Determinants

The correct option is(A): xyz=−1.

Given , $%5Cleft%7C%5Cbegin%7Bmatrix%7Dx%26x%5E%7B2%7D%261%2Bx%5E%7B3%7D%5C%5C%20y%26y%5E%7B2%7D%261%2By%5E%7B3%7D%5C%5C%20z%26z%5E%7B2%7D%261%2Bz%5E%7B3%7D%5Cend%7Bmatrix%7D%5Cright%7C%3D0$
$%5CRightarrow%20%5Cleft%7C%5Cbegin%7Bmatrix%7Dx%26x%5E%7B2%7D%261%5C%5C%20y%26y%5E%7B2%7D%261%5C%5C%20z%26z%5E%7B2%7D%261%5Cend%7Bmatrix%7D%5Cright%7C%2B%5Cleft%7C%5Cbegin%7Bmatrix%7Dx%26x%5E%7B2%7D%26x%5E%7B3%7D%5C%5C%20y%26y%5E%7B2%7D%26y%5E%7B3%7D%5C%5C%20z%26z%5E%7B2%7D%26z%5E%7B3%7D%5Cend%7Bmatrix%7D%5Cright%7C%3D0$
$%5CRightarrow%20%5Cleft%7C%5Cbegin%7Bmatrix%7Dx%26x%5E%7B2%7D%261%5C%5C%20y%26y%5E%7B2%7D%261%5C%5C%20z%26z%5E%7B2%7D%261%5Cend%7Bmatrix%7D%5Cright%7C%2Bxyz%20%5Cleft%7C%5Cbegin%7Bmatrix%7D1%26x%26x%5E%7B2%7D%5C%5C%201%26y%26y%5E%7B2%7D%5C%5C%201%26z%26z%5E%7B2%7D%5Cend%7Bmatrix%7D%5Cright%7C%3D0$
$%5CRightarrow%20%5Cleft%7C%5Cbegin%7Bmatrix%7Dx%26x%5E%7B2%7D%261%5C%5C%20y%26y%5E%7B2%7D%261%5C%5C%20z%26z%5E%7B2%7D%261%5Cend%7Bmatrix%7D%5Cright%7C%2Bxyz%5Cleft%7C%5Cbegin%7Bmatrix%7Dx%26x%5E%7B2%7D%261%5C%5C%20y%26y%5E%7B2%7D%261%5C%5C%20z%26z%5E%7B2%7D%261%5Cend%7Bmatrix%7D%5Cright%7C%3D0$
$%5CRightarrow%20%5Cleft(1%2Bxyz%5Cright)%20%5Cleft%7C%5Cbegin%7Bmatrix%7Dx%26x%5E%7B2%7D%261%5C%5C%20y%26y%5E%7B2%7D%261%5C%5C%20z%26z%5E%7B2%7D%261%5Cend%7Bmatrix%7D%5Cright%7C%3D0$
$%5CRightarrow%5Cleft(1%2Bxyz%5Cright)%5Cleft%7C%5Cbegin%7Bmatrix%7Dx%26x%5E%7B2%7D%261%5C%5C%20y%26y%5E%7B2%7D%261%5C%5C%20z%26z%5E%7B2%7D%261%5Cend%7Bmatrix%7D%5Cright%7C%3D0$
$%5CRightarrow%201%2Bxyz%3D0$
and $%5Cleft%7C%5Cbegin%7Bmatrix%7Dx%26x%5E%7B2%7D%261%5C%5C%20y%26y%5E%7B2%7D%261%5C%5C%20z%26z%5E%7B2%7D%261%5Cend%7Bmatrix%7D%5Cright%7C%5Cne0$
$%5Ctherefore%20xyz%3D-1$

Given,

A28

3B9

and

62C

are divisible by

k

%5Ctherefore%20A28%3Dk

%5CRightarrow%20100A%2B20%2B8%3Dk%20%3A%203B9%20%3D%20k

%5CRightarrow%20300%2B10B%2B9%3Dk

and

62C%3Dk

%5CRightarrow%20600%2B20%2BC%3Dk

Let

%5CDelta%3D%5Cleft%7C%5Cbegin%7Bmatrix%7DA%263%266%5C%5C%208%269%26C%5C%5C%202%26B%262%5Cend%7Bmatrix%7D%5Cright%7C

%3D%5Cfrac%7B1%7D%7B100%7D%5Ctimes%5Cfrac%7B1%7D%7B10%7D%20%5Cleft%7C%5Cbegin%7Bmatrix%7D100%20A%26300%26600%5C%5C%208%269%26C%5C%5C%2020%2610%20B%2620%5Cend%7Bmatrix%7D%5Cright%7C

Applying

R_%7B1%7D%20%5Crightarrow%20R_%7B1%7D%2BR_%7B2%7D%2BR_%7B3%7D

%3D%5Cfrac%7B1%7D%7B1000%7D

%5Cleft%7C%5Cbegin%7Bmatrix%7D100A%2B20%2B8%26300%2B10B%2B9%26600%2B20%2BC%5C%5C%208%269%26C%5C%5C%2020%2610B%2620%5Cend%7Bmatrix%7D%5Cright%7C

%3D%5Cfrac%7B1%7D%7B1000%7D%20%5Cleft%7C%5Cbegin%7Bmatrix%7Dk%26k%26k%5C%5C%208%269%26C%5C%5C%2020%2610B%2620%5Cend%7Bmatrix%7D%5Cright%7C

%3D%5Cfrac%7Bk%7D%7B100%7D%20%5Cleft%7C%5Cbegin%7Bmatrix%7D1%261%261%5C%5C%208%269%26C%5C%5C%2020%2610B%2620%5Cend%7Bmatrix%7D%5Cright%7C

Hence, it is always divisible by

k

About Determinants - AMUEEE

Determinants is a vital chapter for AMUEEE aspirants. Mastering the concepts covered in this chapter is essential for securing a top rank.

By rigorously practicing the previous year questions associated with this chapter, you can identify high-yield topics, understand the examiner's perspective, and boost your confidence during the actual exam.

Frequently Asked Questions

Why focus on Determinants PYQs?

Analyzing PYQs for this specific chapter reveals the most frequently tested concepts and the typical complexity of questions, allowing you to tailor your study plan efficiently.

How to best use this analysis?

Review the topic breakdown to see which sub-topics within Determinants carry the most weight. Then, tackle the questions iteratively to solidify your understanding.

High-Yield Trend

Chapter Questions
7 MCQs

Official Solution

Official Solution

Official Solution

Official Solution

Official Solution

Official Solution

Official Solution

About Determinants - AMUEEE

Frequently Asked Questions

Why focus on Determinants PYQs?

How to best use this analysis?

Determinants

High-Yield Trend

Chapter Questions 7 MCQs

Official Solution

Official Solution

Official Solution

Official Solution

Official Solution

Official Solution

Official Solution

About Determinants - AMUEEE

Frequently Asked Questions

Why focus on Determinants PYQs?

How to best use this analysis?

Chapter Questions
7 MCQs