Advertisements
Advertisements
Question
If the determinant \[\begin{vmatrix}0 & x^2 - a & x^3 - b \\ x^2 + a & 0 & x^2 + c \\ x^4 + b & x - c & 0\end{vmatrix} = 0 \text{ is }\]
Options
a, b, c are in H . P
\[ \alpha\text{ is a root of 4a} x^2 + 12bx + 9c = 0\text{ or a, b, c are in G . P .}\]
a, b, c are in G . P . only
a, b, c are in A . P .
Solution
\[\alpha\text{ is a root of 4a }x^2 + 12bx + 9c = 0\text{ or a, b, c are in G . P .}\]
\[\text{ Let }\Delta = \begin{vmatrix} a & b & 2a\alpha + 3b\\ b & c & 2b\alpha + 3c\\2a\alpha + 3b & 2b\alpha + 3c & 0 \end{vmatrix}\]
\[ = \begin{vmatrix} a - b & b & 2a\alpha + 3b\\ b - c & c & 2b\alpha + 3c\\2a\alpha + 3b - 2b\alpha - 3c & 2b\alpha + 3c & 0 \end{vmatrix} \left[\text{ Applying }C_1 \to C_1 - C_2 \right]\]
\[ = \begin{vmatrix} a - b & b & 2a\alpha + 3b\\ b - c & c & 2b\alpha + 3c\\2\left( a - b \right)\alpha + 3\left( b - c \right) & 2b\alpha + 3c & 0 \end{vmatrix}\]
\[ = \begin{vmatrix} a - b & b & 2a\alpha + 3b\\ b - c & c & 2b\alpha + 3c\\ 0 & 0 & - 2\alpha\left( 2a\alpha + 3b \right) - 3 \left( 2b\alpha + 3c \right) \end{vmatrix} \left[\text{ Applying } R_3 \to R_3 - 2\alpha, R_1 - 3 R_2 \right]\]
\[ = - 2\alpha\left( 2a\alpha + 3b \right) - 3 \left( 2b\alpha + 3c \right)\begin{vmatrix} a - b & b \\ b - c & c \end{vmatrix} \left[\text{ Expanding along }R_3 \right]\]
\[ = - \left( 4a \alpha^2 + 12b\alpha + 9c \right)\left( ac - b^2 \right)\]
\[\text{ But }\Delta = 0 \left[\text{ Given }\right]\]
\[ \Rightarrow - \left( 4a \alpha^2 + 12b\alpha + 9c \right)\left( ac - b^2 \right) = 0\]
\[ \Rightarrow \left( 4a \alpha^2 + 12b\alpha + 9c \right) = 0 \]
\[or \left( ac - b^2 \right) = 0\]
\[ \Rightarrow \alpha\text{ is a root of }4a x^2 + 12bx + 9c = 0\]
\[\text{ or ac }= b^2 \text{ , i . e . a, b, c are in G . P .} \]
APPEARS IN
RELATED QUESTIONS
Evaluate the determinant.
`|(2,4),(-5, -1)|`
Which of the following is correct?
A. Determinant is a square matrix.
B. Determinant is a number associated to a matrix.
C. Determinant is a number associated to a square matrix.
D. None of these
What is the value of the determinant \[\begin{vmatrix}0 & 2 & 0 \\ 2 & 3 & 4 \\ 4 & 5 & 6\end{vmatrix} ?\]
Write the value of the determinant \[\begin{vmatrix}p & p + 1 \\ p - 1 & p\end{vmatrix}\]
Write the value of the determinant \[\begin{vmatrix}x + y & y + z & z + x \\ z & x & y \\ - 3 & - 3 & - 3\end{vmatrix}\]
If \[A_r = \begin{vmatrix}1 & r & 2^r \\ 2 & n & n^2 \\ n & \frac{n \left( n + 1 \right)}{2} & 2^{n + 1}\end{vmatrix}\] , then the value of \[\sum^n_{r = 1} A_r\] is
Given that: A = `((1,-1,0),(2,3,4),(0,1,2))` and B = `((2,2,-4),(-4,2,-4),(2,-1,5))` , find AB. Using this result, solve the following system of equation: x – y = 3, 2x + 3y + 4z = 17 and y + 2z = 7.
If `omega` is a non-real cube root of unity and n is not a multiple of 3, then `Delta = abs ((1, omega^n, omega^(2n)),(omega^(2n), 1, omega^n),(omega^n, omega^(2n), 1))` is equal to ____________.
If ω is non real cube root of unity, then `abs ((2, 2omega, -omega^2),(1,1,1),(1,-1,0))` is equal to ____________.
A non-trivial solution of the system of equations x + λy + 2z = 0, 2x + λz = 0, 2λx - 2y + 3z = 0 is given by x : y : z = ____________.
If 4x + 3y + 6z = 25, x + 5y + 7z = 13, 2x + 9y + z = 1, then z = ____________.
If the equations 2x + 3y + z = 0, 3x + y - 2z = 0 and ax + 2y - bz = 0 has non-trivial solution, then ____________.
Find the area of the triangle with vertices P(4, 5), Q(4, -2), and R(-6, 2).
If `|(2, 4),(5, 1)| = |(2x, 4),(6, x)|`, then the possible value(s) of ‘x’ is/are ______.
Find the value of the determinant given below, without expanding it at any stage.
`|(βγ, 1, α(β + γ)),(γα, 1, β(γ + α)),(αβ, 1, γ(α + β))|`
