if the equations a y z xb z x y and c x y z where a ne

If the equations $$a\left( {y + z} \right) = x,b\left( {z + x} \right) = y$$ and $$c\left( {x + y} \right) = z,$$ where $$a \ne - 1,b \ne - 1,c \ne - 1,$$ admit of non-trivial solutions then $${\left( {1 + a} \right)^{ - 1}} + {\left( {1 + b} \right)^{ - 1}} + {\left( {1 + c} \right)^{ - 1}}$$ is

A. $$2$$

B. $$1$$

C. $$\frac{1}{2}$$

D. None of these

Answer : $$2$$

Solution :
$$a\left( {x + y + z} \right) = \left( {1 + a} \right)x,b\left( {x + y + z} \right) = \left( {1 + b} \right)y,c\left( {x + y + z} \right) = \left( {1 + c} \right)z.$$
In case of non-trivial solution, $$x + y + z \ne 0.$$
$$\therefore \,\,\frac{a}{{1 + a}} = \frac{x}{{x + y + z}},\,{\text{e}}{\text{.t}}{\text{.c}}{\text{.}}$$
Adding, $$\frac{a}{{1 + a}} + \frac{b}{{1 + b}} + \frac{c}{{1 + c}} = 1.$$
$$\therefore \,\,\left( {1 - \frac{1}{{1 + a}}} \right) + \left( {1 - \frac{1}{{1 + b}}} \right) + \left( {1 - \frac{1}{{1 + c}}} \right) = 1.$$

Releted MCQ Question on
Algebra >> Matrices and Determinants

Releted Question 1

Consider the set $$A$$ of all determinants of order 3 with entries 0 or 1 only. Let $$B$$ be the subset of $$A$$ consisting of all determinants with value 1. Let $$C$$ be the subset of $$A$$ consisting of all determinants with value $$- 1.$$ Then

A. $$C$$ is empty

B. $$B$$ has as many elements as $$C$$

C. $$A = B \cup C$$

D. $$B$$ has twice as many elements as elements as $$C$$

View Answer

Releted Question 2

If $$\omega \left( { \ne 1} \right)$$ is a cube root of unity, then
\[\left| {\begin{array}{*{20}{c}} 1&{1 + i + {\omega ^2}}&{{\omega ^2}}\\ {1 - i}&{ - 1}&{{\omega ^2} - 1}\\ { - i}&{ - i + \omega - 1}&{ - 1} \end{array}} \right|=\]

A. 0

B. 1

C. $$i$$

D. $$\omega $$

View Answer

Releted Question 3

Let $$a, b, c$$ be the real numbers. Then following system of equations in $$x, y$$ and $$z$$
$$\frac{{{x^2}}}{{{a^2}}} + \frac{{{y^2}}}{{{b^2}}} - \frac{{{z^2}}}{{{c^2}}} = 1,$$ $$\frac{{{x^2}}}{{{a^2}}} - \frac{{{y^2}}}{{{b^2}}} + \frac{{{z^2}}}{{{c^2}}} = 1,$$ $$ - \frac{{{x^2}}}{{{a^2}}} + \frac{{{y^2}}}{{{b^2}}} + \frac{{{z^2}}}{{{c^2}}} = 1$$ has

A. no solution

B. unique solution

C. infinitely many solutions

D. finitely many solutions

View Answer

Releted Question 4

If $$A$$ and $$B$$ are square matrices of equal degree, then which one is correct among the followings?

A. $$A + B = B + A$$

B. $$A + B = A - B$$

C. $$A - B = B - A$$

D. $$AB=BA$$

View Answer

If the equations $$a\left( {y + z} \right) = x,b\left( {z + x} \right) = y$$ and $$c\left( {x + y} \right) = z,$$ where $$a \ne - 1,b \ne - 1,c \ne - 1,$$ admit of non-trivial solutions then $${\left( {1 + a} \right)^{ - 1}} + {\left( {1 + b} \right)^{ - 1}} + {\left( {1 + c} \right)^{ - 1}}$$ is

Releted MCQ Question on
Algebra >> Matrices and Determinants

Consider the set $$A$$ of all determinants of order 3 with entries 0 or 1 only. Let $$B$$ be the subset of $$A$$ consisting of all determinants with value 1. Let $$C$$ be the subset of $$A$$ consisting of all determinants with value $$- 1.$$ Then

If $$\omega \left( { \ne 1} \right)$$ is a cube root of unity, then
\[\left| {\begin{array}{*{20}{c}} 1&{1 + i + {\omega ^2}}&{{\omega ^2}}\\ {1 - i}&{ - 1}&{{\omega ^2} - 1}\\ { - i}&{ - i + \omega - 1}&{ - 1} \end{array}} \right|=\]

If $$A$$ and $$B$$ are square matrices of equal degree, then which one is correct among the followings?

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Matrices and Determinants

Practice More MCQ Question on Maths Section

If the equations $$a\left( {y + z} \right) = x,b\left( {z + x} \right) = y$$ and $$c\left( {x + y} \right) = z,$$ where $$a \ne - 1,b \ne - 1,c \ne - 1,$$ admit of non-trivial solutions then $${\left( {1 + a} \right)^{ - 1}} + {\left( {1 + b} \right)^{ - 1}} + {\left( {1 + c} \right)^{ - 1}}$$ is

Releted MCQ Question on Algebra >> Matrices and Determinants

Consider the set $$A$$ of all determinants of order 3 with entries 0 or 1 only. Let $$B$$ be the subset of $$A$$ consisting of all determinants with value 1. Let $$C$$ be the subset of $$A$$ consisting of all determinants with value $$- 1.$$ Then

If $$\omega \left( { \ne 1} \right)$$ is a cube root of unity, then \[\left| {\begin{array}{*{20}{c}} 1&{1 + i + {\omega ^2}}&{{\omega ^2}}\\ {1 - i}&{ - 1}&{{\omega ^2} - 1}\\ { - i}&{ - i + \omega - 1}&{ - 1} \end{array}} \right|=\]

If $$A$$ and $$B$$ are square matrices of equal degree, then which one is correct among the followings?

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Releted MCQ Question on
Algebra >> Matrices and Determinants

If $$\omega \left( { \ne 1} \right)$$ is a cube root of unity, then
\[\left| {\begin{array}{*{20}{c}} 1&{1 + i + {\omega ^2}}&{{\omega ^2}}\\ {1 - i}&{ - 1}&{{\omega ^2} - 1}\\ { - i}&{ - i + \omega - 1}&{ - 1} \end{array}} \right|=\]

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Matrices and Determinants