Advertisements
Advertisements
प्रश्न
Prove that :
Advertisements
उत्तर
\[\text{Let LHS }= \Delta = \begin{vmatrix} b + c & a - b & a\\c + a & b - c & b \\a + b & c - a & c \end{vmatrix}\]
\[\Delta = \left( b + c \right) \begin{vmatrix} b - c & b \\c - a & c \end{vmatrix} - \left( a - b \right) \begin{vmatrix} c + a & b \\ a + b & c \end{vmatrix} + a \begin{vmatrix} c + a & b - c \\a + b & c - a \end{vmatrix} \left[\text{ Expanding }\right] \]
\[ = \left( b + c \right)\left\{ bc - c^2 - bc + ab \right\} - \left( a - b \right)\left\{ c^2 + ac - ab - b^2 \right\} + a\left\{ c^2 - a^2 - ab + ac - b^2 + bc \right\}\]
\[ = b c^2 - c^3 + abc - a c^2 - a^2 c + a^2 b + a b^2 + b c^2 + abc - a b^2 - b^3 + a c^2 - a^3 - a^2 c - a b^2 + abc\]
\[ \Rightarrow \Delta = 3abc - a^3 - b^3 - c^3 \left[\text{ Simplyfying }\right]\]
\[ = RHS\]
APPEARS IN
संबंधित प्रश्न
Find the value of a if `[[a-b,2a+c],[2a-b,3c+d]]=[[-1,5],[0,13]]`
Solve the system of linear equations using the matrix method.
2x – y = –2
3x + 4y = 3
Evaluate the following determinant:
\[\begin{vmatrix}\cos 15^\circ & \sin 15^\circ \\ \sin 75^\circ & \cos 75^\circ\end{vmatrix}\]
For what value of x the matrix A is singular?
\[A = \begin{bmatrix}x - 1 & 1 & 1 \\ 1 & x - 1 & 1 \\ 1 & 1 & x - 1\end{bmatrix}\]
Evaluate the following determinant:
\[\begin{vmatrix}1 & 3 & 9 & 27 \\ 3 & 9 & 27 & 1 \\ 9 & 27 & 1 & 3 \\ 27 & 1 & 3 & 9\end{vmatrix}\]
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:
\[\begin{vmatrix}\sin^2 A & \cot A & 1 \\ \sin^2 B & \cot B & 1 \\ \sin^2 C & \cot C & 1\end{vmatrix}, where A, B, C \text{ are the angles of }∆ ABC .\]
Evaluate the following:
\[\begin{vmatrix}a + x & y & z \\ x & a + y & z \\ x & y & a + z\end{vmatrix}\]
\[\begin{vmatrix}b^2 + c^2 & ab & ac \\ ba & c^2 + a^2 & bc \\ ca & cb & a^2 + b^2\end{vmatrix} = 4 a^2 b^2 c^2\]
Prove that
\[\begin{vmatrix}\frac{a^2 + b^2}{c} & c & c \\ a & \frac{b^2 + c^2}{a} & a \\ b & b & \frac{c^2 + a^2}{b}\end{vmatrix} = 4abc\]
Prove the following identity:
\[\begin{vmatrix}2y & y - z - x & 2y \\ 2z & 2z & z - x - y \\ x - y - z & 2x & 2x\end{vmatrix} = \left( x + y + z \right)^3\]
Show that x = 2 is a root of the equation
Solve the following determinant equation:
Solve the following determinant equation:
Find the area of the triangle with vertice at the point:
(2, 7), (1, 1) and (10, 8)
Prove that :
Given: x + 2y = 1
3x + y = 4
x + y − z = 0
x − 2y + z = 0
3x + 6y − 5z = 0
If a, b, c are non-zero real numbers and if the system of equations
(a − 1) x = y + z
(b − 1) y = z + x
(c − 1) z = x + y
has a non-trivial solution, then prove that ab + bc + ca = abc.
Find the value of the determinant \[\begin{vmatrix}2^2 & 2^3 & 2^4 \\ 2^3 & 2^4 & 2^5 \\ 2^4 & 2^5 & 2^6\end{vmatrix}\].
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
There are two values of a which makes the determinant \[∆ = \begin{vmatrix}1 & - 2 & 5 \\ 2 & a & - 1 \\ 0 & 4 & 2a\end{vmatrix}\] equal to 86. The sum of these two values is
Solve the following system of equations by matrix method:
3x + y = 7
5x + 3y = 12
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:
2x + 6y = 2
3x − z = −8
2x − y + z = −3
Show that the following systems of linear equations is consistent and also find their solutions:
x − y + z = 3
2x + y − z = 2
−x −2y + 2z = 1
Show that each one of the following systems of linear equation is inconsistent:
2x + 3y = 5
6x + 9y = 10
Show that each one of the following systems of linear equation is inconsistent:
x + y − 2z = 5
x − 2y + z = −2
−2x + y + z = 4
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
2x − y + 2z = 0
5x + 3y − z = 0
x + 5y − 5z = 0
The system of equation x + y + z = 2, 3x − y + 2z = 6 and 3x + y + z = −18 has
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?
Solve the following by inversion method 2x + y = 5, 3x + 5y = −3
If `|(2x, 5),(8, x)| = |(6, -2),(7, 3)|`, then value of x is ______.
Solve the following system of equations x − y + z = 4, x − 2y + 2z = 9 and 2x + y + 3z = 1.
The existence of unique solution of the system of linear equations x + y + z = a, 5x – y + bz = 10, 2x + 3y – z = 6 depends on
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
The number of real values λ, such that the system of linear equations 2x – 3y + 5z = 9, x + 3y – z = –18 and 3x – y + (λ2 – |λ|z) = 16 has no solution, is ______.
If a, b, c are non-zero real numbers and if the system of equations (a – 1)x = y + z, (b – 1)y = z + x, (c – 1)z = x + y, has a non-trivial solution, then ab + bc + ca equals ______.
Using the matrix method, solve the following system of linear equations:
`2/x + 3/y + 10/z` = 4, `4/x - 6/y + 5/z` = 1, `6/x + 9/y - 20/z` = 2.
