Advertisements
Advertisements
प्रश्न
2x + y − 2z = 4
x − 2y + z = − 2
5x − 5y + z = − 2
Advertisements
उत्तर
Using the equations we get
\[D = \begin{vmatrix}2 & 1 & - 2 \\ 1 & - 2 & 1 \\ 5 & - 5 & 1\end{vmatrix}\]
\[ \Rightarrow 2\left( - 2 + 5 \right) - 1\left( 1 - 5 \right) - 2\left( - 5 + 10 \right) = 0\]
\[ D_1 = \begin{vmatrix}4 & 1 & - 2 \\ - 2 & - 2 & 1 \\ - 2 & - 5 & 1\end{vmatrix}\]
\[ \Rightarrow 4\left( - 2 + 5 \right) - 1\left( - 2 + 2 \right) - 2\left( 10 - 4 \right) = 0\]
\[ D_2 = \begin{vmatrix}2 & 4 & - 2 \\ 1 & - 2 & 1 \\ 5 & - 2 & 1\end{vmatrix}\]
\[ \Rightarrow 2\left( - 2 + 2 \right) - 4\left( 1 - 5 \right) - 2\left( - 2 + 10 \right) = 0\]
\[ D_3 = \begin{vmatrix}2 & 1 & 4 \\ 1 & - 2 & - 2 \\ 5 & - 5 & - 2\end{vmatrix}\]
\[ \Rightarrow 2\left( 4 - 10 \right) - 1\left( - 2 + 10 \right) + 4\left( - 5 + 10 \right) = 0\]
\[\therefore D = D_1 = D_2 = 0\]
Hence, the system of linear equations has infinitely many solutions.
APPEARS IN
संबंधित प्रश्न
Examine the consistency of the system of equations.
3x − y − 2z = 2
2y − z = −1
3x − 5y = 3
Examine the consistency of the system of equations.
5x − y + 4z = 5
2x + 3y + 5z = 2
5x − 2y + 6z = −1
Solve the system of linear equations using the matrix method.
x − y + z = 4
2x + y − 3z = 0
x + y + z = 2
Evaluate the following determinant:
\[\begin{vmatrix}x & - 7 \\ x & 5x + 1\end{vmatrix}\]
If \[A = \begin{bmatrix}2 & 5 \\ 2 & 1\end{bmatrix} \text{ and } B = \begin{bmatrix}4 & - 3 \\ 2 & 5\end{bmatrix}\] , verify that |AB| = |A| |B|.
Evaluate the following determinant:
\[\begin{vmatrix}6 & - 3 & 2 \\ 2 & - 1 & 2 \\ - 10 & 5 & 2\end{vmatrix}\]
Without expanding, show that the value of the following determinant is zero:
\[\begin{vmatrix}\cos\left( x + y \right) & - \sin\left( x + y \right) & \cos2y \\ \sin x & \cos x & \sin y \\ - \cos x & \sin x & - \cos y\end{vmatrix}\]
Evaluate the following:
\[\begin{vmatrix}x & 1 & 1 \\ 1 & x & 1 \\ 1 & 1 & x\end{vmatrix}\]
\[\begin{vmatrix}0 & b^2 a & c^2 a \\ a^2 b & 0 & c^2 b \\ a^2 c & b^2 c & 0\end{vmatrix} = 2 a^3 b^3 c^3\]
Prove the following identities:
\[\begin{vmatrix}y + z & z & y \\ z & z + x & x \\ y & x & x + y\end{vmatrix} = 4xyz\]
Show that
Solve the following determinant equation:
Find the area of the triangle with vertice at the point:
(0, 0), (6, 0) and (4, 3)
If the points (a, 0), (0, b) and (1, 1) are collinear, prove that a + b = ab.
2x − y = 1
7x − 2y = −7
Prove that :
5x − 7y + z = 11
6x − 8y − z = 15
3x + 2y − 6z = 7
3x − y + 2z = 3
2x + y + 3z = 5
x − 2y − z = 1
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|.
If A = [aij] is a 3 × 3 scalar matrix such that a11 = 2, then write the value of |A|.
Write the value of
Find the value of the determinant \[\begin{vmatrix}2^2 & 2^3 & 2^4 \\ 2^3 & 2^4 & 2^5 \\ 2^4 & 2^5 & 2^6\end{vmatrix}\].
If x ∈ N and \[\begin{vmatrix}x + 3 & - 2 \\ - 3x & 2x\end{vmatrix}\] = 8, then find the value of x.
If a > 0 and discriminant of ax2 + 2bx + c is negative, then
\[∆ = \begin{vmatrix}a & b & ax + b \\ b & c & bx + c \\ ax + b & bx + c & 0\end{vmatrix} is\]
Let \[f\left( x \right) = \begin{vmatrix}\cos x & x & 1 \\ 2\sin x & x & 2x \\ \sin x & x & x\end{vmatrix}\] \[\lim_{x \to 0} \frac{f\left( x \right)}{x^2}\] is equal to
If \[\begin{vmatrix}a & p & x \\ b & q & y \\ c & r & z\end{vmatrix} = 16\] , then the value of \[\begin{vmatrix}p + x & a + x & a + p \\ q + y & b + y & b + q \\ r + z & c + z & c + r\end{vmatrix}\] is
Show that the following systems of linear equations is consistent and also find their solutions:
5x + 3y + 7z = 4
3x + 26y + 2z = 9
7x + 2y + 10z = 5
2x + 3y − z = 0
x − y − 2z = 0
3x + y + 3z = 0
Let a, b, c be positive real numbers. The following system of equations in x, y and z
(a) no solution
(b) unique solution
(c) infinitely many solutions
(d) finitely many solutions
Write the value of `|(a-b, b- c, c-a),(b-c, c-a, a-b),(c-a, a-b, b-c)|`
If A = `[(1, -1, 2),(3, 0, -2),(1, 0, 3)]`, verify that A(adj A) = (adj A)A
`abs ((1, "a"^2 + "bc", "a"^3),(1, "b"^2 + "ca", "b"^3),(1, "c"^2 + "ab", "c"^3))`
If the system of equations x + ky - z = 0, 3x - ky - z = 0 & x - 3y + z = 0 has non-zero solution, then k is equal to ____________.
Let `θ∈(0, π/2)`. If the system of linear equations,
(1 + cos2θ)x + sin2θy + 4sin3θz = 0
cos2θx + (1 + sin2θ)y + 4sin3θz = 0
cos2θx + sin2θy + (1 + 4sin3θ)z = 0
has a non-trivial solution, then the value of θ is
______.
If the following equations
x + y – 3 = 0
(1 + λ)x + (2 + λ)y – 8 = 0
x – (1 + λ)y + (2 + λ) = 0
are consistent then the value of λ can be ______.
