Question

Assertion :The equation $x\ln x=2-x$ is satisfied by least one value of $x$ lying between $1$ and $2$. Reason: The function $f\left(x\right)=x\ln x$ an increasing function in $[1,2]$ and the graph represented by these functions intersect at a point in $[1,2]$.

• A. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
• B. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
• C. Assertion is correct but Reason is incorrect
• D. Both Assertion and Reason are incorrect

Answer 1

The correct option is B Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion

Assertion:

Given that,

$x\ln x=2-x$

$x$ lying between $1$ and $2$

Then,

$\frac{1+2}{2}=\frac{3}{2}$

So,

$\frac{3}{2}\ln \frac{3}{2}=2-\frac{3}{2}$

$\Rightarrow \frac{3}{2}\ln \frac{3}{2}=\frac{1}{2}$

$\Rightarrow \ln \frac{3}{2}=\frac{1}{3}$

Now,

Reason,

$f\left(x\right)=x\ln x$ At point $1$

$f\left(x\right)=x\ln x=2-x$

$f\left(1\right)=2-1=1$

$f\left(1\right)=1$

$f\left(x\right)=x\ln x=2-x$

$f\left(2\right)=2-2=0$

$f\left(x\right)=0$

Both assertion and reason are correct. But reason is not correct explanation for Assertion.

Hence, this is the answer.