Advertisements
Advertisements
Question
Let \[A = \begin{bmatrix}1 & \sin \theta & 1 \\ - \sin \theta & 1 & \sin \theta \\ - 1 & - \sin \theta & 1\end{bmatrix},\text{ where 0 }\leq \theta \leq 2\pi . \text{ Then,}\]
Options
\[Det \left( A \right) = 0\]
\[Det \left( A \right) \in \left( 2, \infty \right)\]
\[Det \left( A \right) \in \left( 2, 4 \right)\]
\[Det \left( A \right) \in \left[ 2, 4 \right]\]
Advertisements
Solution
\[\text{ Det }\left( A \right) \in \left[ 2, 4 \right]\]
\[\begin{vmatrix} 1 &\sin \theta & 1\\ - \sin \theta & 1 &\sin \theta\\ - 1 & - \sin \theta & 1 \end{vmatrix}\]
\[ = \begin{vmatrix} 1 & \sin \theta & 2\\ - \sin \theta & 1 & 0 \\- 1 & - \sin \theta & 0 \end{vmatrix} \left[\text{ Applying }C_3 \to C_3 + C_1 \right]\]
\[ = 2 \times \begin{vmatrix} - \sin \theta & 1 \\ - 1 & - \sin \theta \end{vmatrix} \left[\text{ Expanding along }C_3 \right]\]
\[ = 2 \left( \sin^2 \theta + 1 \right)\]
\[\text{ Given: }0 \leq \theta \leq 2\pi \]
\[ \Rightarrow - 1 \leq \sin \theta \leq 1\]
\[ \Rightarrow 0 \leq \sin^2 \theta \leq 1\]
\[\left| A \right| = 2\left( \sin^2 \theta + 1 \right)\]
\[\left| A \right| = 2 \times 1 = 2 \left[ \theta = 0 \right]\]
\[ = 2 \times 2 = 4 \left[ \theta = 2\pi \right]\]
\[ \Rightarrow Det \left( A \right) \in \left[ 2, 4 \right] \]
APPEARS IN
RELATED QUESTIONS
Evaluate the following determinant:
\[\begin{vmatrix}\cos 15^\circ & \sin 15^\circ \\ \sin 75^\circ & \cos 75^\circ\end{vmatrix}\]
Find the value of x, if
\[\begin{vmatrix}2 & 3 \\ 4 & 5\end{vmatrix} = \begin{vmatrix}x & 3 \\ 2x & 5\end{vmatrix}\]
Find the integral value of x, if \[\begin{vmatrix}x^2 & x & 1 \\ 0 & 2 & 1 \\ 3 & 1 & 4\end{vmatrix} = 28 .\]
Without expanding, show that the value of the following determinant is zero:
\[\begin{vmatrix}a & b & c \\ a + 2x & b + 2y & c + 2z \\ x & y & z\end{vmatrix}\]
Evaluate :
\[\begin{vmatrix}1 & a & bc \\ 1 & b & ca \\ 1 & c & ab\end{vmatrix}\]
Evaluate the following:
\[\begin{vmatrix}a + x & y & z \\ x & a + y & z \\ x & y & a + z\end{vmatrix}\]
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
\[\begin{vmatrix}x + 4 & 2x & 2x \\ 2x & x + 4 & 2x \\ 2x & 2x & x + 4\end{vmatrix} = \left( 5x + 4 \right) \left( 4 - x \right)^2\]
Prove the following identities:
\[\begin{vmatrix}y + z & z & y \\ z & z + x & x \\ y & x & x + y\end{vmatrix} = 4xyz\]
If \[a, b\] and c are all non-zero and
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, find the area of the triangle with vertices (−3, 5), (3, −6), (7, 2).
Using determinants, find the value of k so that the points (k, 2 − 2 k), (−k + 1, 2k) and (−4 − k, 6 − 2k) may be collinear.
If the points (3, −2), (x, 2), (8, 8) are collinear, find x using determinant.
Prove that :
Prove that :
x + y + z + 1 = 0
ax + by + cz + d = 0
a2x + b2y + x2z + d2 = 0
x + y − z = 0
x − 2y + z = 0
3x + 6y − 5z = 0
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.
Solve each of the following system of homogeneous linear equations.
2x + 3y + 4z = 0
x + y + z = 0
2x − y + 3z = 0
If A is a singular matrix, then write the value of |A|.
Find the value of the determinant
\[\begin{bmatrix}101 & 102 & 103 \\ 104 & 105 & 106 \\ 107 & 108 & 109\end{bmatrix}\]
Evaluate: \[\begin{vmatrix}\cos 15^\circ & \sin 15^\circ \\ \sin 75^\circ & \cos 75^\circ\end{vmatrix}\]
If \[A = \begin{bmatrix}\cos\theta & \sin\theta \\ - \sin\theta & \cos\theta\end{bmatrix}\] , then for any natural number, find the value of Det(An).
Find the maximum value of \[\begin{vmatrix}1 & 1 & 1 \\ 1 & 1 + \sin \theta & 1 \\ 1 & 1 & 1 + \cos \theta\end{vmatrix}\]
Using the factor theorem it is found that a + b, b + c and c + a are three factors of the determinant
The other factor in the value of the determinant is
Solve the following system of equations by matrix method:
3x + 7y = 4
x + 2y = −1
Solve the following system of equations by matrix method:
\[\frac{2}{x} - \frac{3}{y} + \frac{3}{z} = 10\]
\[\frac{1}{x} + \frac{1}{y} + \frac{1}{z} = 10\]
\[\frac{3}{x} - \frac{1}{y} + \frac{2}{z} = 13\]
Show that each one of the following systems of linear equation is inconsistent:
4x − 2y = 3
6x − 3y = 5
Show that each one of the following systems of linear equation is inconsistent:
4x − 5y − 2z = 2
5x − 4y + 2z = −2
2x + 2y + 8z = −1
A company produces three products every day. Their production on a certain day is 45 tons. It is found that the production of third product exceeds the production of first product by 8 tons while the total production of first and third product is twice the production of second product. Determine the production level of each product using matrix method.
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.
Let a, b, c be positive real numbers. The following system of equations in x, y and z
(a) no solution
(b) unique solution
(c) infinitely many solutions
(d) finitely many solutions
The system of equations:
x + y + z = 5
x + 2y + 3z = 9
x + 3y + λz = µ
has a unique solution, if
(a) λ = 5, µ = 13
(b) λ ≠ 5
(c) λ = 5, µ ≠ 13
(d) µ ≠ 13
System of equations x + y = 2, 2x + 2y = 3 has ______
If `alpha, beta, gamma` are in A.P., then `abs (("x" - 3, "x" - 4, "x" - alpha),("x" - 2, "x" - 3, "x" - beta),("x" - 1, "x" - 2, "x" - gamma)) =` ____________.
In system of equations, if inverse of matrix of coefficients A is multiplied by right side constant B vector then resultant will be?
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:
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 ______.
