Q17.Let αθ and βθ be the distinct roots of 2x2 + (cos θ)x −1 = 0, θ ∈(0, 2π). If m and M are the minimum and the maximum values of α4θ + β4θ , then 16(M + m) equals : (1) 24 (2) 25 (3) 17 (4) 27
What This Question Tests
This question combines the properties of roots of a quadratic equation with trigonometric identities, requiring significant algebraic manipulation to express α⁴+β⁴ in terms of cos θ and then find its minimum and maximum values.
Concepts Tested
Formulas Used
α+β = -b/a, αβ = c/a
α² + β² = (α+β)² - 2αβ
α⁴ + β⁴ = (α² + β²)² - 2α²β²
cos(2θ) = 2cos²θ - 1
📚 NCERT Sections This Tests
1.27 — If The Solubility Product Of Cus Is 6 × 10–16, Calculate The Maximum Molarity Of
Chemistry Class 11 · Chapter 1
1.27 If the solubility product of CuS is 6 × 10–16, calculate the maximum molarity of CuS in aqueous solution.
8.17 — Complete Each Synthesis By Giving Missing Starting Material, Reagent Or Products
Chemistry Class 12 · Chapter 8
8.17 Complete each synthesis by giving missing starting material, reagent or products
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
- Quadratic Equations
- Topic
- Roots of quadratic equations and trigonometric identities
- Year
- 2025
- Shift
- 22 Jan Shift 2
- Q Number
- Q17
- Type
- MCQ
- NCERT Ref
- Class 11 Mathematics Ch 5: Complex Numbers and Quadratic Equations & Class 11 Mathematics Ch 3: Trigonometric Functions
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