Among the statements: I: If vmatrix 1 & α & β α & 1 & γ β & γ & 1 vmatrix = vmatrix 0 & α & β α & 0 & γ β & γ & 0 vmatrix$$ \begin{vmatrix} 1 & \cos \alpha & \cos \beta \\ \cos \alpha & 1 & \cos \gamma \\ \cos \beta & \cos \gamma & 1 \end{vmatrix} = \begin{vmatrix} 0 & \cos \alpha & \cos \beta \\ \cos \alpha & 0 & \cos \gamma \\ \cos \beta & \cos \gamma & 0 \end{vmatrix} $$ , then ²α + ²β + ²γ = (3)/(2)$\cos^2\alpha + \cos^2\beta + \cos^2\gamma = \frac{3}{2}$, and II: If vmatrix x² + x & x + 1 & x - 2 2x² + 3x - 1 & 3x & 3x - 3 x² + 2x + 3 & 2x - 1 & 2x - 1 vmatrix = px + q$$ \begin{vmatrix} x^2 + x & x + 1 & x - 2 \\ 2x^2 + 3x - 1 & 3x & 3x - 3 \\ x^2 + 2x + 3 & 2x - 1 & 2x - 1 \end{vmatrix} = px + q $$ , then p² = 196q²$p^2 = 196q^2$, Choose the correct option:

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Core Logic

Apply the row operations R₂ → R₂ - R₁$R_2 \to R_2 - R_1$ and R₃ → R₃ - R₁$R_3 \to R_3 - R_1$ to simplify the determinant:

Step 1: Expand the Determinant

Expanding along the first row:

Since ² x + ² x = 1$\sin^2 x + \cos^2 x = 1$, we get:

Step 2: Find Maximum and Minimum Values

The range of 4x$\sin 4x$ is [-1, 1]$[-1, 1]$.

Step 3: Calculate M⁴ - m⁴$M^4 - m^4$

Pattern Recognition

Look for repeated structures or cyclic additions in rows. Subtracting rows quickly creates zeros, reducing complex trigonometric matrices into elementary algebraic expressions.

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Class 12 Mathematics: Matrices and Determinants Class 11 Mathematics: Trigonometric Functions

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Core Logic

Evaluate the determinant of matrix A$A$:

Using change of base rules:

Step 1: Compute |C| and evaluate response target

Since matrix properties dictate |C| = |A|²$|C| = |A|^2$:

Evaluate targeted multiplier:

Pattern Recognition

Recognize the core cofactor theorem identity instantly: multiplying rows by cofactors of other rows creates zero elements, yielding basic diagonal scalar structures matching matrix attributes.

Chapter Mix

Class 12 Mathematics: Matrices and Determinants

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Core Logic

Evaluate lower power forms of A$A$ to establish inductive sequence patterns:

This cleanly establishes general power state rule configuration expression:

Step 1: Setup Matrix Power Equation

Using the pattern, find expressions for terms:

Substitute into the targeted equation block:

Step 2: Equate Corresponding Elements

From the bottom-left entry [2,1]$[2,1]$:

Both integer states satisfy all matrix components seamlessly. Therefore, the number of elements n(S) = 2$n(S) = 2$.

Pattern Recognition

Always calculate first 2–3 matrix powers to spot linear arithmetic trends across specific cell values (m+1, -m$m+1, -m$), avoiding tedious full Cayley-Hamilton character expansions.

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Class 12 Mathematics: Matrices

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Properties of Determinants:

• (AB) = (A) · (B)$\det(AB) = \det(A) \cdot \det(B)$
• (A^T) = (A)$\det(A^T) = \det(A)$
• (Aⁿ) = ( (A))ⁿ$\det(A^n) = (\det(A))^n$

Core Logic

First, find the determinant of matrix A$A$ and matrix B$B$:

Step 1: Calculate determinant of C

Given C = ABAT$C = ABA^{T}$, we calculate its determinant:

Using (A^T) = (A) = 3$\det(A^T) = \det(A) = 3$:

Step 2: Calculate determinant of X

Given X = ATC²A$X = A^{T}C^{2}A$, we determine |X|$|X|$:

Substitute the values |A| = 3$|A| = 3$ and |C| = 9$|C| = 9$:

Pattern Recognition

Sees: Transpose and multiplication operations nested inside a determinant statement. Shortcut: Never explicitly compute product matrices like ABA^T$ABA^T$ or A^TC²A$A^TC^2A$. Apply the distributive identity of determinants |XYZ| = |X||Y||Z|$|XYZ| = |X||Y||Z|$ entirely to work strictly with scalar multiplication.

Chapter Mix

Class 12 Mathematics: Matrices and Determinants

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For a system of linear equations representing planes to have infinitely many solutions, the planes must belong to a single family sharing a common line of intersection:

Core Logic

Let the planes be defined as:

• P₁: 2x + 3y - z - 5 = 0$P_1: 2x + 3y - z - 5 = 0$
• P₂: x + α y + 3z + 4 = 0$P_2: x + \alpha y + 3z + 4 = 0$
• P₃: 3x - y + β z - 7 = 0$P_3: 3x - y + \beta z - 7 = 0$

Expressing P₁$P_1$ as a linear combination of P₂$P_2$ and P₃$P_3$:

Step 1: Evaluate parameters k1 and k2

Comparing the coefficients of x$x$ and the constant terms on both sides:

• For x$x$:

• For the constants:

Multiplying equation (1) by 4 gives 4k₁ + 12k₂ = 8$4k_1 + 12k_2 = 8$. Subtracting equation (2) from this result:

Substituting k₂$k_2$ back into equation (1):

Step 2: Calculate alpha, beta and final product

Comparing coefficients for y$y$ and z$z$:

• For y$y$:

• For z$z$:

Now, compute 13 α β$13 \alpha \beta$:

Pattern Recognition

Sees: Infinite solution framework for three linear planes. Shortcut: Using the family of planes equation is significantly less prone to fractional algebraic mistakes compared to establishing Cramer's rule determinants (D = Dₓ = Dy = Dz = 0$D = D_x = D_y = D_z = 0$).

Chapter Mix

Class 12 Mathematics: Matrices and Determinants