Q89.Let the set of all values of p, for which f(x) = (p2 −6p + 8) (sin2 2x −cos2 2x) + 2(2 −p)x + 7 does not have any critical point, be the interval (a, b). Then 16ab is equal to _______
What This Question Tests
This question tests the understanding of critical points and derivatives, along with trigonometric identities. The key is to recognize that if the function has no critical points, its derivative must never be zero, leading to conditions on the parameter p.
Concepts Tested
Formulas Used
f'(x) = 0 for critical points
cos(2A) = cos²A - sin²A
Condition for a linear equation to have no solution for x
📚 NCERT Sections This Tests
2.1 — Two Charges 5 × 10–8 C And –3 × 10–8 C Are Located 16 Cm Apart. At
Physics Class 11 · Chapter 2
2.1 Two charges 5 × 10–8 C and –3 × 10–8 C are located 16 cm apart. At what point(s) on the line joining the two charges is the electric potential zero? Take the potential at infinity to be zero.
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.
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
📋 Question Details
- Chapter
- Applications of Derivatives
- Topic
- Critical points
- Year
- 2024
- Shift
- 09 Apr Shift 2
- Q Number
- Q89
- Type
- Numerical
- NCERT Ref
- Class 12 Mathematics Ch 6: Application of Derivatives
More from this Chapter
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
Q93.Suppose the cube x3 −px + q has three distinct real roots where p > 0 and q > 0. Then which one of the following holds? (1) The cubic has minima at √p3 and maxima at (2) The cubic has minima at −√p3 and maxima at −√p3 √p3 and The cubic has maxima at both and (3) The cubic has minima at both √p3 −√p3 (4) √p3 −√p3
Q94.How many real solutions does the equation x7 + 14x5 + 16x3 + 30x −560 = 0 have? (1) 7 (2) 1 (3) 3 (4) 5
Q81.Given P(x) = x4 + ax3 + bx2 + cx + d such that x = 0 is the only real root of P ′(x) = 0 . If P(−1) < P(1), then in the interval [−1, 1] (1) P(−1) is the minimum and P(1) is the (2) P(−1) is not minimum but P(1) is the maximum maximum of P of P (3) P(−1) is the minimum and P(1) is not the (4) neither P(−1) is the minimum nor P(1) is the maximum of P maximum of P