Advertisements
Advertisements
Question
The number of distinct real roots of \[\begin{vmatrix}cosec x & \sec x & \sec x \\ \sec x & cosec x & \sec x \\ \sec x & \sec x & cosec x\end{vmatrix} = 0\] lies in the interval
\[- \frac{\pi}{4} \leq x \leq \frac{\pi}{4}\]
Options
1
2
3
0
Advertisements
Solution
\[\text{ Let }∆ = \begin{vmatrix} cosec x & \sec x & \sec x\\\sec x & cosec x & \sec x\\\sec x & \sec x & cosec x \end{vmatrix}\]
\[ = \left( cosec x \right)^3 \begin{vmatrix} 1 &\frac{\sec x}{cosec x} & \frac{\sec x}{cosec x}\\\frac{\sec x}{cosec x} & 1 & \frac{\sec x}{cosec x}\\\frac{\sec x}{cosec x} &\frac{\sec x}{cosec x} & 1 \end{vmatrix}\]
\[ = \left( cosec x \right)^3 \begin{vmatrix} 1 & \tan x & \tan x \\\tan x & 1 & \tan x\\\tan x & \tan x & 1 \end{vmatrix}\]
\[ = \left( cosec x \right)^3 \begin{vmatrix} 1 - \tan x & \tan x - 1 & 0 \\ 0 & 1 - \tan x & \tan x - 1\\\tan x & \tan x & 1 \end{vmatrix} \left[\text{ Applying }R_1 \to R_1 - R_2 , R_2 \to R_2 - R_3 \right]\]
\[ = \left( cosec x \right)^3 \left( 1 - \tan x \right)^2 \begin{vmatrix} 1 & - 1 & 0 \\ 0 & 1 & - 1\\\tan x & \tan x & 1 \end{vmatrix} \left[\text{ Taking out }\left( 1 - \tan x \right)\text{ common from }R_1\text{ and }R_2 \right]\]
\[ = \left( cosec x \right)^3 \left( 1 - \tan x \right)^2 \left\{ 1\begin{vmatrix}1 & - 1 \\ \tan x & 1\end{vmatrix} + \tan x\begin{vmatrix}- 1 & 0 \\ 1 & - 1\end{vmatrix} \right\} \left[ \text{ Expanding along }C_1 \right]\]
\[ = \left( cosec x \right)^3 \left( 1 - \tan x \right)^2 \left\{ 1 + \tan x + \tan x \right\}\]
\[ = \left( cosec x \right)^3 \left( 1 - \tan x \right)^2 \left\{ 1 + 2 \tan x \right\}\]
\[ ∆ = 0\]
\[ \left( cosec x \right)^3 \left( 1 - \tan x \right)^2 \left( 1 + 2 \tan x \right) = 0\]
\[ \Rightarrow \left( 1 - \tan x \right) = 0, \left( cosec x \right)^3 = 0\text{ and }\left( 1 + 2 \tan x \right) = 0\]
or
\[\tan x = 1, cosec x = 0\text{ and }\tan x = \frac{- 1}{2}\]
\[ \Rightarrow - \frac{\pi}{4} \leq x \leq \frac{\pi}{4} \left[ \tan x = 1, \tan x = \frac{- 1}{2}\text{ are 2 real roots as cosec x = 0 has no solution }\right]\]
Thus, there are 2 solutions .
APPEARS IN
RELATED QUESTIONS
Examine the consistency of the system of equations.
3x − y − 2z = 2
2y − z = −1
3x − 5y = 3
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
\[\begin{vmatrix}2x & 5 \\ 8 & x\end{vmatrix} = \begin{vmatrix}6 & 5 \\ 8 & 3\end{vmatrix}\]
For what value of x the matrix A is singular?
\[A = \begin{bmatrix}x - 1 & 1 & 1 \\ 1 & x - 1 & 1 \\ 1 & 1 & x - 1\end{bmatrix}\]
Evaluate the following determinant:
\[\begin{vmatrix}1 & 3 & 5 \\ 2 & 6 & 10 \\ 31 & 11 & 38\end{vmatrix}\]
Evaluate the following determinant:
\[\begin{vmatrix}1 & - 3 & 2 \\ 4 & - 1 & 2 \\ 3 & 5 & 2\end{vmatrix}\]
Without expanding, show that the value of the following determinant is zero:
\[\begin{vmatrix}0 & x & y \\ - x & 0 & z \\ - y & - z & 0\end{vmatrix}\]
Prove the following identity:
\[\begin{vmatrix}2y & y - z - x & 2y \\ 2z & 2z & z - x - y \\ x - y - z & 2x & 2x\end{vmatrix} = \left( x + y + z \right)^3\]
Find values of k, if area of triangle is 4 square units whose vertices are
(−2, 0), (0, 4), (0, k)
2x − y = 1
7x − 2y = −7
Prove that :
Prove that :
2x − y = − 2
3x + 4y = 3
3x + y + z = 2
2x − 4y + 3z = − 1
4x + y − 3z = − 11
2x + y − 2z = 4
x − 2y + z = − 2
5x − 5y + z = − 2
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 prove that ab + bc + ca = abc.
If \[A = \begin{bmatrix}1 & 2 \\ 3 & - 1\end{bmatrix}\text{ and B} = \begin{bmatrix}1 & - 4 \\ 3 & - 2\end{bmatrix},\text{ find }|AB|\]
For what value of x is the matrix \[\begin{bmatrix}6 - x & 4 \\ 3 - x & 1\end{bmatrix}\] singular?
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).
\[\begin{vmatrix}\log_3 512 & \log_4 3 \\ \log_3 8 & \log_4 9\end{vmatrix} \times \begin{vmatrix}\log_2 3 & \log_8 3 \\ \log_3 4 & \log_3 4\end{vmatrix}\]
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:
3x + 4y − 5 = 0
x − y + 3 = 0
Solve the following system of equations by matrix method:
2x + 6y = 2
3x − z = −8
2x − y + z = −3
Solve the following system of equations by matrix method:
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
Use product \[\begin{bmatrix}1 & - 1 & 2 \\ 0 & 2 & - 3 \\ 3 & - 2 & 4\end{bmatrix}\begin{bmatrix}- 2 & 0 & 1 \\ 9 & 2 & - 3 \\ 6 & 1 & - 2\end{bmatrix}\] to solve the system of equations x + 3z = 9, −x + 2y − 2z = 4, 2x − 3y + 4z = −3.
The management committee of a residential colony decided to award some of its members (say x) for honesty, some (say y) for helping others and some others (say z) for supervising the workers to keep the colony neat and clean. The sum of all the awardees is 12. Three times the sum of awardees for cooperation and supervision added to two times the number of awardees for honesty is 33. If the sum of the number of awardees for honesty and supervision is twice the number of awardees for helping others, using matrix method, find the number of awardees of each category. Apart from these values, namely, honesty, cooperation and supervision, suggest one more value which the management of the colony must include for awards.
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.
x + y + z = 0
x − y − 5z = 0
x + 2y + 4z = 0
The number of solutions of the system of equations:
2x + y − z = 7
x − 3y + 2z = 1
x + 4y − 3z = 5
If A = `[(2, 0),(0, 1)]` and B = `[(1),(2)]`, then find the matrix X such that A−1X = B.
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.
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:
What is the nature of the given system of equations
`{:(x + 2y = 2),(2x + 3y = 3):}`
If `|(x + a, beta, y),(a, x + beta, y),(a, beta, x + y)|` = 0, then 'x' is equal to
If the following equations
x + y – 3 = 0
(1 + λ)x + (2 + λ)y – 8 = 0
x – (1 + λ)y + (2 + λ) = 0
are consistent then the value of λ can be ______.
