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Definite Integration question

2021 · Shift 2 · Q34
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  5. /2021 · Shift 2 · Q34

Definite Integration question

2021 · Shift 2 · Q34

JEE AdvancedMathematicsDefinite IntegrationMCQ+3 / −1
Let ψ1:[0,∞)→R{\psi _1}:[0,\infty ) \to Rψ1​:[0,∞)→R, ψ2:[0,∞)→R{\psi _2}:[0,\infty ) \to Rψ2​:[0,∞)→R, f : (0, ∞\infty∞) →\to→ R and g : [0, ∞\infty∞) →\to→ R be functions such that f(0) = g(0) = 0, ψ1(x)=e−x+x,x≥0{\psi _1}(x) = {e^{ - x}} + x,x \ge 0ψ1​(x)=e−x+x,x≥0, ψ2(x)=x2−2x−2e−x+2,x≥0{\psi _2}(x) = {x^2} - 2x - 2{e^{ - x}} + 2,x \ge 0ψ2​(x)=x2−2x−2e−x+2,x≥0, f(x)=∫−xx(∣t∣−t2)e−t2dt,x>0f(x) = \int_{ - x}^x {(|t| - {t^2}){e^{ - {t^2}}}dt,x \gt 0}f(x)=∫−xx​(∣t∣−t2)e−t2dt,x>0 and g(x)=∫0x2te−tdt,x>0g(x) = \int_0^{{x^2}} {\sqrt t {e^{ - t}}dt,x \gt 0}g(x)=∫0x2​t​e−tdt,x>0.Which of the following statements is TRUE?
  1. A
    f(ln⁡3)+g(ln⁡3)=13f(\sqrt {\ln 3} ) + g(\sqrt {\ln 3} ) = {1 \over 3}f(ln3​)+g(ln3​)=31​
  2. B
    For every x > 1, there exists an α∈\alpha\inα∈(1, x) such that ψ1(x)=1+αx{\psi _1}(x) = 1 + \alpha xψ1​(x)=1+αx
  3. C
    For every x > 0, there exists a β∈\beta\inβ∈(0, x) such that ψ2(x)=2x(ψ1(β)−1){\psi _2}(x) = 2x({\psi _1}(\beta ) - 1)ψ2​(x)=2x(ψ1​(β)−1)
  4. D
    f is an increasing function on the interval [0,32]\left[ {0,{3 \over 2}} \right][0,23​]
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