if a 2 b 2 c 2 ne 2 and f x beginarray20c 1 a 2 x and 1 b 2

If $${a^2} + {b^2} + {c^2} \ne - 2$$ and \[f\left( x \right) = \left| {\begin{array}{*{20}{c}} {\left( {1 + {a^2}} \right)x}&{\left( {1 + {b^2}} \right)x}&{\left( {1 + {c^2}} \right)x}\\ {\left( {1 + {a^2}} \right)x}&{\left( {1 + {b^2}} \right)x}&{\left( {1 + {c^2}} \right)x}\\ {\left( {1 + {a^2}} \right)x}&{\left( {1 + {b^2}} \right)x}&{\left( {1 + {c^2}} \right)x} \end{array}} \right|,\] then $$f\left( x \right)$$ is a polynomial of degree

A. 1

B. 0

C. 3

D. 2

Answer : 2

Solution :
\[f\left( x \right) = \left| {\begin{array}{*{20}{c}} {1 + \left( {{a^2} + {b^2} + {c^2} + 2} \right)x}&{\left( {1 + {b^2}} \right)x}&{\left( {1 + {c^2}} \right)x}\\ {1 + \left( {{a^2} + {b^2} + {c^2} + 2} \right)x}&{1 + {b^2}x}&{\left( {1 + {c^2}} \right)x}\\ {1 + \left( {{a^2} + {b^2} + {c^2} + 2} \right)x}&{\left( {1 + {b^2}} \right)x}&{1 + {c^2}x} \end{array}} \right|\]
$${\text{Applying, }}{C_1} \to {C_1} + {C_2} + {C_3}$$
\[ = \left| {\begin{array}{*{20}{c}} 1&{\left( {1 + {b^2}} \right)x}&{\left( {1 + {c^2}} \right)x}\\ 1&{1 + {b^2}x}&{\left( {1 + {c^2}} \right)x}\\ 1&{\left( {1 + {b^2}} \right)x}&{1 + {c^2}x} \end{array}} \right|\]
$$\therefore {a^2} + {b^2} + {c^2} + 2 \ne 0$$
\[f\left( x \right) = \left| {\begin{array}{*{20}{c}} 0&{x - 1}&0\\ 0&{1 - x}&{x - 1}\\ 1&{\left( {1 + {b^2}} \right)x}&{1 + {c^2}x} \end{array}} \right|\]
$$\eqalign{ & {\text{Applying }}{R_1} \to {R_1} - {R_2},{R_2} \to {R_2} - {R_3} \cr & f\left( x \right) = {\left( {x - 1} \right)^2}\,\,\,\,{\text{Hence degree = 2}}{\text{.}} \cr} $$

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 $$C$$

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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. $$\omega $$

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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

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

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