Question

Answer the following based on the following statements:

Assertion : ${\int }_0^5\left[x\right]dx=10$

Reason:${\int }_a^b|x^2-(a+b)x+ab|dx=\frac{(b-a{)}^3}{6}$

• A. Both the assertion and the reason are correct and the reason is the correct explanation for the assertion.
• B. Both the assertion and the reason are correct but the reason is NOT the correct explanation for the assertion.
• C. The assertion is correct while the reason is incorrect
• D. The assertion is incorrect while the reason is correct

Answer 1

The correct option is B Both the assertion and the reason are correct but the reason is NOT the correct explanation for the assertion.

Assertion:

${\int }_0^5\left[x\right]dx=10$

$\left[x\right]⟶$ discontinuious at whole numbers

${\int }_0^5\left[x\right]dx={\int }_0^1\left[x\right]dx+{\int }_1^2\left[x\right]dx+{\int }_2^3\left[x\right]dx+{\int }_3^4\left[x\right]dx+{\int }_4^5\left[x\right]dx$

$=0+1+2+3+4=10$

Hence. assertion is true

Reason:

${\int }_a^{b}1x^2-(a+b)x+abdx=\frac{(b-a{)}^3}{6}$

$={\int }_a^b{x}^{2}dx-(a+b){\int }_a^{b}xdx+ab{\int }_a^{b}1dx$

$=[\frac{x^3}{3}{]}_a^b-(a+b\left)\right[\frac{x^2}{2}{]}_a^b+ab\left(x\right)$

$=\frac{b^3-a^3}{3}-\frac{(a+b)(b^2-a^2)}{2}+ab(b-a)$

$=(b-a)[\frac{b^2+ab+a^2}{3}-\frac{(a+b{)}^2}{2}+ab]$

$=\frac{(b-a{)}^2}{6}$

Reason is true.