if g x beginarray20c a x and e xlog ea and x 2 a 3x and e

If \[g\left( x \right) = \left| {\begin{array}{*{20}{c}} {{a^{ - x}}}&{{e^{x{{\log }_e}a}}}&{{x^2}}\\ {{a^{ - 3x}}}&{{e^{3x{{\log }_e}a}}}&{{x^4}}\\ {{a^{ - 5x}}}&{{e^{5x{{\log }_e}a}}}&1 \end{array}} \right|,\] then

A. $$g\left( x \right) + g\left( { - x} \right) = 0$$

B. $$g\left( x \right) - g\left( { - x} \right) = 0$$

C. $$g\left( x \right) \times g\left( { - x} \right) = 0$$

D. None of these

Answer : $$g\left( x \right) + g\left( { - x} \right) = 0$$

Solution :
\[\begin{array}{l} g\left( x \right) = \left| {\begin{array}{*{20}{c}} {{a^{ - x}}}&{{e^{x{{\log }_e}a}}}&{{x^2}}\\ {{a^{ - 3x}}}&{{e^{3x{{\log }_e}a}}}&{{x^4}}\\ {{a^{ - 5x}}}&{{e^{5x{{\log }_e}a}}}&1 \end{array}} \right|\\ = \left| {\begin{array}{*{20}{c}} {{a^{ - x}}}&{{e^x}}&{{x^2}}\\ {{a^{ - 3x}}}&{{e^{3x}}}&{{x^4}}\\ {{a^{ - 5x}}}&{{e^{5x}}}&1 \end{array}} \right|\left( {{e^{\log {a^x}}} = {a^x}} \right)\\ \Rightarrow g\left( { - x} \right) = \left| {\begin{array}{*{20}{c}} {{a^x}}&{{a^{ - x}}}&{{x^2}}\\ {{a^{3x}}}&{{a^{ - 3x}}}&{{x^4}}\\ {{a^{5x}}}&{{a^{ - 5x}}}&1 \end{array}} \right|\\ = - \left| {\begin{array}{*{20}{c}} {{a^{ - x}}}&{{a^x}}&{{x^4}}\\ {{a^{ - 3x}}}&{{a^{3x}}}&{{x^4}}\\ {{a^{ - 5x}}}&{{a^{5x}}}&1 \end{array}} \right|\left( \begin{array}{l} {\rm{Interchanging }}\,\,I\\ {\rm{and }}\,\,II\,\,{\rm{ columns}} \end{array} \right) \end{array}\]
$$\eqalign{ & = - g\left( x \right) \cr & \Rightarrow g\left( x \right) + g\left( { - x} \right) = 0 \cr} $$

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 $$C$$

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 \[g\left( x \right) = \left| {\begin{array}{*{20}{c}} {{a^{ - x}}}&{{e^{x{{\log }_e}a}}}&{{x^2}}\\ {{a^{ - 3x}}}&{{e^{3x{{\log }_e}a}}}&{{x^4}}\\ {{a^{ - 5x}}}&{{e^{5x{{\log }_e}a}}}&1 \end{array}} \right|,\] then

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 \[g\left( x \right) = \left| {\begin{array}{*{20}{c}} {{a^{ - x}}}&{{e^{x{{\log }_e}a}}}&{{x^2}}\\ {{a^{ - 3x}}}&{{e^{3x{{\log }_e}a}}}&{{x^4}}\\ {{a^{ - 5x}}}&{{e^{5x{{\log }_e}a}}}&1 \end{array}} \right|,\] then

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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Practice More MCQ Question on Maths Section

Releted MCQ Question on
Algebra >> Matrices and Determinants

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

Practice More Releted MCQ Question on
Matrices and Determinants