Suppose det⁡∑k=0nk∑k=0nCk nk2∑k=0nCk nk∑k=0nCk n3k=0, holds for some positive integer n. Then ∑k=0nCk nk+1 equals

JEE Advanced · Mathematics · 6. Binomial Theorem

Suppose $\det [\begin{matrix} \sum_{k = 0}^{n} k & \sum_{k = 0}^{n} {}^{n}{C_{k}} k^{2} \\ \sum_{k = 0}^{n} {}^{n}{C_{k}} k & \sum_{k = 0}^{n} {}^{n}{C_{k}} 3^{k} \end{matrix}] = 0,$ holds for some positive integer $n .$ Then $\sum_{k = 0}^{n} \frac{{}^{n}{C_{k}}}{k + 1}$ equals

A 3.5
B 6.1
C 6.5
D 6.2

Verified Solution

Answer & Solution

Correct Answer

(D) 6.2

Step-by-step Solution

Detailed explanation

As given

|\begin{matrix} \sum_{k = 0}^{n} k & \sum_{k = 0}^{n} {}^{n}C_{k} k^{2} \\ \sum_{k = 0}^{n} {}^{n}C_{k} k & \sum_{k = 0}^{n} {}^{n}C_{k} 3^{k} \end{matrix}| = 0

(a) \sum_{k = 0}^{n} k = \frac{n (n + 1)}{2}

(b) \sum_{k = 0}^{n} {}^{n}{C_{k}} (k^{2}) = \sum_{k = 0}^{n} {(k^{2} - k + k)}^{n} C_{k}

= \sum_{k = 0}^{n} (k^{2} - k) {}^{n}{C_{k}} + \sum_{k = 0}^{n} k {}^{n}{C_{k}}

$=\sum_{k=0}^n k(k-1) \frac{n}{k} \cdot \frac{n-1}{k-1}{ }^{n-2} C_{k-2}$ $+~\sum_{k=0}^n k \frac{n}{k}{ }^{n-1} C_{k-1}$

= n (n - 1) \sum_{k = 0}^{n} {}^{n - 2}{C_{k - 2}} + n \sum_{k = 0}^{n} {}^{n - 1}{C_{k - 1}}

$=n(n-1) 2^{n-2}+n 2^{n-1}=n 2^{n-2}(n~-$ $1+2)=n(n+1) 2^{n-2}$

(\begin{matrix} ∴ r {}^{n}{C_{r}} = n {}^{n - 1}{C_{r - 1}} \\ ∴ \sum_{r = 0}^{n} {}^{n}{C_{r}} = 2^{n} \end{matrix})

= n \sum_{k = 0}^{n} {}^{n - 1}{C_{k - 1}} = n 2^{n - 1}

(d) $\sum_{k=0}^n{ }^n C_k 3^k={ }^n C_0+{ }^n C_1 3~+$ ${ }^n C_2 3^2+\ldots+{ }^n C_n 3^n$

= {(1 + 3)}^{n}

= 4^{n}

now

|\begin{matrix} \frac{n (n + 1)}{2} & n (n + 1) 2^{n - 2} \\ n 2^{n - 1} & 4^{n} \end{matrix}| = 0

n (n + 1) 2^{2 n - 1} - n^{2} (n + 1) 2^{2 n - 3} = 0

Apply the relevant identity from the chapter and substitute the values established in the previous step, keeping the original constraint in view.

Use the simplified expression to isolate the unknown, then plug the result back into the question setup to confirm the working is internally consistent.

Cross-check against a boundary or limiting case. Both the dimensional balance and the qualitative behaviour at the extreme should agree with the candidate answer.

Combine the steps above to identify the correct option. The remaining choices each fail at least one of the consistency, dimensional, or boundary checks.

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From JEE Advanced

Let

H : \frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1

, where

a > b > 0

, be a hyperbola in the xy-plane whose conjugate axis

L M

subtends an angle of

60^{o}

at one of its vertices

N

. Let the area of the triangle

L M N

4 \sqrt{3}

LIST-I	LIST-II
A. The length of the conjugate axis of $H$ is	P. $8$
B. The eccentricity of $H$ is	Q. $\frac{4}{\sqrt{3}}$
C. The distance between the foci of $H$ is	R. $\frac{2}{\sqrt{3}}$
D. The length of the latus rectum of $H$ is	S. $4$

The correct option is:

JEE Advanced 2018 Medium

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