(a) If \(^{k}P_{2} = 72\), find the value of k. (b) Solve the equation : \(2\cos^{2} \theta - 5\cos \theta = 3; 0° \leq \theta \leq 360°\)
Question 1 Report
(a) If \(^{k}P_{2} = 72\), find the value of k.
(b) Solve the equation : \(2\cos^{2} \theta - 5\cos \theta = 3; 0° \leq \theta \leq 360°\)
Answer Details
(a) \(^{k}P_2 = k(k - 1) = 72\):
\[k^2 - k - 72 = 0 \ \Rightarrow\ (k - 9)(k + 8) = 0\]Since \(k > 0\), \(k = 9\).
(b) \(2\cos^2\theta - 5\cos\theta = 3\). Let \(c = \cos\theta\):
\[2c^2 - 5c - 3 = 0 \ \Rightarrow\ (2c + 1)(c - 3) = 0\] \[c = -\tfrac{1}{2} \quad\text{or}\quad c = 3\ (\text{rejected, since } |\cos\theta| \le 1)\]So \(\cos\theta = -\tfrac{1}{2}\). In \(0^\circ \le \theta \le 360^\circ\):
\[\theta = 120^\circ \quad\text{and}\quad \theta = 240^\circ\]Download The App On Google Playstore
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