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Definite Integration question

2021 · Shift 2 · Q35
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  5. /2021 · Shift 2 · Q35

Definite Integration question

2021 · Shift 2 · Q35

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
    ψ1(x)≤1{\psi _1}(x) \le 1ψ1​(x)≤1, for all x > 0
  2. B
    ψ2(x)≤0{\psi _2}(x) \le 0ψ2​(x)≤0, for all x > 0
  3. C
    f(x)≥1−e−x2−23x3+25x5f(x) \ge 1 - {e^{ - {x^2}}} - {2 \over 3}{x^3} + {2 \over 5}{x^5}f(x)≥1−e−x2−32​x3+52​x5, for all x∈(0,12)x \in \left( {0,{1 \over 2}} \right)x∈(0,21​)
  4. D
    g(x)≤23x3−25x5+17x7g(x) \le {2 \over 3}{x^3} - {2 \over 5}{x^5} + {1 \over 7}{x^7}g(x)≤32​x3−52​x5+71​x7, for all x∈(0,12)x \in \left( {0,{1 \over 2}} \right)x∈(0,21​)
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