First slide

Introduction to Determinants

Question

If $Δ (x) = |\begin{array}{lll} x^{2} + 4 x - 3 2 x + 4 13 \\ 2 x^{2} + 5 x - 9 4 x + 5 26 \\ 8 x^{2} - 6 x + 1 16 x - 6 104 \end{array}| = {ax}^{3} + {bx}^{2} + cx + d$ , then

Moderate

A

a = 3
B

b = 0
C

c = 0
D

None of these

Solution

$\begin{array}{l} Δ' (x) = |\begin{array}{ccc} 2 x + 4 & 2 x + 4 & 13 \\ 4 x + 5 & 4 x + 5 & 26 \\ 16 x - 6 & 16 x - 6 & 104 \end{array}| + |\begin{array}{ccc} x^{2} + 4 x - 3 & 2 & 13 \\ 2 x^{2} + 5 x - 9 & 4 & 26 \\ 8 x^{2} - 6 x + 1 & 16 & 104 \end{array}| \\ = 0 + 2 \times 13 \times (0) = 0 \end{array}$
$\Rightarrow Δ (x) = constant \Rightarrow a = 0, b = 0, c = 0$

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