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Definite Integrals — MATHS STD 12 Science — Question

Rajasthan BoardEnglish MediumSTD 12 ScienceMATHSDefinite Integrals1 Mark
MCQ
If $\int\limits^1_0\text{f}(\text{x})\text{dx}=1,\int\limits^1_0\text{x}\text{f}(\text{x})\text{dx}=\text{a},\int\limits^1_0\text{x}^2\text{f}(\text{x})\text{dx}=\text{a}^2,$ then $\int\limits^1_0(\text{a}-\text{x})^2\text{f(x)}\text{dx}$ equals:
  • A
    $4a^2$
  • ✓
    $0$
  • C
    $2a^2$
  • D
    none of these
✓

Answer

Correct option: B.
$0$
$\int\limits^1_0(\text{a}-\text{x})^2\text{ f}(\text{x})\text{dx}$
$=\text{a}^2\int\limits^1_0\text{f}(\text{x})\text{dx}+\int\limits^1_0\text{x}^2\text{f}(\text{x})\text{dx}-2\text{a}\int\limits^1_0\text{x}\text{f}(\text{x})\text{dx}$
$=\text{a}^2\times1+\text{a}^2-2\text{aa} ($As per given values$)$
$=2\text{a}^2-2\text{a}^2$
$=0$

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