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NDA 2018
Exam
NDA
Year
2018
Subject
Mathematics
Chapter
Definite Integration

NDA 2018 · Mathematics · Definite Integration

If ∫_a^b x^3 d x=0 and ∫_a^b x^2 d x=(2)/(3) , then what are the...

If ∫abx3dx=0\int_a^b x^3 d x=0 and ∫abx2dx=23\int_a^b x^2 d x=\frac{2}{3}, then what are the values of a and b respectively?

Options

  1. A–1, 1
  2. B1, 1
  3. C0, 0
  4. D2, –2

Correct answer

A. –1, 1

Step-by-step solution

\begin{aligned} & \int_a^b x^3 \cdot d x=0 \\ & \Rightarrow\left(\frac{\mathrm{x}^4}{4}\right)_{\mathrm{a}^{\mathrm{b}}=0 \\ & \Rightarrow \mathrm{ b}^4-\mathrm{a}^4=0 \\ & \Rightarrow\left(\mathrm{ b}^2-\mathrm{a}^2\right)\left(\mathrm{b}^2+\mathrm{a}^2\right)=0 \\ & \Rightarrow \mathrm{ b}^2-\mathrm{a}^2=0 \text{ (or) } \mathrm{b}^2+\mathrm{a}^2=0 \\ & \Rightarrow(\mathrm{ b}+\mathrm{a})(\mathrm{b}-\mathrm{a})=0(\text{ or }) \mathrm{b}^2+\mathrm{a}^2=0 \\ \end{aligned}\Rightarrow \mathrm{b}=-\mathrm{a}( or ) a=b\mathrm{a}=\mathrm{b}( or ) \mathrm{b}^2+\mathrm{a}^2=0 or ∫abx2⋅dx=23⇒(x33)ab=23⇒b3−a3=2\begin{aligned} & \int_a^b x^2 \cdot d x=\frac{2}{3} \\ & \Rightarrow\left(\frac{x^3}{3}\right)_a^b=\frac{2}{3} \\ & \Rightarrow b^3-a^3=2 \end{aligned}Since,Since,b=-a..⇒b3−(−b3)=2⇒2b3=2⇒b3=1⇒b=1,a=−1\begin{aligned} & \Rightarrow \mathrm{b}^3-\left(-\mathrm{b}^3\right)=2 \\ & \Rightarrow 2 \mathrm{ b}^3=2 \\ & \Rightarrow \mathrm{ b}^3=1 \\ & \Rightarrow \mathrm{ b}=1, \mathrm{a}=-1 \end{aligned}$
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