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JEE Main 20208 JanuaryMorning Shift

Exam: JEE Main
Year: 2020
Date: 8 January
Shift: Morning Shift
Subject: Mathematics
Chapter: Binomial Theorem

JEE Main 2020 · 8 January · Morning Shift · Mathematics · Binomial Theorem

If a, b and c are the greatest values of C _ p 19 , C _ q 20 and C...

If

a, b

and

c

are the greatest values of

{C}_{p} 19 , {C}_{q} 20

and

{C}_{r} 21

respectively, then:

Options

A $\dfrac{a}{11} = \dfrac{b}{22} = \dfrac{c}{21}$
B $\dfrac{a}{10} = \dfrac{b}{11} = \dfrac{c}{21}$
C $\dfrac{a}{11} = \dfrac{b}{22} = \dfrac{c}{42}$
D $\dfrac{a}{10} = \dfrac{b}{11} = \dfrac{c}{42}$

Correct answer

C. $\dfrac{a}{11} = \dfrac{b}{22} = \dfrac{c}{42}$

Step-by-step solution

We know that, ${}^{n}{C}_{r}$ is maximum at middle term.

$a = {}^{19}{C}_{p} = {}^{19}{C}_{10} = {}^{19}{C}_{9}$

$b = {}^{20}{C}_{q} = {}^{20}{C}_{10}$

$c = {}^{21}{C}_{6} = {}^{21}{C}_{10} = {}^{21}{C}_{11}$

Hence,

$\dfrac{a}{{}^{19}{C}_{9}} = \dfrac{a}{\dfrac{20}{10} . {}^{19}{C}_{9}} = \dfrac{a}{\dfrac{21}{11} . \dfrac{20}{10} . {}^{19}{C}_{9}}$

$⇒ \dfrac{a}{1} = \dfrac{b}{2} = \dfrac{c}{\left(\dfrac{42}{11}\right)}$

$⇒ \dfrac{a}{11} = \dfrac{b}{22} = \dfrac{c}{42}$

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