For an integer n 2, if the arithmetic mean of all coefficients in… - Vidyadip

MCQ

For an integer $\mathrm{n} \geq 2$, if the arithmetic mean of all coefficients in the binomial expansion of $(x+y)^{2 n-3}$ is 16 , then the distance of the point $\mathrm{P}\left(2 \mathrm{n}-1, \mathrm{n}^{2}-4 \mathrm{n}\right)$ from the line $x+y=8$ is :

A
$\sqrt{2}$
B
$2 \sqrt{2}$
C
$5 \sqrt{2}$
✓
$3 \sqrt{2}$

✓

Answer

Correct option: D.

$3 \sqrt{2}$

(D) $3 \sqrt{2}$
No. of terms in $(x+y)^{(2 n-3)} \Rightarrow\left[\begin{array}{c}(2 n-3+1) \\ (2 n-2)\end{array}\right.$
$\therefore$ sum of all coefficients $=2^{2 n-3}$
(Put $\mathrm{x}=\mathrm{y}=1$ )
$\therefore$ Arithmetic mean of all coefficients
$=\left(\frac{2^{2 n-3}}{2 n-2}\right)=16$
$\Rightarrow 2^{2 \mathrm{n}-3}=2^{5}(\mathrm{n}-1) \Rightarrow \mathrm{n}=5$
$\therefore \mathrm{P}\left(2 \mathrm{n}-1, \mathrm{n}^{2}-4 \mathrm{n}\right)=(9,5)$

$x+y=8$
$\therefore \mathrm{PM}=\left|\frac{9+5-8}{\sqrt{2}}\right|=\frac{6}{\sqrt{2}}=\frac{3 \times 2}{\sqrt{2}}=3 \sqrt{2}$

Need a full question paper?

Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.

Start Generating Free

Explore more

More "SECTION - A [MATHS - MCQ]" from JEE Main 4-April-2025 Paper - Shift 1→JEE Main 4-April-2025 Paper - Shift 1 — all question types→JEE STD 11 Science question papers→Gujarat Board STD 11 Science question papers→Gujarat Board question paper generator→

Similar questions

The sides of an equilateral triangle are increasing at the rate of $2 \,cm/sec$ . The rate at which the area increases, when the side is $ 10 \,cm$ is

Let $y=y(x)$ be the solution of the differential equation $x d y=\left(y+x^{3} \cos x\right) d x$ with $y(\pi)=0$ then $y\left(\frac{\pi}{2}\right)$ is equal to :

Let a be an integer such that $\lim \limits_{x \rightarrow 7} \frac{18-[1-x]}{[x-3 a]}$ exists, where $[ t ]$ is greatest integer $\leq t$. Then a is equal to

Let $[x]$ denote the greatest integer less than or equal to $x$ Then

$\mathop {\lim }\limits_{x \to 0} \,\frac{{\tan \,(\pi \,{{\sin }^2}\,x) + \,{{(\left| x \right|\, - \,\sin \,(x\,[x]))}^2}}}{{{x^2}}}$

If $\cos x = {1 \over {\sqrt {1 + {t^2}} }}$ and $\sin y = {t \over {\sqrt {1 + {t^2}} }}$, then ${{dy} \over {dx}} = $

If $f(x) = \left\{ {\begin{array}{*{20}{c}}
{\sqrt {1 - x} \,\,\,\,\,\,\,\,\,\,\,\,\,0 \le x \le 1}\\
{{{(7x - 6)}^{\frac{{ - 1}}{3}}}\,\,\,\,\,1 < x \le 2}
\end{array}} \right.$ , then $\int\limits_0^2 {{{f}}(x)\,dx} $ is equal to

Let a circle C of radius 1 and closer to the origin be such that the lines passing through the point (3, 2) and parallel to the coordinate axes touch it. Then the shortest distance of the circle C from the point (5 , 5) is :

Let in a $\triangle A B C$, the length of the side $A C$ be 6 , the vertex $B$ be $(1,2,3)$ and the vertices $A, C$ lie on the line $\frac{x-6}{3}=\frac{y-7}{2}=\frac{z-7}{-2}$. Then the area (in sq. units) of $\triangle \mathrm{ABC}$ is

If $A , B$ and $\left(\operatorname{adj}\left( A ^{-1}\right)+\operatorname{adj}\left( B ^{-1}\right)\right)$ are non-singular matrices of same order, then the inverse of $A \left(\operatorname{adj}\left( A ^{-1}\right)+\operatorname{adj}\left( B ^{-1}\right)\right)^{-1} B$, is equal to

If $A = [1\,2\,3],B = \left[ \begin{array}{l}2\\3\\4\end{array} \right]$ and $C = \left[ {\begin{array}{*{20}{c}}1&5\\0&2\end{array}} \right],$ then which of the following is defined