Practicing Success
For all values of $θ∈(0,π/2)$ the determinant of the matrix $\begin{bmatrix}-2&\tan θ+ \sec^2θ&3\\-\sin θ&\cos θ&\sin θ\\-3&-4&3\end{bmatrix}$ always lies in the interval |
$[\frac{7}{2},\frac{21}{4}]$ $(3,5]$ $(4, 6)$ $(\frac{5}{2}, \frac{19}{4})$ |
$(3,5]$ |
Let f(θ) be the determinant of the given matrix. Then $f(θ)=\begin{bmatrix}-2&\tan θ+ \sec^2θ&3\\-\sin θ&\cos θ&\sin θ\\-3&-4&3\end{bmatrix}$ $⇒f(θ)=\begin{bmatrix}1&\tan θ+ \sec^2θ&3\\0&\cos θ&\sin θ\\0&-4&3\end{bmatrix}$ [Applying $C_1 → C_1 +C_3$] $⇒f(θ)= 3 \cos θ+4 \sin θ ⇒f(θ)= 5 \sin θ(θ+\tan^{-1}\frac{3}{4})$ Now, $θ∈(0,\frac{π}{2})$ $⇒θ+\tan^{-1}\frac{3}{4}∈(\tan^{-1}\frac{3}{4},\frac{π}{2}+\tan^{-1}\frac{3}{4})$ $⇒min\,f(θ)= 5\sin(\tan^{-1}\frac{3}{4})$ and, $max\,f(θ)= 5\sin\frac{π}{2}$ $⇒min\,f(θ)= 5\sin(\sin^{-1}\frac{3}{5})=3$ and, $max\,f(θ)= 5$ $⇒f(θ)∈(3,5]$ |