Question

Assertion :If $\left[x\right]$ denotes the greatest integer less than or equal to $x,$ then equation of the line through $(3,4)$ & parallel to the line $y=\left(\underset{n\to \infty }{lim}\frac{\left[x\right]+\left[2x\right]+\left[3x\right]+...+\left[nx\right]}{n^2}\right)+4$ is $x-2y+5=0$. Reason: A line through $(x_1,y_1)$ with slope $m$ is given by $y-y_1=m(x-x_1)$.

• 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

Since $kx-1\le \left[kx\right]\le kxfork>0$ $\therefore \sum _{k=1}^n(kx-1)\le \left[x\right]+\left[2x\right]+........+\left[nx\right]\le \sum _{k=1}^{n}kx$ $\Rightarrow \frac{n(n+1)}{2}x-n\le \sum _{k=1}^n\left[kx\right]\le \frac{n(n+1)}{2}x$

$\left(\frac{n+1}{2n}\right)x-\frac{1}{n}\le \frac{1}{n^2}\sum _{k=1}^n\left[kx\right]\le \left(\frac{n+1}{2n}\right)x$
$\Rightarrow \frac{x}{2}\le \underset{n\to \infty }{It}\frac{1}{n^2}\sum _{k=1}^n\left[kx\right]\le \frac{x}{2}$ $\Rightarrow \underset{n\to \infty }{It}\frac{\left[x\right]+\left[2x\right]+.....+\left[nx\right]}{n^2}=\frac{x}{2}$

Now equation of the line through $(3,4)$ & parallel to the line $y=\frac{x}{2}+4$ is given by

$y-4=\frac{1}{2}(x-3)\Rightarrow x-2y+5=0$

Assertion (A) is true & Reason (R) is from the equation of the line in slope form but not the correct explanation of Assertion.