First slide

Introduction to Determinants

Question

If $Δ (x) = |\begin{array}{ccc} \tan x & \tan (x + h) & \tan (x + 2 h) \\ \tan (x + 2 h) & \tan x & \tan (x + h) \\ \tan (x + h) & \tan (x + 2 h) & \tan x \end{array}|$ , then the value of $lim_{h \to 0} \frac{Δ (π / 3)}{\sqrt{3} h^{2}}$ is

Moderate

A

144
B

81
C

64
D

36

Solution

$\frac{∆}{h^{2}} = |\begin{matrix} tanx & \frac{\tan (x + h) - tanx}{h} & \frac{\tan (x + 2 h) - tanx}{h} \\ \tan (x + 2 h) & \frac{tanx - \tan (x + 2 h)}{h} & \frac{\tan (x + h) - \tan (x + 2 h)}{h} \\ \tan (x + h) & \frac{\tan (x + 2 h) - \tan (x + h)}{h} & \frac{tanx - \tan (x + h)}{h} \end{matrix}|$
$\begin{array}{rcl} \Rightarrow & lim_{h \to 0} \frac{Δ}{h^{2}} = |\begin{array}{ccc} \tan x & \sec^{2} x & 2 \sec^{2} x \\ \tan x & - 2 \sec^{2} x & - \sec^{2} x \\ \tan x & \sec^{2} x & - \sec^{2} x \end{array}| \end{array}$
$\begin{array}{rcl} = & |\begin{array}{ccc} 0 & 0 & 3 \sec^{2} x \\ 0 & - 3 \sec^{2} x & 0 \\ \tan x & \sec^{2} x & - \sec^{2} x \end{array}| \end{array}$
$= 9 \tan x \sec^{4} x$
$\Rightarrow lim_{h \to 0} \frac{Δ (π / 3)}{\sqrt{3} h^{2}} = 144$

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