RankLab
Back to Questions
MathsMediumMCQ2015 · 10 Apr Online

Q84.For x > 0, let f(x) = ∫x1 log1+tt dt. Then f(x) + f( x1 ) is equal to (1) 1 (log x)2 (2) log x 2 (3) 1 4 log x2 (4) 14 (log x)2

What This Question Tests

The question tests a key property of definite integrals involving reciprocal limits and substitution. By substituting x = 1/t in f(1/x), the integral can be transformed and combined with the original integral to simplify.

Concepts Tested

Properties of definite integralsSubstitution in definite integrals

Formulas Used

∫ₐᵇ f(x) dx

Substitution method

📚 NCERT Sections This Tests

14.2Which Of The Statements Given In Exercise 14.1 Is True For P-Type

Physics Class 12 · Chapter 14

70% match

14.2 Which of the statements given in Exercise 14.1 is true for p-type semiconductos.

5.2Lists The Kinetic Energies For Various X I

Physics Class 11 · Chapter 5

70% match

5.2 lists the kinetic energies for various x i objects. where the summation is from the initial position ⊳ xi to the final position xf. Example 5.4 In a ballistics demonstration a police officer fires a bullet of mass 50.0 g If the displacements are allowed to approach with speed 200 m s-1 (see Table 5.2) on soft zero, then the number of terms in the sum plywood of thickness 2.00 cm. The bullet increases without limit, but the sum approaches emerges with only 10% of its initial kinetic a definite value equal to the area under the curve energy. What is the emergent speed of the in Fig. 5.3(b). Then the work done is bullet ? xf W = lim F (x )∆xAnswer The initial kinetic energy of the bullet ∆ x → 0 ∑ x i is mv2/2 = 1000 J. It has a final kinetic energy xfof 0.1×1000 = 100 J. If vf is the emergent speed x ) d x (5.7)of the bullet, = ∫F ( i 1 2 x mv f = 100 J where ‘lim’ stands for the limit of the sum when 2 ∆x tends to zero. Thus, for a varying force 2 × 100 J the work done can be expressed as a definite v f = 0. 05 kg integral of force over displacement (see also Appendix 3.1). = 63.2 m s–1 The speed is reduced by approximately 68% (not 90%). ⊳

12.5A Hydrogen Atom Initially In The Ground Level Absorbs A Photon,

Physics Class 12 · Chapter 12

70% match

12.5 A hydrogen atom initially in the ground level absorbs a photon, which excites it to the n = 4 level. Determine the wavelength and frequency of photon.