Advertisements
Advertisements
Question
Show that x = 2 is a root of the equation
Advertisements
Solution
\[\text{ Let }∆ = \begin{vmatrix}x & - 6 & - 1 \\ 2 & - 3x & x - 3 \\ - 3 & 2x & x + 2\end{vmatrix}\]
\[ = \begin{vmatrix}x & - 6 & - 1 \\ 2 & - 3x & x - 3 \\ - 3 - x & 2x + 6 & x + 3\end{vmatrix} \left[\text{ Applying }R_3\text{ to }R_3 - R_1 \right]\]
\[ = \left( x + 3 \right)\begin{vmatrix}x & - 6 & - 1 \\ 2 & - 3x & x - 3 \\ - 1 & 2 & 1\end{vmatrix} \]
\[ = \left( x + 3 \right)\begin{vmatrix}x - 2 & 3x - 6 & - x + 2 \\ 2 & - 3x & x - 3 \\ - 1 & 2 & 1\end{vmatrix} \left[\text{ Applying } R_1 \text{ to }R_1 - R_2 \right]\]
\[ = \left( x + 3 \right)\left( x - 2 \right)\begin{vmatrix}1 & 3 & - 1 \\ 2 & - 3x & x - 3 \\ - 1 & 2 & 1\end{vmatrix} \]
\[ = \left( x + 3 \right)\left( x - 2 \right)\begin{vmatrix}1 & 3 & 0 \\ 2 & - 3x & x - 1 \\ - 1 & 2 & 0\end{vmatrix} \left[\text{ Applying }C_3 \text{ to }C_3 + C_1 \right]\]
\[ = \left( x + 3 \right)\left( x - 2 \right)\left( x - 1 \right)\begin{vmatrix}1 & 3 & 0 \\ 2 & - 3x & 1 \\ - 1 & 2 & 0\end{vmatrix} \]
\[ = \left( x + 3 \right)\left( x - 2 \right)\left( x - 1 \right)\left\{ - 1\begin{vmatrix}1 & 3 \\ - 1 & 2\end{vmatrix} \right\} \left[ \text{ Expanding along }C_3 \right]\]
\[ = - 5\left( x + 3 \right)\left( x - 2 \right)\left( x - 1 \right)\]
\[x = 2, - 3, 1\]
APPEARS IN
RELATED QUESTIONS
The cost of 4 kg onion, 3 kg wheat and 2 kg rice is Rs. 60. The cost of 2 kg onion, 4 kg wheat and 6 kg rice is Rs. 90. The cost of 6 kg onion 2 kg wheat and 3 kg rice is Rs. 70. Find the cost of each item per kg by matrix method.
If A \[\begin{bmatrix}1 & 0 & 1 \\ 0 & 1 & 2 \\ 0 & 0 & 4\end{bmatrix}\] , then show that |3 A| = 27 |A|.
Find the value of x, if
\[\begin{vmatrix}2 & 4 \\ 5 & 1\end{vmatrix} = \begin{vmatrix}2x & 4 \\ 6 & x\end{vmatrix}\]
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:
\[\begin{vmatrix}a + b & 2a + b & 3a + b \\ 2a + b & 3a + b & 4a + b \\ 4a + b & 5a + b & 6a + b\end{vmatrix}\]
Without expanding, show that the value of the following determinant is zero:
\[\begin{vmatrix}1^2 & 2^2 & 3^2 & 4^2 \\ 2^2 & 3^2 & 4^2 & 5^2 \\ 3^2 & 4^2 & 5^2 & 6^2 \\ 4^2 & 5^2 & 6^2 & 7^2\end{vmatrix}\]
Evaluate :
\[\begin{vmatrix}a & b + c & a^2 \\ b & c + a & b^2 \\ c & a + b & c^2\end{vmatrix}\]
\[\begin{vmatrix}- a \left( b^2 + c^2 - a^2 \right) & 2 b^3 & 2 c^3 \\ 2 a^3 & - b \left( c^2 + a^2 - b^2 \right) & 2 c^3 \\ 2 a^3 & 2 b^3 & - c \left( a^2 + b^2 - c^2 \right)\end{vmatrix} = abc \left( a^2 + b^2 + c^2 \right)^3\]
Prove the following identity:
`|(a^3,2,a),(b^3,2,b),(c^3,2,c)| = 2(a-b) (b-c) (c-a) (a+b+c)`
If the points (x, −2), (5, 2), (8, 8) are collinear, find x using determinants.
If the points (3, −2), (x, 2), (8, 8) are collinear, find x using determinant.
Using determinants, find the equation of the line joining the points
(1, 2) and (3, 6)
Find values of k, if area of triangle is 4 square units whose vertices are
(−2, 0), (0, 4), (0, k)
Prove that :
2x − y = 17
3x + 5y = 6
2x − y = − 2
3x + 4y = 3
2x + 3y = 10
x + 6y = 4
3x − y + 2z = 6
2x − y + z = 2
3x + 6y + 5z = 20.
x − y + z = 3
2x + y − z = 2
− x − 2y + 2z = 1
2x + y − 2z = 4
x − 2y + z = − 2
5x − 5y + z = − 2
x − y + 3z = 6
x + 3y − 3z = − 4
5x + 3y + 3z = 10
An automobile company uses three types of steel S1, S2 and S3 for producing three types of cars C1, C2and C3. Steel requirements (in tons) for each type of cars are given below :
| Cars C1 |
C2 | C3 | |
| Steel S1 | 2 | 3 | 4 |
| S2 | 1 | 1 | 2 |
| S3 | 3 | 2 | 1 |
Using Cramer's rule, find the number of cars of each type which can be produced using 29, 13 and 16 tons of steel of three types respectively.
Find the real values of λ for which the following system of linear equations has non-trivial solutions. Also, find the non-trivial solutions
\[2 \lambda x - 2y + 3z = 0\]
\[ x + \lambda y + 2z = 0\]
\[ 2x + \lambda z = 0\]
If the matrix \[\begin{bmatrix}5x & 2 \\ - 10 & 1\end{bmatrix}\] is singular, find the value of x.
If A and B are non-singular matrices of the same order, write whether AB is singular or non-singular.
If \[\begin{vmatrix}2x & 5 \\ 8 & x\end{vmatrix} = \begin{vmatrix}6 & - 2 \\ 7 & 3\end{vmatrix}\] , write the value of x.
The maximum value of \[∆ = \begin{vmatrix}1 & 1 & 1 \\ 1 & 1 + \sin\theta & 1 \\ 1 + \cos\theta & 1 & 1\end{vmatrix}\] is (θ is real)
The value of \[\begin{vmatrix}1 & 1 & 1 \\ {}^n C_1 & {}^{n + 2} C_1 & {}^{n + 4} C_1 \\ {}^n C_2 & {}^{n + 2} C_2 & {}^{n + 4} C_2\end{vmatrix}\] is
Solve the following system of equations by matrix method:
5x + 2y = 3
3x + 2y = 5
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:
8x + 4y + 3z = 18
2x + y +z = 5
x + 2y + z = 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?
Solve the following equations by using inversion method.
x + y + z = −1, x − y + z = 2 and x + y − z = 3
If the system of equations x + ky - z = 0, 3x - ky - z = 0 & x - 3y + z = 0 has non-zero solution, then k is equal to ____________.
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 `|(x + 1, x + 2, x + a),(x + 2, x + 3, x + b),(x + 3, x + 4, x + c)|` = 0, then a, b, care in
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 ______.
Let the system of linear equations x + y + az = 2; 3x + y + z = 4; x + 2z = 1 have a unique solution (x*, y*, z*). If (α, x*), (y*, α) and (x*, –y*) are collinear points, then the sum of absolute values of all possible values of α 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 ______.
