Advertisements
Advertisements
प्रश्न
Prove that
Advertisements
उत्तर
\[\text{ Let LHS }= ∆ = \begin{vmatrix} a^2 & 2ab & b^2 \\ b^2 & a^2 & 2ab \\2ab & b^2 & a^2 \end{vmatrix}\]
\[ = a^2 \begin{vmatrix} a^2 & 2ab \\ b^2 & a^2 \end{vmatrix} - \left( 2ab \right) \begin{vmatrix} b^2 & 2ab \\2ab & a^2 \end{vmatrix} + b^2 \begin{vmatrix} b^2 & a^2 \\2ab & b^2 \end{vmatrix} \left[\text{ Expanding }\right]\]
\[ = a^2 \left( a^4 - 2a b^3 \right) - \left( 2ab \right)\left( b^2 a^2 - 4 a^2 b^2 \right) + b^2 \left( b^4 - 2 a^3 b \right)\]
\[ = a^6 - 2 a^3 b^3 - 2 a^3 b^3 + 8 a^3 b^3 + b^6 - 2 a^3 b^3 \]
\[ = a^6 + 2 a^3 b^3 + b^6 \]
\[ = \left( a^3 \right)^2 + 2 a^3 b^3 + \left( b^3 \right)^2 \]
\[ = \left( a^3 + b^3 \right)^2 \]
\[ = RHS\]
Hence proved.
APPEARS IN
संबंधित प्रश्न
Solve the system of linear equations using the matrix method.
4x – 3y = 3
3x – 5y = 7
Evaluate the following determinant:
\[\begin{vmatrix}x & - 7 \\ x & 5x + 1\end{vmatrix}\]
Evaluate the following determinant:
\[\begin{vmatrix}\cos \theta & - \sin \theta \\ \sin \theta & \cos \theta\end{vmatrix}\]
Evaluate the following determinant:
\[\begin{vmatrix}a + ib & c + id \\ - c + id & a - ib\end{vmatrix}\]
Find the value of x, if
\[\begin{vmatrix}2x & 5 \\ 8 & x\end{vmatrix} = \begin{vmatrix}6 & 5 \\ 8 & 3\end{vmatrix}\]
Evaluate the following determinant:
\[\begin{vmatrix}1 & 3 & 5 \\ 2 & 6 & 10 \\ 31 & 11 & 38\end{vmatrix}\]
Evaluate the following determinant:
\[\begin{vmatrix}67 & 19 & 21 \\ 39 & 13 & 14 \\ 81 & 24 & 26\end{vmatrix}\]
Without expanding, show that the value of the following determinant is zero:
\[\begin{vmatrix}1 & a & a^2 - bc \\ 1 & b & b^2 - ac \\ 1 & c & c^2 - ab\end{vmatrix}\]
\[\begin{vmatrix}- a \left( b^2 + c^2 - a^2 \right) & 2 b^3 & 2 c^3 \\ 2 a^3 & - b \left( c^2 + a^2 - b^2 \right) & 2 c^3 \\ 2 a^3 & 2 b^3 & - c \left( a^2 + b^2 - c^2 \right)\end{vmatrix} = abc \left( a^2 + b^2 + c^2 \right)^3\]
Solve the following determinant equation:
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
\[a + b + c = 0 \text{ or, } a = b = c\]
If \[a, b\] and c are all non-zero and
Using determinants, find the area of the triangle with vertices (−3, 5), (3, −6), (7, 2).
Using determinants, find the value of k so that the points (k, 2 − 2 k), (−k + 1, 2k) and (−4 − k, 6 − 2k) may be collinear.
If the points (3, −2), (x, 2), (8, 8) are collinear, find x using determinant.
Using determinants, find the equation of the line joining the points
(1, 2) and (3, 6)
2x − y = 1
7x − 2y = −7
Prove that :
2x + 3y = 10
x + 6y = 4
5x + 7y = − 2
4x + 6y = − 3
2x − 3y − 4z = 29
− 2x + 5y − z = − 15
3x − y + 5z = − 11
State whether the matrix
\[\begin{bmatrix}2 & 3 \\ 6 & 4\end{bmatrix}\] is singular or non-singular.
If \[A = \begin{bmatrix}1 & 2 \\ 3 & - 1\end{bmatrix}\text{ and }B = \begin{bmatrix}1 & 0 \\ - 1 & 0\end{bmatrix}\] , find |AB|.
Find the value of the determinant \[\begin{vmatrix}2^2 & 2^3 & 2^4 \\ 2^3 & 2^4 & 2^5 \\ 2^4 & 2^5 & 2^6\end{vmatrix}\].
Write the cofactor of a12 in the following matrix \[\begin{bmatrix}2 & - 3 & 5 \\ 6 & 0 & 4 \\ 1 & 5 & - 7\end{bmatrix} .\]
Write the value of the determinant \[\begin{vmatrix}2 & 3 & 4 \\ 5 & 6 & 8 \\ 6x & 9x & 12x\end{vmatrix}\]
If |A| = 2, where A is 2 × 2 matrix, find |adj A|.
If ω is a non-real cube root of unity and n is not a multiple of 3, then \[∆ = \begin{vmatrix}1 & \omega^n & \omega^{2n} \\ \omega^{2n} & 1 & \omega^n \\ \omega^n & \omega^{2n} & 1\end{vmatrix}\]
The determinant \[\begin{vmatrix}b^2 - ab & b - c & bc - ac \\ ab - a^2 & a - b & b^2 - ab \\ bc - ca & c - a & ab - a^2\end{vmatrix}\]
The value of \[\begin{vmatrix}1 & 1 & 1 \\ {}^n C_1 & {}^{n + 2} C_1 & {}^{n + 4} C_1 \\ {}^n C_2 & {}^{n + 2} C_2 & {}^{n + 4} C_2\end{vmatrix}\] is
Solve the following system of equations by matrix method:
3x + 4y + 2z = 8
2y − 3z = 3
x − 2y + 6z = −2
If \[\begin{bmatrix}1 & 0 & 0 \\ 0 & - 1 & 0 \\ 0 & 0 & - 1\end{bmatrix}\begin{bmatrix}x \\ y \\ z\end{bmatrix} = \begin{bmatrix}1 \\ 0 \\ 1\end{bmatrix}\], find x, y and z.
The system of equation x + y + z = 2, 3x − y + 2z = 6 and 3x + y + z = −18 has
Consider the system of equations:
a1x + b1y + c1z = 0
a2x + b2y + c2z = 0
a3x + b3y + c3z = 0,
if \[\begin{vmatrix}a_1 & b_1 & c_1 \\ a_2 & b_2 & c_2 \\ a_3 & b_3 & c_3\end{vmatrix}\]= 0, then the system has
Solve the following system of equations by using inversion method
x + y = 1, y + z = `5/3`, z + x = `4/3`
The cost of 4 dozen pencils, 3 dozen pens and 2 dozen erasers is ₹ 60. The cost of 2 dozen pencils, 4 dozen pens and 6 dozen erasers is ₹ 90. Whereas the cost of 6 dozen pencils, 2 dozen pens and 3 dozen erasers is ₹ 70. Find the cost of each item per dozen by using matrices
