Advertisements
Advertisements
प्रश्न
Prove that :
Advertisements
उत्तर
\[\text{ Let LHS }= \Delta = \begin{vmatrix} \left( a + 1 \right)\left( a + 2 \right) & a + 2 & 1\\\left( a + 2 \right)\left( a + 3 \right) & a + 3 & 1 \\\left( a + 3 \right)\left( a + 4 \right) & a + 4 & 1 \end{vmatrix}\]
\[ = \begin{vmatrix} \left( a + 1 \right)\left( a + 2 \right) - \left( a + 2 \right)\left( a + 3 \right) & \left( a + 2 \right) - \left( a + 3 \right) & 0\\\left( a + 2 \right)\left( a + 3 \right) - \left( a + 3 \right)\left( a + 4 \right) & \left( a + 3 \right) - \left( a + 4 \right) & 0 \\\left( a + 3 \right)\left( a + 4 \right) & \left( a + 4 \right) & 1 \end{vmatrix} \left[\text{ Applying }C_1 \to \hspace{0.167em} C_1 - C_2\text{ and }C_2 \to C_2 - C_3 \right]\]
\[ = \begin{vmatrix} - 2\left( a + 2 \right) & - 1 & 0\\ - 2\left( a + 3 \right) & - 1 & 0\\\left( a + 3 \right)\left( a + 4 \right) & \left( a + 4 \right) & 1 \end{vmatrix}\]
\[ = \left\{ 1 \times \begin{vmatrix} - 2\left( a + 2 \right) & - 1 \\- 2\left( a + 3 \right) & - 1 \end{vmatrix} \right\} \left[\text{ Expanding along }C_3 \right]\]
\[ = 4 + 2a - 2a - 6\]
\[ = - 2\]
\[ = RHS\]
Hence proved.
APPEARS IN
संबंधित प्रश्न
If `|[2x,5],[8,x]|=|[6,-2],[7,3]|`, write the value of x.
Solve the system of linear equations using the matrix method.
5x + 2y = 3
3x + 2y = 5
Solve the system of linear equations using the matrix method.
2x + y + z = 1
x – 2y – z = `3/2`
3y – 5z = 9
Evaluate the following determinant:
\[\begin{vmatrix}\cos 15^\circ & \sin 15^\circ \\ \sin 75^\circ & \cos 75^\circ\end{vmatrix}\]
Show that
\[\begin{vmatrix}\sin 10^\circ & - \cos 10^\circ \\ \sin 80^\circ & \cos 80^\circ\end{vmatrix} = 1\]
Find the value of x, if
\[\begin{vmatrix}x + 1 & x - 1 \\ x - 3 & x + 2\end{vmatrix} = \begin{vmatrix}4 & - 1 \\ 1 & 3\end{vmatrix}\]
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:
\[\begin{vmatrix}49 & 1 & 6 \\ 39 & 7 & 4 \\ 26 & 2 & 3\end{vmatrix}\]
\[\begin{vmatrix}b^2 + c^2 & ab & ac \\ ba & c^2 + a^2 & bc \\ ca & cb & a^2 + b^2\end{vmatrix} = 4 a^2 b^2 c^2\]
\[\begin{vmatrix}- a \left( b^2 + c^2 - a^2 \right) & 2 b^3 & 2 c^3 \\ 2 a^3 & - b \left( c^2 + a^2 - b^2 \right) & 2 c^3 \\ 2 a^3 & 2 b^3 & - c \left( a^2 + b^2 - c^2 \right)\end{vmatrix} = abc \left( a^2 + b^2 + c^2 \right)^3\]
Prove the following identity:
\[\begin{vmatrix}a + x & y & z \\ x & a + y & z \\ x & y & a + z\end{vmatrix} = a^2 \left( a + x + y + z \right)\]
Find the area of the triangle with vertice at the point:
(2, 7), (1, 1) and (10, 8)
Using determinants show that the following points are collinear:
(5, 5), (−5, 1) and (10, 7)
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)
Prove that :
Prove that
x + 2y = 5
3x + 6y = 15
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 \[\begin{vmatrix}2x & 5 \\ 8 & x\end{vmatrix} = \begin{vmatrix}6 & - 2 \\ 7 & 3\end{vmatrix}\] , write the value of x.
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 maximum value of \[∆ = \begin{vmatrix}1 & 1 & 1 \\ 1 & 1 + \sin\theta & 1 \\ 1 + \cos\theta & 1 & 1\end{vmatrix}\] is (θ is real)
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 + 4y − 5 = 0
x − y + 3 = 0
Solve the following system of equations by matrix method:
x + y + z = 3
2x − y + z = − 1
2x + y − 3z = − 9
Show that each one of the following systems of linear equation is inconsistent:
4x − 2y = 3
6x − 3y = 5
Two schools P and Q want to award their selected students on the values of Discipline, Politeness and Punctuality. The school P wants to award ₹x each, ₹y each and ₹z each the three respectively values to its 3, 2 and 1 students with a total award money of ₹1,000. School Q wants to spend ₹1,500 to award its 4, 1 and 3 students on the respective values (by giving the same award money for three values as before). If the total amount of awards for one prize on each value is ₹600, using matrices, find the award money for each value. Apart from the above three values, suggest one more value for awards.
3x − y + 2z = 0
4x + 3y + 3z = 0
5x + 7y + 4z = 0
x + y − 6z = 0
x − y + 2z = 0
−3x + y + 2z = 0
Solve the following system of equations by using inversion method
x + y = 1, y + z = `5/3`, z + x = `4/3`
Using determinants, find the equation of the line joining the points (1, 2) and (3, 6).
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 ____________.
Let A = `[(1,sin α,1),(-sin α,1,sin α),(-1,-sin α,1)]`, where 0 ≤ α ≤ 2π, then:
If the system of equations x + λy + 2 = 0, λx + y – 2 = 0, λx + λy + 3 = 0 is consistent, then
The value (s) of m does the system of equations 3x + my = m and 2x – 5y = 20 has a solution satisfying the conditions x > 0, y > 0.
If `|(x + a, beta, y),(a, x + beta, y),(a, beta, x + y)|` = 0, then 'x' is equal to
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 ______.
