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

High-Yield Trend

Chapter Questions
7 MCQs

Official Solution

Official Solution

Official Solution

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Official Solution

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Official Solution

Determinants

High-Yield Trend

Chapter Questions 7 MCQs

Official Solution

Official Solution

Official Solution

Official Solution

Official Solution

Official Solution

Official Solution

Chapter Questions
7 MCQs