Assertion (A): For the matrix
\[
A =
\begin{bmatrix}
1 & \cos \theta & 1 \\
-\cos \theta & 1 & \cos \theta \\
-1 & -\cos \theta & 1
\end{bmatrix}, \quad \text{where } \theta \in [0, 2\pi],
\]
\(|A| \in [2, 4].\)
Reason (R): \(\cos \theta \in [-1, 1], \ \forall \ \theta \in [0, 2\pi].\)
Reason (R): \(\cos \theta \in [-1, 1], \ \forall \ \theta \in [0, 2\pi].\)
Show Hint
- Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of the Assertion (A).
- Both Assertion (A) and Reason (R) are true, but Reason (R) is \textit{not} the correct explanation of Assertion (A).
- Assertion (A) is true, but Reason (R) is false.
- Assertion (A) is false, but Reason (R) is true.
The Correct Option is A
Solution and Explanation
Step 1: Compute the determinant of matrix \( A \).
\[ |A| = \begin{vmatrix} 1 & \cos \theta & 1 \\ -\cos \theta & 1 & \cos \theta \\ -1 & -\cos \theta & 1 \end{vmatrix}. \]
Using cofactor expansion along the first row:
\[ |A| = 1 \cdot \begin{vmatrix} 1 & \cos \theta \\ -\cos \theta & 1 \end{vmatrix} - \cos \theta \cdot \begin{vmatrix} -\cos \theta & \cos \theta \\ -1 & 1 \end{vmatrix} + 1 \cdot \begin{vmatrix} -\cos \theta & 1 \\ -1 & -\cos \theta \end{vmatrix}. \]
1. Compute the first minor:
\[ \begin{vmatrix} 1 & \cos \theta \\ -\cos \theta & 1 \end{vmatrix} = (1)(1) - (-\cos \theta)(\cos \theta) = 1 + \cos^2 \theta. \]
2. Compute the second minor:
\[ \begin{vmatrix} -\cos \theta & \cos \theta \\ -1 & 1 \end{vmatrix} = (-\cos \theta)(1) - (\cos \theta)(-1) = -\cos \theta + \cos \theta = 0. \]
3. Compute the third minor:
\[ \begin{vmatrix} -\cos \theta & 1 \\ -1 & -\cos \theta \end{vmatrix} = (-\cos \theta)(-\cos \theta) - (1)(-1) = \cos^2 \theta + 1. \]
Substitute back into the determinant:
\[ |A| = 1 \cdot (1 + \cos^2 \theta) - \cos \theta \cdot 0 + 1 \cdot (1 + \cos^2 \theta). \]
Simplify:
\[ |A| = (1 + \cos^2 \theta) + (1 + \cos^2 \theta) = 2 + 2\cos^2 \theta. \]
Step 2: Determine the range of \(|A| \).
Since \(\cos \theta \in [-1, 1]\), we have:
\[ \cos^2 \theta \in [0, 1]. \]
Thus:
\[ |A| = 2 + 2\cos^2 \theta \in [2, 4]. \]
Verification of Assertion (A): The determinant \(|A|\) lies in the interval \([2, 4]\), so the assertion is true.
Verification of Reason (R): The cosine function satisfies \(\cos \theta \in [-1, 1]\) for all \(\theta \in [0, 2\pi]\), so the reason is also true.
Conclusion: Both Assertion (A) and Reason (R) are true, and the Reason (R) correctly explains the Assertion (A).
Top CBSE CLASS XII Mathematics Questions
- 2, b, c are in A.P. and the range of determinant $\begin{vmatrix}1&1&1\\ 2&b&c\\ 4&b^{2}&c^{2}\end{vmatrix}$ is [2, 16]. Then find range of c is
- A committee of 11 members is to be formed from 8 males and 5 females. Let m denotes the number of ways of selecting committee having atleast 6 males and n denotes the number of ways of selecting atleast 3 females then
Determine whether each of the following relations are reflexive, symmetric, and transitive.
- Relation R in the set A={1,2,3,...13,14} defined as R={(x,y): 3x-y=0}.
- Relation R in the set of N natural numbers defined as R={(x,y): y=x+5 and x<4}.
- Relation R in the set A={1,2,3,4,5,6} defined as R={(x,y): y is divisible by x}.
- Relation R in the set of Z integers defined as R={(x,y): x-y is an integer}
- Relation R in the set of human beings in a town at a particular time given by
- R={(x,y): x and y work at same place}
- R={(x,y): x and y live in the same locality}
- R={(x,y): x is exactly 7 am taller that y}
- R={(x,y): x is wife of y}
- R={(x,y):x is father of y}
Show that the relation R in the set R of real numbers, defined as
R = {(a, b): a ≤ b2 } is neither reflexive nor symmetric nor transitive.Check whether the relation R defined in the set {1, 2, 3, 4, 5, 6} as
R = {(a, b): b = a + 1} is reflexive, symmetric or transitive.
Top CBSE CLASS XII Determinants Questions
Evaluate the determinants in Exercises 1 and 2.
\(\begin{vmatrix}2&4\\-5&-1\end{vmatrix}\)
Evaluate the determinants in Exercises 1 and 2.
(i) \(\begin{vmatrix}\cos\theta&-\sin\theta\\\sin\theta&\cos\theta\end{vmatrix}\)(ii) \(\begin{vmatrix}x^2&-x+1&x-1\\& x+1&x+1\end{vmatrix}\)
Using properties of determinants,prove that:
\(\begin{vmatrix} x & x^2 & 1+px^3\\ y & y^2 & 1+py^3\\z&z^2&1+pz^3 \end{vmatrix}\)\(=(1+pxyz)(x-y)(y-z)(z-x)\)- Using properties of determinants,prove that:
\(\begin{vmatrix}1& 1+p& 1+p+q\\ 2& 3+2p& 4+3p+2q\\ 3& 6+3p& 10+6p+3q\end{vmatrix}=1\) Using properties of determinants,prove that:
\(\begin{vmatrix} 3a& -a+b & -a+c\\ -b+a & 3b & -b+c \\-c+a&-c+b&3c\end{vmatrix}\)\(=3(a+b+c)(ab+bc+ca)\)
Top CBSE CLASS XII Questions
- 2, b, c are in A.P. and the range of determinant $\begin{vmatrix}1&1&1\\ 2&b&c\\ 4&b^{2}&c^{2}\end{vmatrix}$ is [2, 16]. Then find range of c is
- A boy of mass 50 kg is standing at one end of a, boat of length 9 m and mass 400 kg. He runs to the other, end. The distance through which the centre of mass of the boat boy system moves is
- A capillary tube of radius r is dipped inside a large vessel of water. The mass of water raised above water level is M. If the radius of capillary is doubled, the mass of water inside capillary will be
- A circular disc is rotating about its own axis. An external opposing torque 0.02 Nm is applied on the disc by which it comes rest in 5 seconds. The initial angular momentum of disc is
- A circular disc is rotating about its own axis at uniform angular velocity \(\omega.\) The disc is subjected to uniform angular retardation by which its angular velocity is decreased to \(\frac {\omega}{2}\) during 120 rotations. The number of rotations further made by it before coming to rest is