Advertisements
Advertisements
प्रश्न
Without expanding, show that the value of the following determinant is zero:
\[\begin{vmatrix}1 & 43 & 6 \\ 7 & 35 & 4 \\ 3 & 17 & 2\end{vmatrix}\]
Advertisements
उत्तर
\[ ∆ = \begin{vmatrix}1 & 43 & 6 \\ 7 & 35 & 4 \\ 3 & 17 & 2\end{vmatrix}\]
\[ = \begin{vmatrix}1 & 1 & 6 \\ 7 & 7 & 4 \\ 3 & 3 & 2\end{vmatrix} = 0 \left[ \text{ Applying }C_2 \to C_2 - 7 C_3 \right]\]
APPEARS IN
संबंधित प्रश्न
Evaluate the following determinant:
\[\begin{vmatrix}\cos \theta & - \sin \theta \\ \sin \theta & \cos \theta\end{vmatrix}\]
Evaluate the following determinant:
\[\begin{vmatrix}\cos 15^\circ & \sin 15^\circ \\ \sin 75^\circ & \cos 75^\circ\end{vmatrix}\]
Find the value of x, if
\[\begin{vmatrix}x + 1 & x - 1 \\ x - 3 & x + 2\end{vmatrix} = \begin{vmatrix}4 & - 1 \\ 1 & 3\end{vmatrix}\]
For what value of x the matrix A is singular?
\[A = \begin{bmatrix}x - 1 & 1 & 1 \\ 1 & x - 1 & 1 \\ 1 & 1 & x - 1\end{bmatrix}\]
Without expanding, show that the value of the following determinant is zero:
\[\begin{vmatrix}1 & a & a^2 - bc \\ 1 & b & b^2 - ac \\ 1 & c & c^2 - ab\end{vmatrix}\]
Without expanding, show that the value of the following determinant is zero:
\[\begin{vmatrix}\left( 2^x + 2^{- x} \right)^2 & \left( 2^x - 2^{- x} \right)^2 & 1 \\ \left( 3^x + 3^{- x} \right)^2 & \left( 3^x - 3^{- x} \right)^2 & 1 \\ \left( 4^x + 4^{- x} \right)^2 & \left( 4^x - 4^{- x} \right)^2 & 1\end{vmatrix}\]
Evaluate :
\[\begin{vmatrix}a & b & c \\ c & a & b \\ b & c & a\end{vmatrix}\]
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)`
Show that x = 2 is a root of the equation
Find the area of the triangle with vertice at the point:
(2, 7), (1, 1) and (10, 8)
Find values of k, if area of triangle is 4 square units whose vertices are
(k, 0), (4, 0), (0, 2)
Find values of k, if area of triangle is 4 square units whose vertices are
(−2, 0), (0, 4), (0, k)
9x + 5y = 10
3y − 2x = 8
2y − 3z = 0
x + 3y = − 4
3x + 4y = 3
3x + y = 5
− 6x − 2y = 9
3x − y + 2z = 6
2x − y + z = 2
3x + 6y + 5z = 20.
If A is a singular matrix, then write the value of |A|.
For what value of x, the following matrix is singular?
If \[A = \left[ a_{ij} \right]\] is a 3 × 3 diagonal matrix such that a11 = 1, a22 = 2 a33 = 3, then find |A|.
Write the value of the determinant \[\begin{vmatrix}2 & 3 & 4 \\ 5 & 6 & 8 \\ 6x & 9x & 12x\end{vmatrix}\]
Find the maximum value of \[\begin{vmatrix}1 & 1 & 1 \\ 1 & 1 + \sin \theta & 1 \\ 1 & 1 & 1 + \cos \theta\end{vmatrix}\]
Let \[\begin{vmatrix}x & 2 & x \\ x^2 & x & 6 \\ x & x & 6\end{vmatrix} = a x^4 + b x^3 + c x^2 + dx + e\]
Then, the value of \[5a + 4b + 3c + 2d + e\] is equal to
Using the factor theorem it is found that a + b, b + c and c + a are three factors of the determinant
The other factor in the value of the determinant is
If \[A + B + C = \pi\], then the value of \[\begin{vmatrix}\sin \left( A + B + C \right) & \sin \left( A + C \right) & \cos C \\ - \sin B & 0 & \tan A \\ \cos \left( A + B \right) & \tan \left( B + C \right) & 0\end{vmatrix}\] is equal to
The value of the determinant \[\begin{vmatrix}x & x + y & x + 2y \\ x + 2y & x & x + y \\ x + y & x + 2y & x\end{vmatrix}\] is
Solve the following system of equations by matrix method:
3x + y = 19
3x − y = 23
Show that each one of the following systems of linear equation is inconsistent:
2x + 3y = 5
6x + 9y = 10
A total amount of ₹7000 is deposited in three different saving bank accounts with annual interest rates 5%, 8% and \[8\frac{1}{2}\] % respectively. The total annual interest from these three accounts is ₹550. Equal amounts have been deposited in the 5% and 8% saving accounts. Find the amount deposited in each of the three accounts, with the help of matrices.
The existence of the unique solution of the system of equations:
x + y + z = λ
5x − y + µz = 10
2x + 3y − z = 6
depends on
x + y = 1
x + z = − 6
x − y − 2z = 3
Find the inverse of the following matrix, using elementary transformations:
`A= [[2 , 3 , 1 ],[2 , 4 , 1],[3 , 7 ,2]]`
The value of x, y, z for the following system of equations x + y + z = 6, x − y+ 2z = 5, 2x + y − z = 1 are ______
Solve the following equations by using inversion method.
x + y + z = −1, x − y + z = 2 and x + y − z = 3
If `|(2x, 5),(8, x)| = |(6, 5),(8, 3)|`, then find x
`abs (("a"^2, 2"ab", "b"^2),("b"^2, "a"^2, 2"ab"),(2"ab", "b"^2, "a"^2))` is equal to ____________.
Let A = `[(1,sin α,1),(-sin α,1,sin α),(-1,-sin α,1)]`, where 0 ≤ α ≤ 2π, then:
If the system of linear equations x + 2ay + az = 0; x + 3by + bz = 0; x + 4cy + cz = 0 has a non-zero solution, then a, b, c ______.
