EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR MATHEMATICS>Chapter 8 - Matrices and Determinants>JEE Main - 25 July 2022 Shift 2>Q 181

HARD

JEE Main

IMPORTANT

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Let $A = [\begin{array}{l} 1 & a & a \\ 0 & 1 & b \\ 0 & 0 & 1 \end{array}], a, b \in ℝ$ . If for some $n \in N, A^{n} = [\begin{matrix} 1 & 48 & 2160 \\ 0 & 1 & 96 \\ 0 & 0 & 1 \end{matrix}]$ then $n + a + b$ is equal to _______.

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Important Questions on Matrices and Determinants

MEDIUM

JEE Main

IMPORTANT

EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR MATHEMATICS>Chapter 8 - Matrices and Determinants>JEE Main - 26 July 2022 Shift 1>Q 182

Let

A

be a

2 \times 2

matrix with

det (A) = - 1

and

\det ((A + I) Adj (A) + I) = 4

. Then the sum of the diagonal elements of

A

can be:

MEDIUM

JEE Main

IMPORTANT

EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR MATHEMATICS>Chapter 8 - Matrices and Determinants>JEE Main - 26 July 2022 Shift 2>Q 183

Let

A = [\begin{array}{l} 1 \\ 1 \\ 1 \end{array}]

and

B = [\begin{matrix} 9^{2} & - 10^{2} & 11^{2} \\ 12^{2} & 13^{2} & - 14^{2} \\ - 15^{2} & 16^{2} & 17^{2} \end{matrix}]

, then the value of

A^{'} B A

is;

HARD

JEE Main

IMPORTANT

EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR MATHEMATICS>Chapter 8 - Matrices and Determinants>JEE Main - 26 July 2022 Shift 2>Q 184

The number of matrices

A = [\begin{array}{l} a & b \\ c & d \end{array}]

, where

a, b, c, d \in \{- 1, 0, 1, 2, 3, \dots \dots, 10\}

, such that

A = A^{- 1}

, is ______.

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JEE Main

IMPORTANT

EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR MATHEMATICS>Chapter 8 - Matrices and Determinants>JEE Main - 27 July 2022 Shift 1>Q 185

Let

A = (\begin{matrix} 1 & 2 \\ - 2 & - 5 \end{matrix})

. Let

α, β \in ℝ

be such that

α A^{2} + β A = 2 I

. Then

α + β

is equal to

HARD

JEE Main

IMPORTANT

EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR MATHEMATICS>Chapter 8 - Matrices and Determinants>JEE Main - 27 July 2022 Shift 2>Q 186

Let

A = (\begin{matrix} 4 & - 2 \\ α & β \end{matrix})

. If

A^{2} + γ A + 18 I = O

, then

det (A)

is equal to _______.

HARD

JEE Main

IMPORTANT

EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR MATHEMATICS>Chapter 8 - Matrices and Determinants>JEE Main - 27 July 2022 Shift 2>Q 187

Consider a matrix

A = [\begin{matrix} α & β & γ \\ α^{2} & β^{2} & γ^{2} \\ β + γ & γ + α & α + β \end{matrix}]

, where

α, β, γ

are three distinct natural numbers.
If

\frac{det (adj (adj (adj (adj A)))}{{(α - β)}^{16} {(β - γ)}^{16} {(γ - α)}^{16}} = 2^{32} \times 3^{16}

, then the number of such

3

- tuples

(α, β, γ)

is _______.

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JEE Main

IMPORTANT

EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR MATHEMATICS>Chapter 8 - Matrices and Determinants>JEE Main - 28 July 2022 Shift 1>Q 188

Let the matrix

A = [\begin{matrix} 0 & 1 & 0 \\ 0 & 0 & 1 \\ 1 & 0 & 0 \end{matrix}]

and the matrix

B_{0} = A^{49} + 2 A^{98}

. If

B_{n} = Adj (B_{a - 1})

for all

n \geq 1

, then

det (B_{4})

is equal to

MEDIUM

JEE Main

IMPORTANT

EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR MATHEMATICS>Chapter 8 - Matrices and Determinants>JEE Main - 28 July 2022 Shift 1>Q 189

Let

A = [\begin{matrix} 1 & - 1 \\ 2 & α \end{matrix}]

and

B = [\begin{array}{l} β & 1 \\ 1 & 0 \end{array}], α, β \in R

. Let

α_{1}

be the value of

α

which satisfies

{(A + B)}^{2} = A^{2} + [\begin{array}{l} 2 & 2 \\ 2 & 2 \end{array}]

and

α_{2}

be the value of

α

which satisfies

{(A + B)}^{2} = B^{2}

. Then

|α_{1} - α_{2}|

is equal to