$\left| {\begin{array}{*{20}{c}} {\rm{\alpha }}&{ - {\rm{\beta }}}&0\\ 0&{\rm{\alpha }}&{\rm{\beta }}\\ {\rm{\beta }}&0&{\rm{\alpha }} \end{array}} \right| = 0$, then

${\rm{\alpha }}$ is one of the cube roots of unity

${\rm{\alpha \beta }}$ is one of the cube roots of unity

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JEE • MATHEMATICS • MATRICES

Matrices practice · Mixed difficulty

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These sets use single-correct-answer MCQs. They do not cover every JEE Main or JEE Advanced question format and are not full-length mock papers.

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Showing 991-995 of 995 questions

991 Easy

991 of 995

Application +4 | -1

A

is an involuntary matrix given by

A=\left[\begin{array}{rrr}0 & 1 & -1 \\ 4 & -3 & 4 \\ 3 & -3 & 4\end{array}\right]

, then the inverse of

\frac{1}{2} A

will be

View answer and explanation

Correct option: A

2 A

Option A (2A) is correct because, for an involutory matrix A, A² = I, which implies A = A⁻¹. Therefore, the inverse of (1/2)A is 2A.

992 Medium

992 of 995

Analyzing +4 | -1

\alpha, \beta, \gamma

are different from 1 and are the roots of

a x^3+b x^2+c x+d=0

and

(\beta-\gamma)(\gamma-\alpha)(\alpha-\beta)=25 / 2

, then the determinant

\Delta=\left|\begin{array}{ccc}\frac{\alpha}{1-\alpha} & \frac{\beta}{1-\beta} & \frac{\gamma}{1-\gamma} \\ \alpha & \beta & \gamma \\ \alpha^2 & \beta^2 & \gamma^2\end{array}\right|

equals

View answer and explanation

Correct option: D

\frac{{ - 25d}}{{2(a + b + c + d)}}

Option D (

\frac{{ - 25d}}{{2(a + b + c + d)}}

) is correct because after simplification and using Vieta's formulas, the determinant equals this expression.

993 Easy

993 of 995

Analyzing +4 | -1

If for some

\alpha

and

\beta

R

, the intersection of the following three planes

\begin{array}{l} x + 4y - 2z = 1\\ x + 7y - 5z = \beta \\ x + 5y + az = 5 \end{array}

is a line in

{R^3}

, then

\alpha + \beta

is equal to

View answer and explanation

Correct option: B

Option B (10) is correct because it is the sum of alpha and beta after finding their values based on the condition that the intersection of the three planes is a line.

994 Easy

994 of 995

Analyzing +4 | -1

The rank of the matrix

\left[ {\begin{array}{cccccccccccccccccccc} 1\\ 2\\ 3\\ 6 \end{array}\;\,\,\begin{array}{cccccccccccccccccccc} 2\\ 4\\ 2\\ 8 \end{array}\;\,\,\begin{array}{cccccccccccccccccccc} 3\\ 3\\ 1\\ 7 \end{array}\;\,\,\begin{array}{cccccccccccccccccccc} 0\\ 2\\ 3\\ 5 \end{array}} \right]

View answer and explanation

Correct option: C

Option C (3) is correct because after row reduction, there are 3 non-zero rows, indicating a rank of 3.

995 Easy

995 of 995

Application +4 | -1

\left| {\begin{array}{cccccccccccccccccccc} \mathrm{\alpha }&{ - \mathrm{\beta }}&0\\ 0&\mathrm{\alpha }&\mathrm{\beta }\\ \mathrm{\beta }&0&\mathrm{\alpha } \end{array}} \right| = 0

, then

View answer and explanation

Correct option: A

\frac{\mathrm{\alpha }}{\mathrm{\beta }}

is one of the cube roots of unity

Option A (

\frac{{\rm{{\alpha }}}}{{\rm{{\beta }}}}}

is one of the cube roots of unity) is correct because the determinant equation simplifies to

(\frac{{\rm{{\alpha }}}}{{\rm{{\beta }}}})^3 = 1

, indicating that

\frac{{\rm{{\alpha }}}}{{\rm{{\beta }}}}

is a cube root of unity.

Revision notes and key takeaways

Review MATRICES in your textbook or class notes, then use this set to check your understanding.

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