Advertisements
Advertisements
प्रश्न
If a, b, c are in A.P., then the determinant
\[\begin{vmatrix}x + 2 & x + 3 & x + 2a \\ x + 3 & x + 4 & x + 2b \\ x + 4 & x + 5 & x + 2c\end{vmatrix}\]
पर्याय
0
1
x
2x
Advertisements
उत्तर
\[\begin{vmatrix} x + 2 & x + 3 & x + 2a\\x + 3 & x + 4 & x + 2b\\x + 4 & x + 5 & x + 2c \end{vmatrix}\]
\[ = \begin{vmatrix} 0 & 0 & 2\left( a + c - 2b \right)\\x + 3 & x + 4 & x + 2b\\x + 4 & x + 5 & x + 2c \end{vmatrix} \left[\text{ Applying }R_1 \to R_1 + R_3 - R_2 , R_1 \to R_1 - R_2 \right]\]
\[ = \begin{vmatrix} 0 & 0 & 0\\x + 3 & x + 4 & x + 2b\\x + 4 & x + 5 & x + 2c \end{vmatrix} \left[ \because\text{ a, b, c are in A . P . }\right]\]
\[ = 0\]
APPEARS IN
संबंधित प्रश्न
Let A be a nonsingular square matrix of order 3 × 3. Then |adj A| is equal to ______.
Examine the consistency of the system of equations.
x + 2y = 2
2x + 3y = 3
Solve the system of linear equations using the matrix method.
2x – y = –2
3x + 4y = 3
Solve the system of linear equations using the matrix method.
2x + 3y + 3z = 5
x − 2y + z = −4
3x − y − 2z = 3
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
\[\begin{vmatrix}2x & 5 \\ 8 & x\end{vmatrix} = \begin{vmatrix}6 & 5 \\ 8 & 3\end{vmatrix}\]
Evaluate the following determinant:
\[\begin{vmatrix}6 & - 3 & 2 \\ 2 & - 1 & 2 \\ - 10 & 5 & 2\end{vmatrix}\]
\[\begin{vmatrix}0 & b^2 a & c^2 a \\ a^2 b & 0 & c^2 b \\ a^2 c & b^2 c & 0\end{vmatrix} = 2 a^3 b^3 c^3\]
Show that x = 2 is a root of the equation
Using determinants, find the area of the triangle with vertices (−3, 5), (3, −6), (7, 2).
Using determinants, find the equation of the line joining the points
(1, 2) and (3, 6)
5x + 7y = − 2
4x + 6y = − 3
Find the value of the determinant
\[\begin{bmatrix}101 & 102 & 103 \\ 104 & 105 & 106 \\ 107 & 108 & 109\end{bmatrix}\]
If \[A = \begin{bmatrix}1 & 2 \\ 3 & - 1\end{bmatrix}\text{ and B} = \begin{bmatrix}1 & - 4 \\ 3 & - 2\end{bmatrix},\text{ find }|AB|\]
Write the value of
Find the value of x from the following : \[\begin{vmatrix}x & 4 \\ 2 & 2x\end{vmatrix} = 0\]
If \[\begin{vmatrix}2x & 5 \\ 8 & x\end{vmatrix} = \begin{vmatrix}6 & - 2 \\ 7 & 3\end{vmatrix}\] , write the value of x.
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).
The value of the determinant
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
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
If \[A = \begin{bmatrix}2 & 3 & 1 \\ 1 & 2 & 2 \\ 3 & 1 & - 1\end{bmatrix}\] , find A–1 and hence solve the system of equations 2x + y – 3z = 13, 3x + 2y + z = 4, x + 2y – z = 8.
Use product \[\begin{bmatrix}1 & - 1 & 2 \\ 0 & 2 & - 3 \\ 3 & - 2 & 4\end{bmatrix}\begin{bmatrix}- 2 & 0 & 1 \\ 9 & 2 & - 3 \\ 6 & 1 & - 2\end{bmatrix}\] to solve the system of equations x + 3z = 9, −x + 2y − 2z = 4, 2x − 3y + 4z = −3.
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.
A company produces three products every day. Their production on a certain day is 45 tons. It is found that the production of third product exceeds the production of first product by 8 tons while the total production of first and third product is twice the production of second product. Determine the production level of each product using matrix method.
x + y − 6z = 0
x − y + 2z = 0
−3x + y + 2z = 0
x + y + z = 0
x − y − 5z = 0
x + 2y + 4z = 0
2x + 3y − z = 0
x − y − 2z = 0
3x + y + 3z = 0
The number of solutions of the system of equations:
2x + y − z = 7
x − 3y + 2z = 1
x + 4y − 3z = 5
The system of linear equations:
x + y + z = 2
2x + y − z = 3
3x + 2y + kz = 4 has a unique solution if
x + y = 1
x + z = − 6
x − y − 2z = 3
System of equations x + y = 2, 2x + 2y = 3 has ______
Three chairs and two tables cost ₹ 1850. Five chairs and three tables cost ₹2850. Find the cost of four chairs and one table by using matrices
If `|(2x, 5),(8, x)| = |(6, 5),(8, 3)|`, then find x
`abs ((1, "a"^2 + "bc", "a"^3),(1, "b"^2 + "ca", "b"^3),(1, "c"^2 + "ab", "c"^3))`
If A = `[(1,-1,0),(2,3,4),(0,1,2)]` and B = `[(2,2,-4),(-4,2,-4),(2,-1,5)]`, then:
