Advertisements
Advertisements
प्रश्न
Prove that :
Advertisements
उत्तर
\[\text{ Let LHS }= \Delta = \begin{vmatrix} a + b + 2c & a & b\\c & b + c + 2a & b\\c & a & c + a + 2b \end{vmatrix}\]
\[ \Rightarrow ∆ = \begin{vmatrix} 2a + 2b + 2c & a & b \\2a + 2b + 2c & b + c + 2a & b\\2a + 2b + 2c & a & c + a + 2b \end{vmatrix}\left[\text{ Applying }C_1 \to C_1 + C_2 + C_3 \right] \]
\[ = 2 \left( a + b + c \right) \begin{vmatrix} 1 & a & b \\1 & b + c + 2a & b\\1 & a & c + a + 2b \end{vmatrix} \left[\text{ Taking out 2(a + b + c) common from }C_1 \right]\]
\[ ∆ = 2 \left( a + b + c \right) \begin{vmatrix} 1 & a & b \\0 & b + c + a & 0\\0 & - b - c - a & c + a + b \end{vmatrix} \left[\text{ Applying }R_2 \to R_2 - R_1\text{ and }R_2 \to R_2 - R_3 \right]\]
\[ = 2\left( a + b + c \right)\left( a + b + c \right)\left( a + b + c \right)\begin{vmatrix} 1 & a & b \\0 & 1 & 0\\0 & - 1 & 1 \end{vmatrix} \left[\text{ Taking out (a + b + c) common from }R_2\text{ and }R_3 \right]\]
\[ = 2 \left( a + b + c \right)^3 \left\{ 1\left( 1 - 0 \right) \right\} \left[\text{ Expanding along }C_1 \right]\]
\[ = 2 \left( a + b + c \right)^3 \]
\[ = RHS\]
APPEARS IN
संबंधित प्रश्न
Examine the consistency of the system of equations.
x + y + z = 1
2x + 3y + 2z = 2
ax + ay + 2az = 4
Examine the consistency of the system of equations.
5x − y + 4z = 5
2x + 3y + 5z = 2
5x − 2y + 6z = −1
Evaluate
\[\begin{vmatrix}2 & 3 & 7 \\ 13 & 17 & 5 \\ 15 & 20 & 12\end{vmatrix}^2 .\]
Find the value of x, if
\[\begin{vmatrix}3 & x \\ x & 1\end{vmatrix} = \begin{vmatrix}3 & 2 \\ 4 & 1\end{vmatrix}\]
Without expanding, show that the value of the following determinant is zero:
\[\begin{vmatrix}1 & a & a^2 - bc \\ 1 & b & b^2 - ac \\ 1 & c & c^2 - ab\end{vmatrix}\]
Without expanding, show that the value of the following determinant is zero:
\[\begin{vmatrix}\sin\alpha & \cos\alpha & \cos(\alpha + \delta) \\ \sin\beta & \cos\beta & \cos(\beta + \delta) \\ \sin\gamma & \cos\gamma & \cos(\gamma + \delta)\end{vmatrix}\]
Evaluate the following:
\[\begin{vmatrix}a + x & y & z \\ x & a + y & z \\ x & y & a + z\end{vmatrix}\]
\[\begin{vmatrix}1 + a & 1 & 1 \\ 1 & 1 + a & a \\ 1 & 1 & 1 + a\end{vmatrix} = a^3 + 3 a^2\]
Solve the following determinant equation:
Solve the following determinant equation:
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
Using determinants show that the following points are collinear:
(3, −2), (8, 8) and (5, 2)
x − 2y = 4
−3x + 5y = −7
2x − y = 17
3x + 5y = 6
5x + 7y = − 2
4x + 6y = − 3
x+ y = 5
y + z = 3
x + z = 4
2y − 3z = 0
x + 3y = − 4
3x + 4y = 3
For what value of x, the following matrix is singular?
Write the value of \[\begin{vmatrix}a + ib & c + id \\ - c + id & a - ib\end{vmatrix} .\]
Evaluate: \[\begin{vmatrix}\cos 15^\circ & \sin 15^\circ \\ \sin 75^\circ & \cos 75^\circ\end{vmatrix}\]
If \[∆_1 = \begin{vmatrix}1 & 1 & 1 \\ a & b & c \\ a^2 & b^2 & c^2\end{vmatrix}, ∆_2 = \begin{vmatrix}1 & bc & a \\ 1 & ca & b \\ 1 & ab & c\end{vmatrix},\text{ then }\]}
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
\[- \frac{\pi}{4} \leq x \leq \frac{\pi}{4}\]
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
Show that the following systems of linear equations is consistent and also find their solutions:
5x + 3y + 7z = 4
3x + 26y + 2z = 9
7x + 2y + 10z = 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
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.
Two schools P and Q want to award their selected students on the values of Tolerance, Kindness and Leadership. The school P wants to award ₹x each, ₹y each and ₹z each for the three respective values to 3, 2 and 1 students respectively with a total award money of ₹2,200. School Q wants to spend ₹3,100 to award its 4, 1 and 3 students on the respective values (by giving the same award money to the three values as school P). If the total amount of award for one prize on each values is ₹1,200, using matrices, find the award money for each value.
Apart from these three values, suggest one more value which should be considered for award.
2x − y + 2z = 0
5x + 3y − z = 0
x + 5y − 5z = 0
x + y − z = 0
x − 2y + z = 0
3x + 6y − 5z = 0
The number of solutions of the system of equations:
2x + y − z = 7
x − 3y + 2z = 1
x + 4y − 3z = 5
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 = `[(2, 0),(0, 1)]` and B = `[(1),(2)]`, then find the matrix X such that A−1X = B.
The system of simultaneous linear equations kx + 2y – z = 1, (k – 1)y – 2z = 2 and (k + 2)z = 3 have a unique solution if k equals:
For what value of p, is the system of equations:
p3x + (p + 1)3y = (p + 2)3
px + (p + 1)y = p + 2
x + y = 1
consistent?
What is the nature of the given system of equations
`{:(x + 2y = 2),(2x + 3y = 3):}`
