Advertisements
Advertisements
प्रश्न
Without expanding, show that the value of the following determinant is zero:
\[\begin{vmatrix}\sqrt{23} + \sqrt{3} & \sqrt{5} & \sqrt{5} \\ \sqrt{15} + \sqrt{46} & 5 & \sqrt{10} \\ 3 + \sqrt{115} & \sqrt{15} & 5\end{vmatrix}\]
Advertisements
उत्तर
\[\begin{vmatrix}\sqrt{23} + \sqrt{3} & \sqrt{5} & \sqrt{5} \\ \sqrt{15} + \sqrt{46} & 5 & \sqrt{10} \\ 3 + \sqrt{115} & \sqrt{15} & 5\end{vmatrix}\]
\[ = \begin{vmatrix}\sqrt{3} & \sqrt{5} & \sqrt{5} \\ \sqrt{15} & 5 & \sqrt{10} \\ 3 & \sqrt{15} & 5\end{vmatrix} + \begin{vmatrix}\sqrt{23} & \sqrt{5} & \sqrt{5} \\ \sqrt{46} & 5 & \sqrt{10} \\ \sqrt{115} & \sqrt{15} & 5\end{vmatrix}\]
\[ = \sqrt{3}\begin{vmatrix}1 & \sqrt{5} & \sqrt{5} \\ \sqrt{5} & 5 & \sqrt{10} \\ \sqrt{3} & \sqrt{15} & 5\end{vmatrix} + \sqrt{23}\begin{vmatrix}1 & \sqrt{5} & \sqrt{5} \\ \sqrt{2} & 5 & \sqrt{10} \\ \sqrt{5} & \sqrt{15} & 5\end{vmatrix}\]
\[ = \sqrt{3} \times \sqrt{5}\begin{vmatrix}1 & 1 & \sqrt{5} \\ \sqrt{5} & \sqrt{5} & \sqrt{10} \\ \sqrt{3} & \sqrt{3} & 5\end{vmatrix} + \sqrt{23} \times \sqrt{5}\begin{vmatrix}1 & \sqrt{5} & 1 \\ \sqrt{2} & 5 & \sqrt{2} \\ \sqrt{5} & \sqrt{15} & \sqrt{5}\end{vmatrix}\]
\[ = 0 + 0\]
\[ = 0\]
APPEARS IN
संबंधित प्रश्न
If `|[x+1,x-1],[x-3,x+2]|=|[4,-1],[1,3]|`, then write the value of x.
Examine the consistency of the system of equations.
2x − y = 5
x + y = 4
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.
Evaluate the following determinant:
\[\begin{vmatrix}a + ib & c + id \\ - c + id & a - ib\end{vmatrix}\]
If A \[\begin{bmatrix}1 & 0 & 1 \\ 0 & 1 & 2 \\ 0 & 0 & 4\end{bmatrix}\] , then show that |3 A| = 27 |A|.
Without expanding, show that the value of the following determinant is zero:
\[\begin{vmatrix}1 & 43 & 6 \\ 7 & 35 & 4 \\ 3 & 17 & 2\end{vmatrix}\]
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 :
\[\begin{vmatrix}a & b & c \\ c & a & b \\ b & c & a\end{vmatrix}\]
Show that
Prove that :
Prove that :
Prove that :
Prove that :
2x − 3y − 4z = 29
− 2x + 5y − z = − 15
3x − y + 5z = − 11
3x − y + 2z = 6
2x − y + z = 2
3x + 6y + 5z = 20.
x + y − z = 0
x − 2y + z = 0
3x + 6y − 5z = 0
2x + y − 2z = 4
x − 2y + z = − 2
5x − 5y + z = − 2
A salesman has the following record of sales during three months for three items A, B and C which have different rates of commission
| Month | Sale of units | Total commission drawn (in Rs) |
||
| A | B | C | ||
| Jan | 90 | 100 | 20 | 800 |
| Feb | 130 | 50 | 40 | 900 |
| March | 60 | 100 | 30 | 850 |
Find out the rates of commission on items A, B and C by using determinant method.
If A is a singular matrix, then write the value of |A|.
If \[\begin{vmatrix}2x + 5 & 3 \\ 5x + 2 & 9\end{vmatrix} = 0\]
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 value of \[\begin{vmatrix}5^2 & 5^3 & 5^4 \\ 5^3 & 5^4 & 5^5 \\ 5^4 & 5^5 & 5^6\end{vmatrix}\]
Solve the following system of equations by matrix method:
8x + 4y + 3z = 18
2x + y +z = 5
x + 2y + z = 5
Show that the following systems of linear equations is consistent and also find their solutions:
x + y + z = 6
x + 2y + 3z = 14
x + 4y + 7z = 30
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
Transform `[(1, 2, 4),(3, -1, 5),(2, 4, 6)]` into an upper triangular matrix by using suitable row transformations
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:
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.
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?
Let A = `[(i, -i),(-i, i)], i = sqrt(-1)`. Then, the system of linear equations `A^8[(x),(y)] = [(8),(64)]` has ______.
The greatest value of c ε R for which the system of linear equations, x – cy – cz = 0, cx – y + cz = 0, cx + cy – z = 0 has a non-trivial solution, is ______.
