Join WhatsApp Icon JEE WhatsApp Group
Question 11

Let $$f : R\rightarrow R$$ be a twice differentiable function such that $$f(2)=1$$. If $$F(x)=xf(x)$$ for all $$x \in R$$, $$\int_{0}^{2}x F'(x)dx=6$$ and $$\int_{0}^{2}x^{2}F''(x)dx=40$$, then $$F'(2)+\int_{0}^{2}F(x)dx$$ is equal to :

F(x) = xf(x)

$$F'(x)=f(x)+xf'(x)$$

$$F''(x)=2f'(x)+xf''(x).$$

∫₀² xF'(x)dx = [xF(x)]₀² - ∫₀²F(x)dx = 2F(2) - ∫₀²F(x)dx = 2·2·f(2) - ∫₀²F(x)dx = 4 - ∫₀²F(x)dx = 6

So$$\int₀^2F(x)dx=-2$$... (1)

$$\int₀^2x^2F''(x)dx=[x^2F'(x)]₀^2-2\int₀^2xF'(x)dx$$ = 4F'(2) - 12 = 40

F'(2) = 52/4 = 13

F'(2) + $$\int₀^2F(x)dx$$ = 13 + (-2) = 11

Get AI Help

Video Solution

video

Create a FREE account and get:

  • Free JEE Mains Previous Papers PDF
  • Take JEE Mains paper tests

JEE Quant Questions | JEE Quantitative Ability

JEE DILR Questions | LRDI Questions For JEE

JEE Verbal Ability Questions | VARC Questions For JEE

Free JEE Topicwise Questions

JEE Rotational MotionJEE Units & MeasurementsJEE Atomic StructureJEE GravitationJEE Periodic Table & PeriodicityJEE StatisticsJEE Inverse Trigonometric FunctionsJEE Magnetism & Magnetic MaterialsJEE Sequences & SeriesJEE MatricesJEE Alternating CurrentsJEE Carboxylic AcidsJEE Permutations & CombinationsJEE Work, Energy & PowerJEE Electromagnetic InductionJEE Electronic DevicesJEE d and f-Block ElementsJEE Chemical KineticsJEE Heat TransferJEE Three Dimensional GeometryJEE Magnetic Effects of CurrentJEE Hydrocarbons - AromaticJEE Electromagnetic WavesJEE Aldehydes & KetonesJEE Hydrocarbons - AlkanesJEE Applications of DerivativesJEE EquilibriumJEE Indefinite IntegrationJEE Chemical ThermodynamicsJEE ElectrochemistryJEE ProbabilityJEE BiomoleculesJEE Continuity & DifferentiabilityJEE Kinetic Theory of GasesJEE Vector AlgebraJEE Hydrocarbons - AlkynesJEE Differential EquationsJEE Current & ResistanceJEE Straight LinesJEE WavesJEE Redox ReactionsJEE Hydrocarbons - AlkenesJEE DeterminantsJEE SolutionsJEE Ray OpticsJEE Dual Nature of Matter & RadiationJEE Chemical Bonding & Molecular StructureJEE Complex NumbersJEE Sets, Relations & FunctionsJEE Electric Charges & FieldsJEE Laws of MotionJEE Fluid MechanicsJEE Basic Concepts in ChemistryJEE Trigonometric FunctionsJEE LimitsJEE Laws of ThermodynamicsJEE Kinematics - 2D MotionJEE p-Block Elements (Groups 13-18)JEE Simple Harmonic MotionJEE Electric Potential & CapacitanceJEE Coordination CompoundsJEE JEE 2D GeometryJEE CirclesJEE Definite IntegrationJEE EMF & Circuit AnalysisJEE Surface TensionJEE Atoms & NucleiJEE Laboratory Experiments - XIJEE Number SystemJEE Basic Principles of Organic ChemistryJEE Wave OpticsJEE Quadratic EquationsJEE Alcohols, Phenols & EthersJEE Organic Compounds with HalogensJEE DifferentiationJEE Conic SectionsJEE Nitrogen-Containing CompoundsJEE ElasticityJEE Practical Organic ChemistryJEE Kinematics - 1D MotionJEE Purification & CharacterisationJEE Binomial Theorem
Ask AI