SATHEE: Matrices and Determinants 2 Question 11

Matrices and Determinants 2 Question 11

13. Let $ω$ be a complex number such that $2 ω + 1 = z$ , where $z = \sqrt{- 3}$ . If $| \begin{matrix} 1 & 1 & 1 \\ 1 & - ω^{2} - 1 & ω^{2} \\ 1 & ω^{2} & ω^{7} \end{matrix} | = 3 k$ , then $k$ is equal to

(a) $- z$

(b) $z$

(d) 1

(2017 Main)

Show Answer

Answer:

Correct Answer: 13. (a)

Solution:

Given, $2 ω + 1 = z$

$\begin{aligned} \Rightarrow & 2 ω + 1 = \sqrt{- 3} & [∵ z = \sqrt{- 3}] \\ \Rightarrow & ω = \frac{- 1 + \sqrt{3} i}{2} \end{aligned}$

Since, $ω$ is cube root of unity.

$∴ ω^{2} = \frac{- 1 - \sqrt{3} i}{2}$ and $ω^{3 n} = 1$

Now, $| \begin{matrix} 1 & 1 & 1 \\ 1 & - ω^{2} - 1 & ω^{2} \\ 1 & ω^{2} & ω^{7} \end{matrix} | = 3 k$

$\Rightarrow | \begin{matrix} 1 & 1 & 1 \\ 1 & ω & ω^{2} \\ 1 & ω^{2} & ω \end{matrix} | = 3 k$

$[∵ 1 + ω + ω^{2} = 0 and ω^{7} = {(ω^{3})}^{2} \cdot ω = ω]$

On applying $R_{1} \to R_{1} + R_{2} + R_{3}$ , we get

$| \begin{matrix} 3 & 1 + ω + ω^{2} & 1 + ω + ω^{2} \\ 1 & ω & ω^{2} \\ 1 & ω^{2} & ω \end{matrix} | = 3 k$

$\Rightarrow$ $| \begin{matrix} 3 & 0 & 0 \\ 1 & ω & ω^{2} \\ 1 & ω^{2} & ω \end{matrix} | = 3 k$

$\Rightarrow 3 (ω^{2} - ω^{4}) = 3 k$

$\Rightarrow (ω^{2} - ω) = k$

$∴ k = \frac{- 1 - \sqrt{3} i}{2} - \frac{- 1 + \sqrt{3} i}{2} = - \sqrt{3} i = - z$

Matrices and Determinants 2 Question 11

13. Let $ω$ be a complex number such that $2 ω + 1 = z$ , where $z = \sqrt{- 3}$ . If $| \begin{matrix} 1 & 1 & 1 \\ 1 & - ω^{2} - 1 & ω^{2} \\ 1 & ω^{2} & ω^{7} \end{matrix} | = 3 k$ , then $k$ is equal to

Answer:

Engineering Preparation

Medical Preparation

NCERT Books & Solutions

Important Resources

Other Resources

Syllabus

Test Platform

Sample Mock Test

Mentorship

NCERT Exercise Video Solution

Matrices and Determinants 2 Question 11

13. Let ω be a complex number such that 2ω+1=z, where z=−3. If |1111−ω2−1ω21ω2ω7|=3k, then k is equal to

Answer:

Engineering Preparation

Medical Preparation

NCERT Books & Solutions

Important Resources

Other Resources

Syllabus

Test Platform

Sample Mock Test

Mentorship

NCERT Exercise Video Solution

Menu

13. Let $ω$ be a complex number such that $2 ω + 1 = z$ , where $z = \sqrt{- 3}$ . If $| \begin{matrix} 1 & 1 & 1 \\ 1 & - ω^{2} - 1 & ω^{2} \\ 1 & ω^{2} & ω^{7} \end{matrix} | = 3 k$ , then $k$ is equal to