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Match each entry in List - I to the correct entries in List - II
LIST-I LIST-II
I) The tangents of two acute angles $α and β$ are 3 and 2. The difference of $\sqrt{50} | \cos (α - β) - \sin (α - β) |$ P) 18
II) $\sin θ + \cos θ = a and \sin^{3} θ + \cos^{3} θ = b$ then we get $λ a^{3} + μ b + V a = 0$ then the value $| λ^{3} + μ^{3} + V^{3} |$ where $λ, μ, V$ are independent of $θ$ Q) 24
III) $\begin{matrix} \sin α = 4 / 5 & \cos β = 3 / 5 \\ 0 < α < π / 2 & 0 < β < π / 2 \end{matrix}$
The value $25 \sin (α + β)$ R) 6
IV) Using this $a \cos x + b \sin x + c = 0$ range is $[c - \sqrt{a^{2} + b^{2}}, c + \sqrt{a^{2} + b^{2}}]$ . The number of integral value of $k$ for which the equation $7 \cos x + 5 \sin x = 2 k + 1$ has a solution S) 8

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$I - R, I I - S, I I I - P, I V - Q$

$I - R, I I - P, I I I - Q, I V - S$

$I - P, I I - Q, I I I - R, I V - S$

$I - Q, I I - R, I I I - P, I V - S$

answer is B.

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Detailed Solution

I. $\begin{matrix} \tan α = 3 & \tan β = 2 \end{matrix}$
$\begin{matrix} \sin α = \frac{3}{\sqrt{10}} & \cos α = \frac{1}{\sqrt{10}} & \sin β = \frac{2}{\sqrt{5}} & \cos β = \frac{1}{\sqrt{5}} \end{matrix}$
$\sqrt{50} | \frac{1}{\sqrt{50}} + \frac{60}{\sqrt{50}} - \frac{3}{\sqrt{50}} + \frac{2}{\sqrt{50}} | = 6$
II. $\begin{matrix} \sin θ + \cos θ = a & \sin^{3} θ + \cos^{3} θ = b \end{matrix}$
$\sin^{2} θ + \cos^{2} θ + 2 \sin θ \cos θ = a^{2}$
$\sin θ \cos θ = \frac{a^{2} - 1}{2} \sin^{3} θ + \cos^{3} θ + 3 \sin θ \cos θ (\sin θ + \cos θ) = a^{3} b + 3 a \sin θ \cos θ = a^{3} \sin θ \cos θ = \frac{a^{3} - b}{3 a} \begin{matrix} \frac{a^{3} - b}{3 a} = \frac{a^{2} - 1}{2} & 2 a^{3} - 2 b = \end{matrix} 3 a^{3} - 3 a λ = 1, μ = 2, V = - 3 λ^{3} + μ^{3} + V = - 18$
III. $\sin (α + β) = \frac{24}{25}$
IV. $- \sqrt{74} \leq 7 \cos x + 5 \sin x \leq \sqrt{74}$
$- \sqrt{74} < 2 k + 1 < \sqrt{74}$
$K = - 4, \pm 3, \pm 2, \pm 1, 0$
No. of values of $K$ is 8

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