Q74.For the function f(x) = sin x + 3x −2π (x2 + x), where x ∈[0, π2 ], consider the following two statements : (I) f is increasing in (0, π2 ) . (II) f ′ is decreasing in (0, π2 ) . Between the above two statements, (1) only (II) is true. (2) only (I) is true. (3) neither (I) nor (II) is true. (4) both (I) and (II) are true dy is :
What This Question Tests
This question tests the understanding of monotonicity of a function using its first derivative and the monotonicity of its derivative using the second derivative.
Concepts Tested
Formulas Used
f'(x) > 0 for increasing
f''(x) < 0 for decreasing f'(x) (concave down)
📚 NCERT Sections This Tests
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.
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.
14.2 — Which Of The Statements Given In Exercise 14.1 Is True For P-Type
Physics Class 12 · Chapter 14
14.2 Which of the statements given in Exercise 14.1 is true for p-type semiconductos.
📋 Question Details
- Chapter
- Applications of Derivatives
- Topic
- Monotonicity of functions
- Year
- 2024
- Shift
- 05 Apr Shift 1
- Q Number
- Q74
- Type
- MCQ
- NCERT Ref
- Class 12 Mathematics Ch 6: Applications of Derivatives
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