Prove the following identity:
`|(a^3,2,a),(b^3,2,b),(c^3,2,c)| = 2(a-b) (b-c) (c-a) (a+b+c)`
[4] Determinants
Chapter: [4] Determinants
Concept: undefined >> undefined
Without expanding, prove that
\[\begin{vmatrix}a & b & c \\ x & y & z \\ p & q & r\end{vmatrix} = \begin{vmatrix}x & y & z \\ p & q & r \\ a & b & c\end{vmatrix} = \begin{vmatrix}y & b & q \\ x & a & p \\ z & c & r\end{vmatrix}\]
[4] Determinants
Chapter: [4] Determinants
Concept: undefined >> undefined
Show that
\[\begin{vmatrix}x + 1 & x + 2 & x + a \\ x + 2 & x + 3 & x + b \\ x + 3 & x + 4 & x + c\end{vmatrix} =\text{ 0 where a, b, c are in A . P .}\]
[4] Determinants
Chapter: [4] Determinants
Concept: undefined >> undefined
\[If \begin{vmatrix}p & b & c \\ a & q & c \\ a & b & r\end{vmatrix} = 0,\text{ find the value of }\frac{p}{p - a} + \frac{q}{q - b} + \frac{r}{r - c}, p \neq a, q \neq b, r \neq c .\]
[4] Determinants
Chapter: [4] Determinants
Concept: undefined >> undefined
Show that x = 2 is a root of the equation
\[\begin{vmatrix}x & - 6 & - 1 \\ 2 & - 3x & x - 3 \\ - 3 & 2x & x + 2\end{vmatrix} = 0\] and solve it completely.
[4] Determinants
Chapter: [4] Determinants
Concept: undefined >> undefined
Solve the following determinant equation:
\[\begin{vmatrix}x + a & b & c \\ a & x + b & c \\ a & b & x + c\end{vmatrix} = 0\]
[4] Determinants
Chapter: [4] Determinants
Concept: undefined >> undefined
Solve the following determinant equation:
\[\begin{vmatrix}x + a & x & x \\ x & x + a & x \\ x & x & x + a\end{vmatrix} = 0, a \neq 0\]
[4] Determinants
Chapter: [4] Determinants
Concept: undefined >> undefined
Solve the following determinant equation:
\[\begin{vmatrix}3x - 8 & 3 & 3 \\ 3 & 3x - 8 & 3 \\ 3 & 3 & 3x - 8\end{vmatrix} = 0\]
[4] Determinants
Chapter: [4] Determinants
Concept: undefined >> undefined
Solve the following determinant equation:
\[\begin{vmatrix}1 & x & x^2 \\ 1 & a & a^2 \\ 1 & b & b^2\end{vmatrix} = 0, a \neq b\]
[4] Determinants
Chapter: [4] Determinants
Concept: undefined >> undefined
Solve the following determinant equation:
\[\begin{vmatrix}x + 1 & 3 & 5 \\ 2 & x + 2 & 5 \\ 2 & 3 & x + 4\end{vmatrix} = 0\]
[4] Determinants
Chapter: [4] Determinants
Concept: undefined >> undefined
Solve the following determinant equation:
\[\begin{vmatrix}1 & x & x^3 \\ 1 & b & b^3 \\ 1 & c & c^3\end{vmatrix} = 0, b \neq c\]
[4] Determinants
Chapter: [4] Determinants
Concept: undefined >> undefined
Solve the following determinant equation:
\[\begin{vmatrix}1 & 1 & x \\ p + 1 & p + 1 & p + x \\ 3 & x + 1 & x + 2\end{vmatrix} = 0\]
[4] Determinants
Chapter: [4] Determinants
Concept: undefined >> undefined
Solve the following determinant equation:
\[\begin{vmatrix}15 - 2x & 11 - 3x & 7 - x \\ 11 & 17 & 14 \\ 10 & 16 & 13\end{vmatrix} = 0\]
[4] Determinants
Chapter: [4] Determinants
Concept: undefined >> undefined
Solve the following determinant equation:
\[\begin{vmatrix}3 & - 2 & \sin\left( 3\theta \right) \\ - 7 & 8 & \cos\left( 2\theta \right) \\ - 11 & 14 & 2\end{vmatrix} = 0\]
[4] Determinants
Chapter: [4] Determinants
Concept: undefined >> undefined
Show that
`|(x-3,x-4,x-alpha),(x-2,x-3,x-beta),(x-1,x-2,x-gamma)|=0`, where α, β, γ are in A.P.
[4] Determinants
Chapter: [4] Determinants
Concept: undefined >> undefined
If a, b, c are real numbers such that
\[\begin{vmatrix}b + c & c + a & a + b \\ c + a & a + b & b + c \\ a + b & b + c & c + a\end{vmatrix} = 0\] , then show that either
\[a + b + c = 0 \text{ or, } a = b = c\]
[4] Determinants
Chapter: [4] Determinants
Concept: undefined >> undefined
If \[a, b\] and c are all non-zero and
\[\begin{vmatrix}1 + a & 1 & 1 \\ 1 & 1 + b & 1 \\ 1 & 1 & 1 + c\end{vmatrix} =\] 0, then prove that
\[\frac{1}{a} + \frac{1}{b} + \frac{1}{c} +\]1
= 0
[4] Determinants
Chapter: [4] Determinants
Concept: undefined >> undefined
If \[\begin{vmatrix}a & b - y & c - z \\ a - x & b & c - z \\ a - x & b - y & c\end{vmatrix} =\] 0, then using properties of determinants, find the value of \[\frac{a}{x} + \frac{b}{y} + \frac{c}{z}\] , where \[x, y, z \neq\] 0
[4] Determinants
Chapter: [4] Determinants
Concept: undefined >> undefined
Find the area of the triangle with vertice at the point:
(3, 8), (−4, 2) and (5, −1)
[4] Determinants
Chapter: [4] Determinants
Concept: undefined >> undefined
Find the area of the triangle with vertice at the point:
(2, 7), (1, 1) and (10, 8)
[4] Determinants
Chapter: [4] Determinants
Concept: undefined >> undefined
