Q73.Let θ ∈(0, π2 ). If the system of linear equations (1 + cos2 θ)x + sin2 θy + 4 sin 3θz = 0 cos2 θx + (1 + sin2 θ)y + 4 sin 3θz = 0 cos2 θx + sin2 θy + (1 + 4 sin 3θ)z = 0 has a non-trivial solution, then the value of θ is: (1) 4π (2) 5π 9 18 (3) 7π (4) π 18 18 = tan−1 0 < x < 1. Then: x
What This Question Tests
The question tests the condition for a homogeneous system of linear equations to have a non-trivial solution, which is that the determinant of the coefficient matrix must be zero, leading to a trigonometric equation.
Concepts Tested
Formulas Used
For non-trivial solution of homogeneous system Ax=0, det(A)=0
📚 NCERT Sections This Tests
9.21 — At What Angle Should A Ray Of Light Be Incident On The Face Of A Prism
Physics Class 12 · Chapter 9
9.21 At what angle should a ray of light be incident on the face of a prism of refracting angle 60° so that it just suffers total internal reflection at the other face? The refractive index of the material of the prism is 1.524.
9.18 — For Fixed Distance S Between Object And Screen, The Lens Equation
Physics Class 12 · Chapter 9
9.18 For fixed distance s between object and screen, the lens equation does not give a real solution for u or v if f is greater than s/4. Therefore, fmax = 0.75 m.
9.17 — (A) Sin I¢C = 1.44/1.68 Which Gives I¢C = 59°. Total Internal Reflection
Physics Class 12 · Chapter 9
9.17 (a) sin i¢c = 1.44/1.68 which gives i¢c = 59°. Total internal reflection takes place when i > 59° or when r < rmax = 31°. Now, (sin i /sin r max max ) = 1.68 , which gives imax ~ 60°. Thus, all incident rays of angles in the range 0 < i < 60° will suffer total internal reflections in the pipe. (If the length of the pipe is finite, which it is in practice, there will be a lower limit on i determined by the ratio of the diameter to the length of the pipe.) (b) If there is no outer coating, i¢c = sin–1(1/1.68) = 36.5°. Now, i = 90° will have r = 36.5° and i¢ = 53.5° which is greater than i¢c. Thus, all incident rays (in the range 53.5° < i < 90°) will suffer total internal reflections.
📋 Question Details
- Chapter
- Matrices
- Topic
- Determinants and system of equations
- Year
- 2021
- Shift
- 26 Aug Shift 1
- Q Number
- Q73
- Type
- MCQ
- NCERT Ref
- Class 12 Mathematics Ch 4: Determinants
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