $
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Question No: #5
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} $
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Question No: #6
Without expanding prove that: $ \begin{vmatrix} \log_x\ xyz & \log_x\ y & \log_x\ z \\ \log_y\ xyz & 1 & \log_y\ z \\ \log_z\ xyz & \log_z\ y & 1 \end{vmatrix} = 0 $
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Question No: #7
Without expanding prove that: $ \begin{vmatrix} bc & a^2 & a^2 \\ b^2 & ca & b^2 \\ c^2 & c^2 & ab \end{vmatrix} = \begin{vmatrix} bc & ab & ca \\ ab & ca & bc \\ ca & bc & ab \end{vmatrix} $
Ans: N.A.
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Question No: #8
If $ A = \begin{bmatrix} 1 & -1 & 0 \\ 1 & 2 & -1 \\ 3 & 2 & -2 \end{bmatrix} $ then find the roots of `\det\ (A-xI_3)=0` where `I` is the unit matrix of order 3.
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Question No: #9
If `x^3=1`, then prove that $ \begin{vmatrix} a & b & c \\ b & c & a \\ c & a & b \end{vmatrix} = (a+bx+cx^2) \begin{vmatrix} 1 & b & c \\ x^2 & c & a \\ x & a & b \end{vmatrix} $
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Question No: #10
Prove that: $ \begin{vmatrix} x^2+y^2+1 & x^2+2y^2+3 & x^2+3y^2+4 \\ y^2+2 & 2y^2+6 & 3y^2+8 \\ y^2+1 & 2y^2+3 & 3y^2+4 \end{vmatrix} = x^2y^2 $
Ans:
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