Matrices & DeterminantshardFree

Matrices & Determinants: Vmatrix Vmatrix

JEE Maths question with a full step-by-step solution.

Question

f(x) = \begin{vmatrix} x^2-4x+6 & 2x^2+4x+10 & 3x^2-2x+16 \\ x-2 & 2x+2 & 3x-1 \\ 1 & 2 & 3 \end{vmatrix}

, then

f(x)

is:

AIncreasing when

x \in (3, 5)

BIncreasing when

x \in (-3, -5)

CDecreasing when

x \in (3, 5)

DConstant for all real

x

correct

Solution

Step 1: Differentiate using the row-differentiation rule

f'(x) = \begin{vmatrix} R_1' \\ R_2 \\ R_3 \end{vmatrix} + \begin{vmatrix} R_1 \\ R_2' \\ R_3 \end{vmatrix} + \begin{vmatrix} R_1 \\ R_2 \\ R_3' \end{vmatrix}

Step 2: Observe proportionality of derivative rows Row 1 differentiated:

(2x-4,\ 4x+4,\ 6x-2) = 2(x-2,\ 2x+2,\ 3x-1) = 2R_2

. A determinant with two proportional rows is zero. Row 2 differentiated:

(1, 2, 3) = R_3

. A determinant with two identical rows is zero. Row 3 differentiated:

(0, 0, 0)

. This determinant is also zero. Step 3: Conclude

f'(x) = 0

for all

x \in \mathbb{R}

, so

f

is constant. Evaluating at

x = 0

gives

f(0) = 2

. Answer: (4)

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