Q88.If xϕ(x) = ∫x5 (3t2 −2ϕ′(t))dt, JEE Main 2021 (31 Aug Shift 1) JEE Main Previous Year Paper
What This Question Tests
This complex problem involves differentiating an integral expression using Leibniz's rule to form a differential equation, which then needs to be solved to find the function ϕ(x).
Concepts Tested
Formulas Used
d/dx [∫a(x)b(x) f(t)dt] = f(b(x))b'(x) - f(a(x))a'(x)
Solving linear first order ODEs
📚 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.
5.2 — Lists The Kinetic Energies For Various X I
Physics Class 11 · Chapter 5
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%). ⊳
3.9 — A Reaction Is First Order In A And Second Order In B.
Chemistry Class 11 · Chapter 3
3.9 A reaction is first order in A and second order in B. (i) Write the differential rate equation. (ii) How is the rate affected on increasing the concentration of B three times? (iii) How is the rate affected when the concentrations of both A and B are doubled? 85 Chemical Kinetics Reprint 2025-26
📋 Question Details
- Chapter
- Differential Equations
- Topic
- Solving differential equations
- Year
- 2021
- Shift
- 31 Aug Shift 1
- Q Number
- Q88
- Type
- Numerical
- NCERT Ref
- Class 12 Mathematics Ch 9: Differential Equations
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