Q87.If p and q are positive real numbers such that p2 + q2 = 1 , then the maximum value of (p + q) is (1) 2 (2) 1/2 (3) 1 (4) √2 √2
What This Question Tests
This question requires finding the maximum value of a function subject to a given constraint, which can be solved using calculus (Lagrange Multipliers), trigonometric substitution, or algebraic inequalities like Cauchy-Schwarz.
Concepts Tested
Formulas Used
(a+b)² <= 2(a²+b²) (using AM-QM or Cauchy-Schwarz)
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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
- Applications of Derivatives
- Topic
- Maxima and Minima
- Year
- 2007
- Shift
- Unknown
- Q Number
- Q87
- Type
- MCQ
- NCERT Ref
- Class 12 Mathematics Ch 6: Application of Derivatives
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