Advertisements
Advertisements
प्रश्न
Without expanding, show that the value of the following determinant is zero:
\[\begin{vmatrix}49 & 1 & 6 \\ 39 & 7 & 4 \\ 26 & 2 & 3\end{vmatrix}\]
उत्तर
\[ ∆ = \begin{vmatrix}49 & 1 & 6 \\ 39 & 7 & 4 \\ 26 & 2 & 3\end{vmatrix}\]
\[ = \begin{vmatrix}1 & 1 & 6 \\ 7 & 7 & 4 \\ 2 & 2 & 3\end{vmatrix} \left[ \text{ Applying } C_1 \to C_1 - 8 C_3 \right]\]
\[ \Rightarrow ∆ = 0\]
APPEARS IN
संबंधित प्रश्न
If `|[2x,5],[8,x]|=|[6,-2],[7,3]|`, write the value of x.
Show that
\[\begin{vmatrix}\sin 10^\circ & - \cos 10^\circ \\ \sin 80^\circ & \cos 80^\circ\end{vmatrix} = 1\]
Evaluate the following determinant:
\[\begin{vmatrix}1 & - 3 & 2 \\ 4 & - 1 & 2 \\ 3 & 5 & 2\end{vmatrix}\]
Without expanding, show that the value of the following determinant is zero:
\[\begin{vmatrix}6 & - 3 & 2 \\ 2 & - 1 & 2 \\ - 10 & 5 & 2\end{vmatrix}\]
\[\begin{vmatrix}0 & b^2 a & c^2 a \\ a^2 b & 0 & c^2 b \\ a^2 c & b^2 c & 0\end{vmatrix} = 2 a^3 b^3 c^3\]
Prove the following identity:
\[\begin{vmatrix}a + x & y & z \\ x & a + y & z \\ x & y & a + z\end{vmatrix} = a^2 \left( a + x + y + z \right)\]
Solve the following determinant equation:
If \[a, b\] and c are all non-zero and
If \[\begin{vmatrix}a & b - y & c - z \\ a - x & b & c - z \\ a - x & b - y & c\end{vmatrix} =\] 0, then using properties of determinants, find the value of \[\frac{a}{x} + \frac{b}{y} + \frac{c}{z}\] , where \[x, y, z \neq\] 0
9x + 5y = 10
3y − 2x = 8
3x + y + z = 2
2x − 4y + 3z = − 1
4x + y − 3z = − 11
3x + y = 5
− 6x − 2y = 9
3x − y + 2z = 6
2x − y + z = 2
3x + 6y + 5z = 20.
x + 2y = 5
3x + 6y = 15
x + y − z = 0
x − 2y + z = 0
3x + 6y − 5z = 0
2x + y − 2z = 4
x − 2y + z = − 2
5x − 5y + z = − 2
Solve each of the following system of homogeneous linear equations.
x + y − 2z = 0
2x + y − 3z = 0
5x + 4y − 9z = 0
Write the value of the determinant
\[\begin{bmatrix}2 & 3 & 4 \\ 2x & 3x & 4x \\ 5 & 6 & 8\end{bmatrix} .\]
If \[A = \begin{bmatrix}1 & 2 \\ 3 & - 1\end{bmatrix}\text{ and }B = \begin{bmatrix}1 & 0 \\ - 1 & 0\end{bmatrix}\] , find |AB|.
Find the value of x from the following : \[\begin{vmatrix}x & 4 \\ 2 & 2x\end{vmatrix} = 0\]
Write the value of the determinant \[\begin{vmatrix}2 & 3 & 4 \\ 5 & 6 & 8 \\ 6x & 9x & 12x\end{vmatrix}\]
For what value of x is the matrix \[\begin{bmatrix}6 - x & 4 \\ 3 - x & 1\end{bmatrix}\] singular?
The value of the determinant
Let \[\begin{vmatrix}x^2 + 3x & x - 1 & x + 3 \\ x + 1 & - 2x & x - 4 \\ x - 3 & x + 4 & 3x\end{vmatrix} = a x^4 + b x^3 + c x^2 + dx + e\]
be an identity in x, where a, b, c, d, e are independent of x. Then the value of e is
If a > 0 and discriminant of ax2 + 2bx + c is negative, then
\[∆ = \begin{vmatrix}a & b & ax + b \\ b & c & bx + c \\ ax + b & bx + c & 0\end{vmatrix} is\]
If a, b, c are in A.P., then the determinant
\[\begin{vmatrix}x + 2 & x + 3 & x + 2a \\ x + 3 & x + 4 & x + 2b \\ x + 4 & x + 5 & x + 2c\end{vmatrix}\]
The number of distinct real roots of \[\begin{vmatrix}cosec x & \sec x & \sec x \\ \sec x & cosec x & \sec x \\ \sec x & \sec x & cosec x\end{vmatrix} = 0\] lies in the interval
\[- \frac{\pi}{4} \leq x \leq \frac{\pi}{4}\]
The determinant \[\begin{vmatrix}b^2 - ab & b - c & bc - ac \\ ab - a^2 & a - b & b^2 - ab \\ bc - ca & c - a & ab - a^2\end{vmatrix}\]
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 + y = 19
3x − y = 23
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
Solve the following system of equations by matrix method:
Show that each one of the following systems of linear equation is inconsistent:
2x + 3y = 5
6x + 9y = 10
2x − y + z = 0
3x + 2y − z = 0
x + 4y + 3z = 0
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?
Transform `[(1, 2, 4),(3, -1, 5),(2, 4, 6)]` into an upper triangular matrix by using suitable row transformations
If c < 1 and the system of equations x + y – 1 = 0, 2x – y – c = 0 and – bx+ 3by – c = 0 is consistent, then the possible real values of b are
