JEE MATHEMATICS MCQ: MATRICES

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Analyzing +4 | -1

Let

a_1, a_2, a_3, \ldots, a_{10}

be in GP. with

a_i>0

for

i=1,2, \ldots, 10

and

\mathrm{S}

be the set of pairs (r, k),

r, k \in N

(the set of natural numbers) for which

\left|\begin{array}{llll} \log _e \mathrm{a}_1^{\mathrm{r}} \mathrm{a}_2^{\mathrm{k}} & \log _e \mathrm{a}_2^{\mathrm{r}} \mathrm{a}_3^{\mathrm{k}} & \log _e \mathrm{a}_3^{\mathrm{r}} \mathrm{a}_4^{\mathrm{k}} \\ \log _e \mathrm{a}_4^{\mathrm{r}} \mathrm{a}_5^{\mathrm{k}} & \log _e \mathrm{a}_5^{\mathrm{r}} \mathrm{a}_6^{\mathrm{k}} & \log _e \mathrm{a}_6^{\mathrm{r}} \mathrm{a}_7^{\mathrm{k}} \\ \log _e \mathrm{a}_7^{\mathrm{r}} \mathrm{a}_8^{\mathrm{k}} & \log _e \mathrm{a}_8^{\mathrm{r}} \mathrm{a}_9^{\mathrm{k}} & \log _e \mathrm{a}_9^{\mathrm{r}} \mathrm{a}_{10}^{\mathrm{k}} \end{array}\right|=0

Then the number of elements in S, is :

View answer and explanation

Correct option: B

infinitelymany

Option B (infinitelymany) is correct because the determinant becomes zero regardless of the values of r and k, as the first two columns become proportional after simplification.

More Questions from MATRICES

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\left|\begin{array}{ccc}a & b-c & b+c \\ c+a & b & c-a \\ a-b & a+b & c\end{array}\right|=0

and

a, b, c

are real numbers then the line

a x+b y+c=0

passes through the fixed point is The number of solutions of the system of equations

{x_2} - {x_3} = 1,\; - {x_1} + 2{x_3} = - 2,\;{x_1} - 2,\;{x_1} - 2{x_2} = 3

is If the equations

2x + 3y + 1 = 0,3x + y - 2 = 0

and

ax + 2y - b = 0

are consistent, then The value of the determinant

\left|\begin{array}{ccc}1 & \left(\alpha^{2 x}-\alpha^{-2 x}\right)^2 & \left(\alpha^{2 x}+\alpha^{-2 x}\right)^2 \\ 1 & \left(\beta^{2 x}-\beta^{-2 x}\right)^2 & \left(\beta^{2 x}+\beta^{-2 x}\right)^2 \\ 1 & \left(\gamma^{2 x}-\gamma^{-2 x}\right)^2 & \left(\gamma^{2 x}+\gamma^{-2 x}\right)^2\end{array}\right|

, is If

\omega

is the cube root of unity, then

\left| {\begin{array}{cccccccccccccccccccc} 1&\omega &{{\omega ^2}}\\ \omega &{{\omega ^2}}&1\\ {{\omega ^2}}&1&\omega \end{array}} \right|

is equal to For any matrix

A

of order

2 \times 2

, if

A(\operatorname{adj} A)=\left[\begin{array}{cc}10 & 0 \\ 0 & 10\end{array}\right]

, then

|A|

is equal to