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विषय
मुख्य विषय
अध्याय
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Mathematics
Using determinants, find the equation of the line joining the points
(3, 1) and (9, 3)
Chapter: [4] Determinants
Concept: undefined >> undefined
Concept: undefined >> undefined
Find values of k, if area of triangle is 4 square units whose vertices are
(k, 0), (4, 0), (0, 2)
Chapter: [4] Determinants
Concept: undefined >> undefined
Concept: undefined >> undefined
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Find values of k, if area of triangle is 4 square units whose vertices are
(−2, 0), (0, 4), (0, k)
Chapter: [4] Determinants
Concept: undefined >> undefined
Concept: undefined >> undefined
Prove that :
\[\begin{vmatrix}b + c & a - b & a \\ c + a & b - c & b \\ a + b & c - a & c\end{vmatrix} = 3abc - a^3 - b - c^3\]
Chapter: [4] Determinants
Concept: undefined >> undefined
Concept: undefined >> undefined
Prove that :
\[\begin{vmatrix}a + b & b + c & c + a \\ b + c & c + a & a + b \\ c + a & a + b & b + c\end{vmatrix} = 2\begin{vmatrix}a & b & c \\ b & c & a \\ c & a & b\end{vmatrix}\]
Chapter: [4] Determinants
Concept: undefined >> undefined
Concept: undefined >> undefined
Prove that :
\[\begin{vmatrix}a + b + 2c & a & b \\ c & b + c + 2a & b \\ c & a & c + a + 2b\end{vmatrix} = 2 \left( a + b + c \right)^3\]
Chapter: [4] Determinants
Concept: undefined >> undefined
Concept: undefined >> undefined
Prove that :
\[\begin{vmatrix}a - b - c & 2a & 2a \\ 2b & b - c - a & 2b \\ 2c & 2c & c - a - b\end{vmatrix} = \left( a + b + c \right)^3\]
Chapter: [4] Determinants
Concept: undefined >> undefined
Concept: undefined >> undefined
Prove that :
\[\begin{vmatrix}1 & b + c & b^2 + c^2 \\ 1 & c + a & c^2 + a^2 \\ 1 & a + b & a^2 + b^2\end{vmatrix} = \left( a - b \right) \left( b - c \right) \left( c - a \right)\]
Chapter: [4] Determinants
Concept: undefined >> undefined
Concept: undefined >> undefined
Prove that :
\[\begin{vmatrix}1 & a & bc \\ 1 & b & ca \\ 1 & c & ab\end{vmatrix} = \begin{vmatrix}1 & a & a^2 \\ 1 & b & b^2 \\ 1 & c & c^2\end{vmatrix}\]
Chapter: [4] Determinants
Concept: undefined >> undefined
Concept: undefined >> undefined
Prove that :
\[\begin{vmatrix}z & x & y \\ z^2 & x^2 & y^2 \\ z^4 & x^4 & y^4\end{vmatrix} = \begin{vmatrix}x & y & z \\ x^2 & y^2 & z^2 \\ x^4 & y^4 & z^4\end{vmatrix} = \begin{vmatrix}x^2 & y^2 & z^2 \\ x^4 & y^4 & z^4 \\ x & y & z\end{vmatrix} = xyz \left( x - y \right) \left( y - z \right) \left( z - x \right) \left( x + y + z \right) .\]
Chapter: [4] Determinants
Concept: undefined >> undefined
Concept: undefined >> undefined
Prove that :
\[\begin{vmatrix}\left( b + c \right)^2 & a^2 & bc \\ \left( c + a \right)^2 & b^2 & ca \\ \left( a + b \right)^2 & c^2 & ab\end{vmatrix} = \left( a - b \right) \left( b - c \right) \left( c - a \right) \left( a + b + c \right) \left( a^2 + b^2 + c^2 \right)\]
Chapter: [4] Determinants
Concept: undefined >> undefined
Concept: undefined >> undefined
Prove that :
\[\begin{vmatrix}\left( a + 1 \right) \left( a + 2 \right) & a + 2 & 1 \\ \left( a + 2 \right) \left( a + 3 \right) & a + 3 & 1 \\ \left( a + 3 \right) \left( a + 4 \right) & a + 4 & 1\end{vmatrix} = - 2\]
Chapter: [4] Determinants
Concept: undefined >> undefined
Concept: undefined >> undefined
Prove that :
\[\begin{vmatrix}a^2 & a^2 - \left( b - c \right)^2 & bc \\ b^2 & b^2 - \left( c - a \right)^2 & ca \\ c^2 & c^2 - \left( a - b \right)^2 & ab\end{vmatrix} = \left( a - b \right) \left( b - c \right) \left( c - a \right) \left( a + b + c \right) \left( a^2 + b^2 + c^2 \right)\]
Chapter: [4] Determinants
Concept: undefined >> undefined
Concept: undefined >> undefined
Prove that :
\[\begin{vmatrix}1 & a^2 + bc & a^3 \\ 1 & b^2 + ca & b^3 \\ 1 & c^2 + ab & c^3\end{vmatrix} = - \left( a - b \right) \left( b - c \right) \left( c - a \right) \left( a^2 + b^2 + c^2 \right)\]
Chapter: [4] Determinants
Concept: undefined >> undefined
Concept: undefined >> undefined
Prove that :
\[\begin{vmatrix}a^2 & bc & ac + c^2 \\ a^2 + ab & b^2 & ac \\ ab & b^2 + bc & c^2\end{vmatrix} = 4 a^2 b^2 c^2\]
Chapter: [4] Determinants
Concept: undefined >> undefined
Concept: undefined >> undefined
Prove that :
\[\begin{vmatrix}x + 4 & x & x \\ x & x + 4 & x \\ x & x & x + 4\end{vmatrix} = 16 \left( 3x + 4 \right)\]
Chapter: [4] Determinants
Concept: undefined >> undefined
Concept: undefined >> undefined
Prove that :
\[\begin{vmatrix}1 & 1 + p & 1 + p + q \\ 2 & 3 + 2p & 4 + 3p + 2q \\ 3 & 6 + 3p & 10 + 6p + 3q\end{vmatrix} = 1\]
Chapter: [4] Determinants
Concept: undefined >> undefined
Concept: undefined >> undefined
Prove that :
\[\begin{vmatrix}a & b - c & c - b \\ a - c & b & c - a \\ a - b & b - a & c\end{vmatrix} = \left( a + b - c \right) \left( b + c - a \right) \left( c + a - b \right)\]
Chapter: [4] Determinants
Concept: undefined >> undefined
Concept: undefined >> undefined
