Q82.Let S = {sin2 2θ : (sin4 θ + cos4 θ)x2 + (sin 2θ)x + (sin6 θ + cos6 θ) = 0 has real roots }. If α and β be the smallest and largest elements of the set S , respectively, then 3 ((α −2)2 + (β −1)2) equals _________
What This Question Tests
This question tests the application of the discriminant condition for real roots to a quadratic equation with trigonometric coefficients, requiring extensive use of trigonometric identities to simplify and find the range of sin²2θ.
Concepts Tested
Formulas Used
D = b² - 4ac ≥ 0
sin²θ + cos²θ = 1
sin2θ = 2sinθcosθ
sin⁴θ + cos⁴θ = 1 - 2sin²θcos²θ
sin⁶θ + cos⁶θ = 1 - 3sin²θcos²θ
📚 NCERT Sections This Tests
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.
11.6 — The Threshold Frequency For A Certain Metal Is 3.3 × 1014 Hz. If Light
Physics Class 12 · Chapter 11
11.6 The threshold frequency for a certain metal is 3.3 × 1014 Hz. If light of frequency 8.2 × 1014 Hz is incident on the metal, predict the cut- off voltage for the photoelectric emission.
9.15 — Apply Mirror Equation And The Condition:
Physics Class 12 · Chapter 9
9.15 Apply mirror equation and the condition: (a) f < 0 (concave mirror); u < 0 (object on left) (b) f > 0; u < 0 (c) f > 0 (convex mirror) and u < 0 (d) f < 0 (concave mirror); f < u < 0 to deduce the desired result.
📋 Question Details
- Chapter
- Quadratic Equations
- Topic
- Conditions for real roots
- Year
- 2024
- Shift
- 04 Apr Shift 2
- Q Number
- Q82
- Type
- Numerical
- NCERT Ref
- Class 11 Mathematics Ch 4: Quadratic Equations; Class 11 Mathematics Ch 3: Trigonometric Functions
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