If 2 sin sin + 3 cos+ 5 cos sin = 38 R , then det ( adj ( adj A ) ) is equal to (where A = [ tan- 1 3 1 sin] ) is equal to ______

Question

If $2 \sin α \sin β + 3 \cos β + 5 \cos α \sin β = \sqrt{38} \forall α, β \in R,$ then $\det (adj (adj A))$ is equal to (where $A = [\begin{matrix} \tan β & - \frac{1}{3} \\ 1 & \sin α \end{matrix}]$ ) is equal to ______

Smart Achievers · Accepted Answer

(1)

$Let \vec{p} = 2 \hat{i} + 3 \hat{j} + 5 \hat{k}; \vec{q} = \sin α \sin β \hat{i} + \cos β \hat{j} + \cos α \sin β \hat{k}$

$| \vec{q} |$ $= \sqrt{\sin^{2} α \sin^{2} β + \cos^{2} β + \cos^{2} α \sin^{2} β} = 1$

$\Rightarrow \sin α \sin β = 2 λ; \cos β = 3 λ; \cos α \sin β = 5 λ$

$1 = 38 λ^{2}; λ = \frac{1}{\sqrt{38}}$

$\det A = [\frac{\sin α \sin β}{\cos β} + \frac{1}{3}] = 1$

Solution

Q.
If $2 \sin α \sin β + 3 \cos β + 5 \cos α \sin β = \sqrt{38} \forall α, β \in R,$ then $\det (adj (adj A))$ is equal to (where $A = [\begin{matrix} \tan β & - \frac{1}{3} \\ 1 & \sin α \end{matrix}]$ ) is equal to ______

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Solution

Q. If 2sinα sinβ+3cosβ+5cosα sinβ=38 ∀α,β∈R, then det(adj(adjA)) is equal to (where A=[tanβ-131sinα]) is equal to ______

Ans. (1) Let p→=2i^+3j^+5k^; q→=sinα sinβi^+cosβj^+cosα sinβk^ |q→|=sin2αsin2β+cos2β+cos2αsin2β=1 ⇒ sinαsinβ=2λ; cosβ=3λ; cosαsinβ=5λ 1=38λ2; λ=138 detA=[sinα sinβcosβ+13]=1

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Q.
If $2 \sin α \sin β + 3 \cos β + 5 \cos α \sin β = \sqrt{38} \forall α, β \in R,$ then $\det (adj (adj A))$ is equal to (where $A = [\begin{matrix} \tan β & - \frac{1}{3} \\ 1 & \sin α \end{matrix}]$ ) is equal to ______