Let A be a 3 x 3 matrix such that If S = n Z : (|adj(adj A)|)((n - 1)2/2) = |A|3n2 - 5n

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Q52 jee_main_2025_08_april_evening System of Linear Equations

Let

α

be a solution of

x² + x + 1 = 0

, and for some

a

and

b

in

R

,

[4 a b] bmatrix1 & 16 & 13 -1 & -1 & 2 -2 & -14 & -8 bmatrix = [0 0 0]

. If

(4)/(α⁴) + (m)/(α^a) + (n)/(α^b) = 3

, then

m + n

is equal to

A. $3$ $3$
B. $11$ $11$
C. $7$ $7$
D. $8$ $8$

Solution

Related Formula

α² + α + 1 = 0 α = ω where ω³ = 1

Core Logic

Perform row-matrix vector multiplication to generate a system of linear equations in $a$ $a$ and $b$ $b$. Solve for the powers and reduce the algebraic equation using complex roots of unity.

Step 1: Solve Matrix Vector Multiplication

$4 - a - 2b = 0$ $4 - a - 2b = 0$

64 - a - 14b = 0

52 + 2a - 8b = 0

From the first two equations, subtracting them gives:

60 - 12b = 0 b = 5

Substituting $b = 5$ $b = 5$ into the first equation:

4 - a - 10 = 0 a = -6

Step 2: Evaluate Exponential Equation with Roots of Unity

Substitute $a = -6, b = 5$ $a = -6, b = 5$ into the given equation:

(4)/(α⁴) + mα⁻⁶ + (n)/(α⁵) = 3 (4)/(ω) + m + (n)/(ω²) = 3

4ω² + m + nω = 3

Step 3: Resolve Real and Imaginary Components

Substitute standard values $ω = -(1)/(2) + \sqrt(3)2i$ and $ω² = -(1)/(2) - \sqrt(3)2i$ :

4(-(1)/(2) - \sqrt(3)2i) + m + n(-(1)/(2) + \sqrt(3)2i) = 3

Equating the imaginary components:

-4\sqrt(3)2 + n\sqrt(3)2 = 0 n = 4

Equating the real components:

-2 + m - (n)/(2) = 3 -2 + m - 2 = 3 m = 7

m + n = 7 + 4 = 11

Pattern Recognition

Whenever an expression satisfies $Aω² + Bω + C = 0$ $A\omega^2 + B\omega + C = 0$, it directly maps to a comparison with the standard identity $ω² + ω + 1 = 0$ $\omega^2 + \omega + 1 = 0$ up to a linear translation shift.

Chapter Mix

Class 12 Mathematics: Matrices and Determinants Class 11 Mathematics: Complex Numbers

Q69 jee_main_2025_08_april_evening Determinant Properties of Adjoint

Let

A = bmatrix 2 & 2 + p & 2 + p + q 4 & 6 + 2p & 8 + 3p + 2q 6 & 12 + 3p & 20 + 6p + 3q bmatrix

. If

( adj(adj(3A)) ) = 2^m \cdot 3ⁿ

,

m, n in N

, then

m + n

is equal to

A. $22$ $22$
B. $24$ $24$
C. $26$ $26$
D. $20$ $20$

Solution

Related Formula

|adj(adj(M))| = |M|(n-1)²

$|kM| = kⁿ|M|$ $|kM| = k^n|M|$

Core Logic

Perform determinant row reduction transforms to decouple tracking metrics $p$ $p$ and $q$ $q$, leaving a baseline numerical determinant value behind.

Step 1: Simplify the Determinant of Matrix A

Execute columns adjustments: $C₃ arrow C₃ - C₂ - C₁ \times (q)/(2)$ , then $C₂ arrow C₂ - C₁ \times (1 + (p)/(2))$ :

|A| = vmatrix 2 & 0 & 0 4 & 2 & 2+p 6 & 6 & 8+3p vmatrix = 2(16 + 6p - 12 - 6p) = 8 = 2³

Step 2: Apply Adjoint Exponent Transforms

For a matrix of dimensional profile size 3:

|adj(adj(3A))| = |3A|(3-1)² = |3A|⁴

Step 3: Resolve Exponential System Size

|3A| = 3³ |A| = 3³ \times 2³

|3A|⁴ = (3³ \times 2³)⁴ = 2¹² \times 3¹²

Matching base parameter targets: $m = 12, n = 12 m + n = 24$ $m = 12, n = 12 \implies m + n = 24$

Pattern Recognition

Linear parameter shifts down secondary column profiles usually dissolve cleanly during forward element elimination column steps.

Chapter Mix

Class 12 Mathematics: Matrices and Determinants

Q61 jee_main_2025_29_jan_evening System of Linear Equations

Let

α, β

(

α \neq β

) be the values of

m

, for which the equations

x + y + z = 1

;

x + 2y + 4z = m

and

x + 4y + 10z = m²

have infinitely many solutions. Then the value of

Σn=1¹⁰ (n^α + n^β)

is equal to:

A. $440$ $440$
B. $3080$ $3080$
C. $3410$ $3410$
D. $560$ $560$

Solution

Related Formula

Cramer's rule for infinite solutions in a 3 variable system requires:

Δ = Δₓ = Δy = Δz = 0

Core Logic

Set up the primary matrix determinant $Δ$ $\Delta$:

Δ = vmatrix 1 & 1 & 1 1 & 2 & 4 1 & 4 & 10 vmatrix

= 1(20 - 16) - 1(10 - 4) + 1(4 - 2) = 4 - 6 + 2 = 0

Since $Δ = 0$ $\Delta = 0$ is true independent of $m$ $m$, analyze secondary delta constraints to maintain consistency for infinite paths.

Step 1: Compute Dependent Variable Constraints

Evaluate $Δₓ = 0$ $\Delta_x = 0$:

Δₓ = vmatrix 1 & 1 & 1 m & 2 & 4 m² & 4 & 10 vmatrix = 0

1(20 - 16) - 1(10m - 4m²) + 1(4m - 2m²) = 0

4 - 10m + 4m² + 4m - 2m² = 0

2m² - 6m + 4 = 0 m² - 3m + 2 = 0

Thus, $m = 1, 2$ $m = 1, 2$, which gives $α = 1, β = 2$ $\alpha = 1, \beta = 2$.

Step 2: Calculate Sigma Expression

Σn=1¹⁰ (n¹ + n²) = Σn=1¹⁰ n + Σn=1¹⁰ n²

= (10(11))/(2) + (10(11)(21))/(6) = 55 + 385 = 440

Pattern Recognition

When infinitely many solutions are required, solving the determinant created by replacing one column with the constant vector provides parameter roots directly.

Chapter Mix

Class 12 Mathematics: Matrices and Determinants

Q62 jee_main_2025_29_jan_evening Matrix Multiplication and Powers

Let

A = [aᵢⱼ]

be a matrix of order

3 \times 3

, with

aᵢⱼ = (\sqrt(2))i+j

. If the sum of all the elements in the third row of

A²

is

α + β \sqrt(2)

,

α, β in Z

, then

α + β

is equal to

A. $280$ $280$
B. $168$ $168$
C. $210$ $210$
D. $224$ $224$

Solution

Related Formula

Element formula entry rule:

aᵢⱼ = (\sqrt(2))i+j

Core Logic

Constructing the initial matrix structure from the formula entries:

A = bmatrix 2 & 2\sqrt(2) & 4 2\sqrt(2) & 4 & 4\sqrt(2) 4 & 4\sqrt(2) & 8 bmatrix

Factoring scalar factor 4 out to ease squaring multiplication lines:

A = 2 bmatrix 1 & \sqrt(2) & 2 \sqrt(2) & 2 & 2\sqrt(2) 2 & 2\sqrt(2) & 4 bmatrix

Step 1: Calculate Rows of Power Matrix

Squaring matrix $A²$ $A^2$ matches scalar multipliers:

A² = 4 bmatrix 1 & \sqrt(2) & 2 \sqrt(2) & 2 & 2\sqrt(2) 2 & 2\sqrt(2) & 4 bmatrix bmatrix 1 & \sqrt(2) & 2 \sqrt(2) & 2 & 2\sqrt(2) 2 & 2\sqrt(2) & 4 bmatrix

Extract third row entries explicitly:

Row 3 = 4 bmatrix (2+4+8) & (2\sqrt(2)+4\sqrt(2)+8\sqrt(2)) & (4+8+16) bmatrix

= 4 bmatrix 14 & 14\sqrt(2) & 28 bmatrix

Step 2: Aggregate Entries

Sum of row elements = 4(14 + 14\sqrt(2) + 28)

= 4(42 + 14\sqrt(2)) = 168 + 56\sqrt(2)

Matching structural parameters:

α = 168, β = 56

α + β = 168 + 56 = 224

Pattern Recognition

Pull common scaling scalar integers out of matrices before running large multiplications. It eliminates algebraic tracking errors across geometric indices.

Chapter Mix

Class 12 Mathematics: Matrices and Determinants

Q71 jee_main_2025_28_jan_morning Symmetric and Skew Symmetric Matrices

Let

M

denote the set of all real matrices of order

3 \times 3

and let

S = -3, -2, -1, 1, 2

. Let

S₁ = A = [ aᵢⱼ ] in M: A = A^T and aᵢⱼ in S, i, j

S₂ = A = [ aᵢⱼ ] in M: A = -A^T and aᵢⱼ in S, i, j

S₃ = A = [ aᵢⱼ ] in M: a₁₁ + a₂₂ + a₃₃ = 0 and aᵢⱼ in S, i, j

If

n(S₁ S₂ S₃) = 125α

, then

α

equals.

Numerical Answer. Answer: 1613 to 1613

Solution

Related Formula

Set Principle of Inclusion-Exclusion:

n(S₁ S₂ S₃) = n(S₁) + n(S₂) + n(S₃) - n(S₁ S₂) - n(S₂ S₃) - n(S₁ S₃) + n(S₁ S₂ S₃)

Core Logic

Let's count each subset based on the 5 elements available in $S$ $S$:

For $S₁$ $S_1$ (Symmetric matrices): 6 independent element choices $n(S₁) = 5⁶$ $\implies n(S_1) = 5^6$.
For $S₂$ $S_2$ (Skew-symmetric matrices): Diagonal elements must be 0, but $0 S$ $0 \notin S$, so $n(S₂) = 0$ $n(S_2) = 0$.

Since $n(S₂) = 0$ $n(S_2) = 0$, any intersection term involving $S₂$ $S_2$ also becomes 0.

Step 1: Calculating Trace Matrix Variations

For $S₃$ $S_3$ (Trace equal to zero conditions): The condition $a₁₁ + a₂₂ + a₃₃ = 0$ $a_{11} + a_{22} + a_{33} = 0$ over $S = -3, -2, -1, 1, 2$ $S = \{-3, -2, -1, 1, 2\}$ has exactly 12 valid tuple combinations. The remaining 6 elements can be chosen freely.

n(S₃) = 12 \times 5⁶

For the intersection $n(S₁ S₃)$ $n(S_1 \cap S_3)$:

n(S₁ S₃) = 12 \times 5³

Step 2: Final Inclusion-Exclusion Assembly

n(S₁ S₂ S₃) = 5⁶ + 12 \times 5⁶ - 12 \times 5³

= 5³ \times [13 \times 5³ - 12] = 125 \times 1613

Thus, $α = 1613$ $\alpha = 1613$.

Pattern Recognition

Always check if the set contains $0$ $0$. Missing zero elements in skew-symmetric matrix setups instantly zeros out large blocks of permutations.

Chapter Mix

Class 12 Maths: Matrices and Determinants

Solution & Explanation

Related Formula

Core Logic

Step 1: Solve for Exponent Parameter

Step 2: Calculate the Target Summation

Pattern Recognition

Chapter Mix

More Matrices and Determinants Previous-Year Questions — Page 6

Solution

Related Formula

Core Logic

Step 1: Solve Matrix Vector Multiplication

Step 2: Evaluate Exponential Equation with Roots of Unity

Step 3: Resolve Real and Imaginary Components

Pattern Recognition

Chapter Mix

Solution

Related Formula

Core Logic

Step 1: Simplify the Determinant of Matrix A

Step 2: Apply Adjoint Exponent Transforms

Step 3: Resolve Exponential System Size

Pattern Recognition

Chapter Mix

Solution

Related Formula

Core Logic

Step 1: Compute Dependent Variable Constraints

Step 2: Calculate Sigma Expression

Pattern Recognition

Chapter Mix

Solution

Related Formula

Core Logic

Step 1: Calculate Rows of Power Matrix

Step 2: Aggregate Entries

Pattern Recognition

Chapter Mix

Solution

Related Formula

Core Logic

Step 1: Calculating Trace Matrix Variations

Step 2: Final Inclusion-Exclusion Assembly

Pattern Recognition

Chapter Mix

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