Advertisements
Advertisements
प्रश्न
Without expanding, show that the value of the following determinant is zero:
\[\begin{vmatrix}\sin^2 A & \cot A & 1 \\ \sin^2 B & \cot B & 1 \\ \sin^2 C & \cot C & 1\end{vmatrix}, where A, B, C \text{ are the angles of }∆ ABC .\]
Advertisements
उत्तर
\[\begin{vmatrix}\sin^2 A & \cot A & 1 \\ \sin^2 B & \cot B & 1 \\ \sin^2 C & \cot C & 1\end{vmatrix}\]
\[ = \begin{vmatrix}\sin^2 A - \sin^2 B & \cot A - \cot B & 0 \\ \sin^2 B & \cot B & 1 \\ \sin^2 C - \sin^2 B & \cot C - \cot B & 0\end{vmatrix} \left[ \text{ Applying } R_1 \to R_1 - R_2 \text{ and }R_3 \to R_3 - R_2 \right]\]
\[ = \begin{vmatrix}\sin\left( A + B \right)\sin\left( A - B \right) & \frac{\cos A\sin B - \cos B\sin A}{\sin A\sin B} & 0 \\ \sin^2 B & \cot B & 1 \\ \sin\left( C + B \right)\sin\left( C - B \right) & \frac{\cos C\sin B - \cos B\sin C}{\sin B\sin C} & 0\end{vmatrix}\]
\[ = \begin{vmatrix}\sin\left( \pi - C \right)\sin\left( A - B \right) & \frac{- \sin\left( A - B \right)}{\sin A\sin B} & 0 \\ \sin^2 B & cot B & 1 \\ \sin\left( \pi - A \right)\sin\left( C - B \right) & \frac{- \sin\left( C - B \right)}{\sin B\sin C} & 0\end{vmatrix} \left[ \because A + B + C = \pi \right]\]
\[ = \begin{vmatrix}\sin C\sin\left( A - B \right) & \frac{- \sin\left( A - B \right)}{\sin A\sin B} & 0 \\ \sin^2 B & \frac{\cos B}{\sin B} & 1 \\ \sin A\sin\left( C - B \right) & \frac{- \sin\left( C - B \right)}{\sin B\sin C} & 0\end{vmatrix}\]
\[ = \frac{\sin\left( A - B \right)\sin\left( C - B \right)}{\sin B}\begin{vmatrix}\sin C & \frac{- 1}{\sin A} & 0 \\ \sin^2 B & \cos B & 1 \\ \sin A & \frac{- 1}{\sin C} & 0\end{vmatrix}\]
\[ = \frac{\sin\left( A - B \right)\sin\left( C - B \right)}{\sin B\sin A\sin C}\begin{vmatrix}\sin C\sin A & - 1 & 0 \\ \sin^2 B & \cos B & 1 \\ \sin A\sin C & - 1 & 0\end{vmatrix} \left[ \text{ Applying }R_1 \to \sin A R_1\text{ and }R_3 \to \sin C R_3 \right]\]
\[ = \frac{\sin\left( A - B \right)\sin\left( C - B \right)}{\sin B\sin A\sin C}\begin{vmatrix}0 & 0 & 0 \\ \sin^2 B & \cos B & 1 \\ \sin A\sin C & - 1 & 0\end{vmatrix} \left[ \text{ Applying }R_1 \to R_1 - R_3 \right]\]
\[ = 0\]
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 = 4
7x + 3y = 5
Solve the system of the following equations:
`2/x+3/y+10/z = 4`
`4/x-6/y + 5/z = 1`
`6/x + 9/y - 20/x = 2`
Evaluate the following determinant:
\[\begin{vmatrix}\cos \theta & - \sin \theta \\ \sin \theta & \cos \theta\end{vmatrix}\]
If \[A = \begin{bmatrix}2 & 5 \\ 2 & 1\end{bmatrix} \text{ and } B = \begin{bmatrix}4 & - 3 \\ 2 & 5\end{bmatrix}\] , verify that |AB| = |A| |B|.
Find the value of x, if
\[\begin{vmatrix}2 & 3 \\ 4 & 5\end{vmatrix} = \begin{vmatrix}x & 3 \\ 2x & 5\end{vmatrix}\]
Find the value of x, if
\[\begin{vmatrix}3x & 7 \\ 2 & 4\end{vmatrix} = 10\] , find the value of x.
Evaluate the following:
\[\begin{vmatrix}x & 1 & 1 \\ 1 & x & 1 \\ 1 & 1 & x\end{vmatrix}\]
Prove the following identities:
\[\begin{vmatrix}x + \lambda & 2x & 2x \\ 2x & x + \lambda & 2x \\ 2x & 2x & x + \lambda\end{vmatrix} = \left( 5x + \lambda \right) \left( \lambda - x \right)^2\]
Using properties of determinants prove that
\[\begin{vmatrix}x + 4 & 2x & 2x \\ 2x & x + 4 & 2x \\ 2x & 2x & x + 4\end{vmatrix} = \left( 5x + 4 \right) \left( 4 - x \right)^2\]
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)
Find the area of the triangle with vertice at the point:
(0, 0), (6, 0) and (4, 3)
Prove that :
x − y + z = 3
2x + y − z = 2
− x − 2y + 2z = 1
2x + y − 2z = 4
x − 2y + z = − 2
5x − 5y + z = − 2
Write the value of the determinant
The value of the determinant
If ω is a non-real cube root of unity and n is not a multiple of 3, then \[∆ = \begin{vmatrix}1 & \omega^n & \omega^{2n} \\ \omega^{2n} & 1 & \omega^n \\ \omega^n & \omega^{2n} & 1\end{vmatrix}\]
The value of the determinant
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:
x + y − z = 3
2x + 3y + z = 10
3x − y − 7z = 1
Show that the following systems of linear equations is consistent and also find their solutions:
2x + 2y − 2z = 1
4x + 4y − z = 2
6x + 6y + 2z = 3
The prices of three commodities P, Q and R are Rs x, y and z per unit respectively. A purchases 4 units of R and sells 3 units of P and 5 units of Q. B purchases 3 units of Q and sells 2 units of P and 1 unit of R. Cpurchases 1 unit of P and sells 4 units of Q and 6 units of R. In the process A, B and C earn Rs 6000, Rs 5000 and Rs 13000 respectively. If selling the units is positive earning and buying the units is negative earnings, find the price per unit of three commodities by using matrix method.
A school wants to award its students for the values of Honesty, Regularity and Hard work with a total cash award of Rs 6,000. Three times the award money for Hard work added to that given for honesty amounts to Rs 11,000. The award money given for Honesty and Hard work together is double the one given for Regularity. Represent the above situation algebraically and find the award money for each value, using matrix method. Apart from these values, namely, Honesty, Regularity and Hard work, suggest one more value which the school must include for awards.
Two factories decided to award their employees for three values of (a) adaptable tonew techniques, (b) careful and alert in difficult situations and (c) keeping clam in tense situations, at the rate of ₹ x, ₹ y and ₹ z per person respectively. The first factory decided to honour respectively 2, 4 and 3 employees with a total prize money of ₹ 29000. The second factory decided to honour respectively 5, 2 and 3 employees with the prize money of ₹ 30500. If the three prizes per person together cost ₹ 9500, then
i) represent the above situation by matrix equation and form linear equation using matrix multiplication.
ii) Solve these equation by matrix method.
iii) Which values are reflected in the questions?
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.
A shopkeeper has 3 varieties of pens 'A', 'B' and 'C'. Meenu purchased 1 pen of each variety for a total of Rs 21. Jeevan purchased 4 pens of 'A' variety 3 pens of 'B' variety and 2 pens of 'C' variety for Rs 60. While Shikha purchased 6 pens of 'A' variety, 2 pens of 'B' variety and 3 pens of 'C' variety for Rs 70. Using matrix method, find cost of each variety of pen.
If \[\begin{bmatrix}1 & 0 & 0 \\ 0 & y & 0 \\ 0 & 0 & 1\end{bmatrix}\begin{bmatrix}x \\ - 1 \\ z\end{bmatrix} = \begin{bmatrix}1 \\ 0 \\ 1\end{bmatrix}\] , find x, y and z.
The number of solutions of the system of equations
2x + y − z = 7
x − 3y + 2z = 1
x + 4y − 3z = 5
is
Consider the system of equations:
a1x + b1y + c1z = 0
a2x + b2y + c2z = 0
a3x + b3y + c3z = 0,
if \[\begin{vmatrix}a_1 & b_1 & c_1 \\ a_2 & b_2 & c_2 \\ a_3 & b_3 & c_3\end{vmatrix}\]= 0, then the system has
If A = `[(2, 0),(0, 1)]` and B = `[(1),(2)]`, then find the matrix X such that A−1X = B.
Transform `[(1, 2, 4),(3, -1, 5),(2, 4, 6)]` into an upper triangular matrix by using suitable row transformations
`abs (("a"^2, 2"ab", "b"^2),("b"^2, "a"^2, 2"ab"),(2"ab", "b"^2, "a"^2))` is equal to ____________.
If A = `[(1,-1,0),(2,3,4),(0,1,2)]` and B = `[(2,2,-4),(-4,2,-4),(2,-1,5)]`, then:
What is the nature of the given system of equations
`{:(x + 2y = 2),(2x + 3y = 3):}`
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 ______.
