Advertisements
Advertisements
Question
Find the value of \[\lambda\] so that the points (1, −5), (−4, 5) and \[\lambda\] are collinear.
Solution
If the points (1, −5), (−4, 5) and \[\left( \lambda, 7 \right)\] are collinear, then
\[\begin{vmatrix}1 & - 5 & 1 \\ - 4 & 5 & 1 \\ \lambda & 7 & 1\end{vmatrix} = 0\]
\[ \Rightarrow \begin{vmatrix}1 & - 5 & 1 \\ - 5 & 10 & 0 \\ \lambda & 7 & 1\end{vmatrix} = 0 \left[\text{ Applying }R_2 \to R_2 - R_1 \right]\]
\[ \Rightarrow \begin{vmatrix}1 & - 5 & 1 \\ - 5 & 10 & 0 \\ \lambda - 1 & 12 & 0\end{vmatrix} = 0 \left[\text{ Applying }R_3 \to R_3 - R_1 \right]\]
\[ \Rightarrow ∆ = \begin{vmatrix}- 5 & 10 \\ \lambda - 1 & 12\end{vmatrix} = 0\]
\[ \Rightarrow - 60 - 10\left( \lambda - 1 \right) = 0\]
\[ \Rightarrow - 60 - 10\lambda + 10 = 0\]
\[ \Rightarrow - 10\lambda = 50\]
\[ \Rightarrow \lambda = - 5\]
APPEARS IN
RELATED QUESTIONS
Let A be a nonsingular square matrix of order 3 × 3. Then |adj A| is equal to ______.
Solve system of linear equations, using matrix method.
5x + 2y = 3
3x + 2y = 5
Solve the system of linear equations using the matrix method.
x − y + 2z = 7
3x + 4y − 5z = −5
2x − y + 3z = 12
Show that
\[\begin{vmatrix}\sin 10^\circ & - \cos 10^\circ \\ \sin 80^\circ & \cos 80^\circ\end{vmatrix} = 1\]
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:
\[\begin{vmatrix}0 & x & y \\ - x & 0 & z \\ - y & - z & 0\end{vmatrix}\]
Without expanding, show that the value of the following determinant is zero:
\[\begin{vmatrix}a & b & c \\ a + 2x & b + 2y & c + 2z \\ x & y & z\end{vmatrix}\]
\[\begin{vmatrix}1 + a & 1 & 1 \\ 1 & 1 + a & a \\ 1 & 1 & 1 + a\end{vmatrix} = a^3 + 3 a^2\]
Solve the following determinant equation:
Find the area of the triangle with vertice at the point:
(2, 7), (1, 1) and (10, 8)
If the points (a, 0), (0, b) and (1, 1) are collinear, prove that a + b = ab.
9x + 5y = 10
3y − 2x = 8
3x + y + z = 2
2x − 4y + 3z = − 1
4x + y − 3z = − 11
x − 4y − z = 11
2x − 5y + 2z = 39
− 3x + 2y + z = 1
3x + y = 5
− 6x − 2y = 9
Find the value of the determinant
\[\begin{bmatrix}4200 & 4201 \\ 4205 & 4203\end{bmatrix}\]
If A and B are non-singular matrices of the same order, write whether AB is singular or non-singular.
Write the cofactor of a12 in the following matrix \[\begin{bmatrix}2 & - 3 & 5 \\ 6 & 0 & 4 \\ 1 & 5 & - 7\end{bmatrix} .\]
If \[A = \begin{bmatrix}\cos\theta & \sin\theta \\ - \sin\theta & \cos\theta\end{bmatrix}\] , then for any natural number, find the value of Det(An).
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
Let \[A = \begin{bmatrix}1 & \sin \theta & 1 \\ - \sin \theta & 1 & \sin \theta \\ - 1 & - \sin \theta & 1\end{bmatrix},\text{ where 0 }\leq \theta \leq 2\pi . \text{ Then,}\]
If \[x, y \in \mathbb{R}\], then the determinant
Solve the following system of equations by matrix method:
5x + 2y = 3
3x + 2y = 5
Solve the following system of equations by matrix method:
3x + 4y − 5 = 0
x − y + 3 = 0
Solve the following system of equations by matrix method:
3x + y = 19
3x − y = 23
Solve the following system of equations by matrix method:
Show that the following systems of linear equations is consistent and also find their solutions:
x + y + z = 6
x + 2y + 3z = 14
x + 4y + 7z = 30
Show that each one of the following systems of linear equation is inconsistent:
2x + 5y = 7
6x + 15y = 13
Show that each one of the following systems of linear equation is inconsistent:
4x − 2y = 3
6x − 3y = 5
Given \[A = \begin{bmatrix}2 & 2 & - 4 \\ - 4 & 2 & - 4 \\ 2 & - 1 & 5\end{bmatrix}, B = \begin{bmatrix}1 & - 1 & 0 \\ 2 & 3 & 4 \\ 0 & 1 & 2\end{bmatrix}\] , find BA and use this to solve the system of equations y + 2z = 7, x − y = 3, 2x + 3y + 4z = 17
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?
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.
x + y − 6z = 0
x − y + 2z = 0
−3x + y + 2z = 0
Let \[X = \begin{bmatrix}x_1 \\ x_2 \\ x_3\end{bmatrix}, A = \begin{bmatrix}1 & - 1 & 2 \\ 2 & 0 & 1 \\ 3 & 2 & 1\end{bmatrix}\text{ and }B = \begin{bmatrix}3 \\ 1 \\ 4\end{bmatrix}\] . If AX = B, then X is equal to
If ` abs((1 + "a"^2 "x", (1 + "b"^2)"x", (1 + "c"^2)"x"),((1 + "a"^2) "x", 1 + "b"^2 "x", (1 + "c"^2) "x"), ((1 + "a"^2) "x", (1 + "b"^2) "x", 1 + "c"^2 "x"))`, then f(x) is apolynomial of degree ____________.
The number of values of k for which the linear equations 4x + ky + 2z = 0, kx + 4y + z = 0 and 2x + 2y + z = 0 possess a non-zero solution is
If a, b, c are non-zeros, then the system of equations (α + a)x + αy + αz = 0, αx + (α + b)y + αz = 0, αx+ αy + (α + c)z = 0 has a non-trivial solution if
Choose the correct option:
If a, b, c are in A.P. then the determinant `[(x + 2, x + 3, x + 2a),(x + 3, x + 4, x + 2b),(x + 4, x + 5, x + 2c)]` is
If the system of linear equations
2x + y – z = 7
x – 3y + 2z = 1
x + 4y + δz = k, where δ, k ∈ R has infinitely many solutions, then δ + k is equal to ______.
