Let a _ 1 , a _ 2 , a _ 3 , a _ 10 be in G . P . with a _ i > 0 for i = 1, 2, , 10 and S be the set of pairs (r, k), r, k ∈ N (the set of natural numbers) for which | cases _ e a _ 1 ^ r a _ 2 ^ k & _ e a _ 2 ^ r a _ 3

Correct answer: A. Infinitely many C _ 3 → C _ 3 - C _ 2 , C _ 2 → C _ 2 - C _ 1 ( Let α is common ratio of GP ) | cases _ e a _ 1 ^ r a _ 2 ^ k & _ e ( α ^ r + k ) & _ e ( α ^ r + k ) _ e a _ 4 ^ r a _ 5 ^ k & _ e ( α ^ r + k ) & _ e ( α ^ r + k ) _ e a _ 7 ^ r a _ 8 ^ k & _ e ( α ^ r + k ) & _ e ( α ^ r + k ) cases | = 0 which is always true because C _ 2 and C _ 3 are identical.

JEE Main 201910 JanuaryEvening Shift

Exam: JEE Main
Year: 2019
Date: 10 January
Shift: Evening Shift
Subject: Mathematics
Chapter: Determinants

JEE Main 2019 · 10 January · Evening Shift · Mathematics · Determinants

Let a _ 1 , a _ 2 , a _ 3 , a _ 10 be in G . P . with a _ i > 0 for...

Let

{a}_{1}, {a}_{2}, {a}_{3} \ldots , {a}_{10}

be in

G . P .

with

{a}_{i} > 0

for

i = 1, 2, \ldots , 10 and S

be the set of pairs \left(r, k\right), r, k \in N

(the set of natural numbers) for which<br /><br />\left|

\begin{cases}\log_{e} {a}_{1}^{r}{a}_{2}^{k} & \log_{e} {a}_{2}^{r}{a}_{3}^{k} & \log_{e} {a}_{3}^{r}{a}_{4}^{k} \\ \log_{e} {a}_{4}^{r}{a}_{5}^{k} & \log_{e} {a}_{5}^{r}{a}_{6}^{k} & \log_{e} {a}_{6}^{r}{a}_{7}^{k} \\ \log_{e} {a}_{7}^{r}{a}_{8}^{k} & \log_{e} {a}_{8}^{r}{a}_{9}^{k} & \log_{e} {a}_{9}^{r}{a}_{10}^{k}\end{cases}

\right| = 0

Then the number of elements in

S,

is:

Options

AInfinitely many
B $4$
C $10$
D $2$

Correct answer

A. Infinitely many

Step-by-step solution

{C}_{3} \to {C}_{3} - {C}_{2}, {C}_{2} \to {C}_{2} - {C}_{1} (

Let

\alpha

is common ratio of

GP )

<br/><br/> \left|

\begin{cases}\log_{e}{a}_{1}^{r}{a}_{2}^{k} & \log_{e}\left({\alpha}^{r + k}\right) & \log_{e}\left({\alpha}^{r + k}\right) \\ \log_{e}{a}_{4}^{r}{a}_{5}^{k} & \log_{e}\left({\alpha}^{r + k}\right) & \log_{e} \left({\alpha}^{r + k}\right) \\ \log_{e}{a}_{7}^{r}{a}_{8}^{k} & \log_{e} \left({\alpha}^{r + k}\right) & \log_{e} \left({\alpha}^{r + k}\right)\end{cases}

\right| = 0

which is always true because

{C}_{2} \text{\text{ and } }

{C}_{3} are identical.

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