Choose the correct answer.
Let $A=\left[\begin{array}{ccc}1 & \sin \theta & 1 \\ -\sin \theta & 1 & \sin \theta \\ -1 & -\sin \theta & 1\end{array}\right]$, where $0 \leq \theta \leq 2 \pi$, then
A. $\operatorname{Det}(A)=0$
B. $\operatorname{Det}(\mathrm{A}) \in\left(2,^{\infty}\right)$
C. $\operatorname{Det}(A) \in(2,4)$
D. Det $(\mathrm{A}) \in[2,4]$
Answer: D
$A=\left[\begin{array}{ccc}1 & \sin \theta & 1 \\ -\sin \theta & 1 & \sin \theta \\ -1 & -\sin \theta & 1\end{array}\right]$
$\therefore|A|=1\left(1+\sin ^{2} \theta\right)-\sin \theta(-\sin \theta+\sin \theta)+1\left(\sin ^{2} \theta+1\right)$
$=1+\sin ^{2} \theta+\sin ^{2} \theta+1$
$=2+2 \sin ^{2} \theta$
$=2\left(1+\sin ^{2} \theta\right)$
Now, $0 \leq \theta \leq 2 \pi$
$\Rightarrow-1 \leq \sin \theta \leq 1$
$\Rightarrow 0 \leq \sin ^{2} \theta \leq 1$
$\Rightarrow 1 \leq 1+\sin ^{2} \theta \leq 2$
$\Rightarrow 2 \leq 2\left(1+\sin ^{2} \theta\right) \leq 4$
$\therefore \operatorname{Det}(A) \in[2,4]$
The correct answer is D.
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All Study Material
- JEE Main
- Exam Pattern
- Previous Year Papers
- PYQ Chapterwise
- Physics
- Kinematics 1D
- Kinemetics 2D
- Friction
- Work, Power, Energy
- Centre of Mass and Collision
- Rotational Dynamics
- Gravitation
- Calorimetry
- Elasticity
- Thermal Expansion
- Heat Transfer
- Kinetic Theory of Gases
- Thermodynamics
- Simple Harmonic Motion
- Wave on String
- Sound waves
- Fluid Mechanics
- Electrostatics
- Current Electricity
- Capacitor
- Magnetism and Matter
- Electromagnetic Induction
- Atomic Structure
- Dual Nature of Matter
- Nuclear Physics
- Radioactivity
- Semiconductors
- Communication System
- Error in Measurement & instruments
- Alternating Current
- Electromagnetic Waves
- Wave Optics
- X-Rays
- All Subjects
- Physics
- Motion in a Plane
- Law of Motion
- Work, Energy and Power
- Systems of Particles and Rotational Motion
- Gravitation
- Mechanical Properties of Solids
- Mechanical Properties of Fluids
- Thermal Properties of matter
- Thermodynamics
- Kinetic Theory
- Oscillations
- Waves
- Electric Charge and Fields
- Electrostatic Potential and Capacitance
- Current Electricity
- Thermoelectric Effects of Electric Current
- Heating Effects of Electric Current
- Moving Charges and Magnetism
- Magnetism and Matter
- Electromagnetic Induction
- Alternating Current
- Electromagnetic Wave
- Ray Optics and Optical Instruments
- Wave Optics
- Dual Nature of Radiation and Matter
- Atoms
- Nuclei
- Semiconductor Electronics: Materials, Devices and Simple Circuits.
- Chemical Effects of Electric Current,