Advertisements
Advertisements
प्रश्न
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.
उत्तर
If the points (k, 2 − 2 k), (− k + 1, 2k) and (− 4 − k, 6 − 2k) are collinear, then
\[∆ = \begin{vmatrix}k & 2 - 2k & 1 \\ - k + 1 & 2k & 1 \\ - 4 - k & 6 - 2k & 1\end{vmatrix} = 0\]
\[ \Rightarrow \begin{vmatrix}k & 2 - 2k & 1 \\ - 2k + 1 & 4k - 2 & 0 \\ - 4 - k & 6 - 2k & 1\end{vmatrix} = 0 \left[\text{ Applying }R_2 \to R_2 - R_1 \right]\]
\[ \Rightarrow \begin{vmatrix}k & 2 - 2k & 1 \\ - 2k + 1 & 4k - 2 & 0 \\ - 4 - 2k & 4 & 0\end{vmatrix} = 0 \left[\text{ Applying } R_3 \to R_3 - R_1 \right]\]
\[ \Rightarrow \begin{vmatrix}- 2k + 1 & 4k - 2 \\ - 4 - 2k & 4\end{vmatrix} = 0\]
\[ \Rightarrow - 8k + 4 + 16k - 8 + 8 k^2 - 4k = 0 \]
\[ \Rightarrow 8 k^2 + 4k - 4 = 0\]
\[ \Rightarrow \left( 8k - 4 \right)\left( k + 1 \right) = 0\]
\[ \Rightarrow k = - 1\text{ or }k = \frac{1}{2}\]
APPEARS IN
संबंधित प्रश्न
Evaluate the following determinant:
\[\begin{vmatrix}\cos \theta & - \sin \theta \\ \sin \theta & \cos \theta\end{vmatrix}\]
Show that
\[\begin{vmatrix}\sin 10^\circ & - \cos 10^\circ \\ \sin 80^\circ & \cos 80^\circ\end{vmatrix} = 1\]
Find the value of x, if
\[\begin{vmatrix}2 & 4 \\ 5 & 1\end{vmatrix} = \begin{vmatrix}2x & 4 \\ 6 & x\end{vmatrix}\]
For what value of x the matrix A is singular?
\[ A = \begin{bmatrix}1 + x & 7 \\ 3 - x & 8\end{bmatrix}\]
Without expanding, show that the value of the following determinant is zero:
\[\begin{vmatrix}1^2 & 2^2 & 3^2 & 4^2 \\ 2^2 & 3^2 & 4^2 & 5^2 \\ 3^2 & 4^2 & 5^2 & 6^2 \\ 4^2 & 5^2 & 6^2 & 7^2\end{vmatrix}\]
Without expanding, show that the value of the following determinant is zero:
\[\begin{vmatrix}\left( 2^x + 2^{- x} \right)^2 & \left( 2^x - 2^{- x} \right)^2 & 1 \\ \left( 3^x + 3^{- x} \right)^2 & \left( 3^x - 3^{- x} \right)^2 & 1 \\ \left( 4^x + 4^{- x} \right)^2 & \left( 4^x - 4^{- x} \right)^2 & 1\end{vmatrix}\]
Without expanding, show that the value of the following determinant is zero:
\[\begin{vmatrix}\cos\left( x + y \right) & - \sin\left( x + y \right) & \cos2y \\ \sin x & \cos x & \sin y \\ - \cos x & \sin x & - \cos y\end{vmatrix}\]
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)`
Solve the following determinant equation:
Using determinants show that the following points are collinear:
(5, 5), (−5, 1) and (10, 7)
Using determinants show that the following points are collinear:
(3, −2), (8, 8) and (5, 2)
x − 2y = 4
−3x + 5y = −7
Prove that :
2x − 3y − 4z = 29
− 2x + 5y − z = − 15
3x − y + 5z = − 11
x − y + 3z = 6
x + 3y − 3z = − 4
5x + 3y + 3z = 10
Find the real values of λ for which the following system of linear equations has non-trivial solutions. Also, find the non-trivial solutions
\[2 \lambda x - 2y + 3z = 0\]
\[ x + \lambda y + 2z = 0\]
\[ 2x + \lambda z = 0\]
If A is a singular matrix, then write the value of |A|.
Write the value of the determinant \[\begin{vmatrix}2 & - 3 & 5 \\ 4 & - 6 & 10 \\ 6 & - 9 & 15\end{vmatrix} .\]
If \[\begin{vmatrix}3x & 7 \\ - 2 & 4\end{vmatrix} = \begin{vmatrix}8 & 7 \\ 6 & 4\end{vmatrix}\] , find the value of x.
If \[\begin{vmatrix}2x & 5 \\ 8 & x\end{vmatrix} = \begin{vmatrix}6 & - 2 \\ 7 & 3\end{vmatrix}\] , write the value of x.
Solve the following system of equations by matrix method:
x + y − z = 3
2x + 3y + z = 10
3x − y − 7z = 1
Solve the following system of equations by matrix method:
6x − 12y + 25z = 4
4x + 15y − 20z = 3
2x + 18y + 15z = 10
Solve the following system of equations by matrix method:
5x + 3y + z = 16
2x + y + 3z = 19
x + 2y + 4z = 25
Solve the following system of equations by matrix method:
8x + 4y + 3z = 18
2x + y +z = 5
x + 2y + z = 5
The sum of three numbers is 2. If twice the second number is added to the sum of first and third, the sum is 1. By adding second and third number to five times the first number, we get 6. Find the three numbers by using matrices.
Two institutions decided to award their employees for the three values of resourcefulness, competence and determination in the form of prices at the rate of Rs. x, y and z respectively per person. The first institution decided to award respectively 4, 3 and 2 employees with a total price money of Rs. 37000 and the second institution decided to award respectively 5, 3 and 4 employees with a total price money of Rs. 47000. If all the three prices per person together amount to Rs. 12000 then using matrix method find the value of x, y and z. What values are described in this equations?
3x + y − 2z = 0
x + y + z = 0
x − 2y + z = 0
The system of linear equations:
x + y + z = 2
2x + y − z = 3
3x + 2y + kz = 4 has a unique solution if
On her birthday Seema decided to donate some money to children of an orphanage home. If there were 8 children less, everyone would have got ₹ 10 more. However, if there were 16 children more, everyone would have got ₹ 10 less. Using the matrix method, find the number of children and the amount distributed by Seema. What values are reflected by Seema’s decision?
If A = `[(1, -1, 2),(3, 0, -2),(1, 0, 3)]`, verify that A(adj A) = (adj A)A
Solve the following system of equations x - y + z = 4, x - 2y + 2z = 9 and 2x + y + 3z = 1.
A set of linear equations is represented by the matrix equation Ax = b. The necessary condition for the existence of a solution for this system is
In system of equations, if inverse of matrix of coefficients A is multiplied by right side constant B vector then resultant will be?
If the system of equations x + λy + 2 = 0, λx + y – 2 = 0, λx + λy + 3 = 0 is consistent, then
What is the nature of the given system of equations
`{:(x + 2y = 2),(2x + 3y = 3):}`
