Complex Numbers
Max-min of real-imaginary expression
nta_pyq_2025_apr
Grade 12
Question:
Let the product of$\$omega = (8 + i)$sin$$\$theta + (7 + 4i)$cos$$\theta and$$\$omega = (1 + 8i)$sin$$\$theta + (4 + 7i)$cos$$\theta be$$\alpha + i$$\beta, 1 2 i =$$\$sqrt-1$. Let p and q be the maximum and the minimum values of$$\alpha +$$\beta respectively.$
Step-by-Step Solution
Key Concept: Expand the product, express$\alpha+\beta$in terms of$\sin2\theta$and$\cos2\theta$, and use amplitude for$max-min.$
$\$omega1 = (8$sin$$\$theta + 7$cos$$\theta) + i(sin$$\$theta + 4$cos$$\theta)$$\$omega2 = (si$n$$\$theta + 4$cos$$\theta) + i(8 sin$$\$theta + 7$cos$$\theta)$(2)$2$$\omega1$$\$omega2 = 8$sin$$\$theta + 7$sin$$\theta cos$$\$theta + 32$sin$$\theta cos$$\$theta+ 2$2 28 cos$$\$theta - 8$sin$$\$theta - 32$sin$$\theta cos$$\$theta - 7$sin$$\theta cos$$\theta 2$2 - 28$cos$$\$theta + i$(sin$$\$theta + 4$sin$$\theta cos$$\$theta + 4$sin$$\theta cos$$\theta 2$2 + 16$cos$$\$theta + 64$sin$$\$theta + 56$sin$$\theta cos$$\$theta + 56$sin$$\theta 2 cos$$\$theta + 49$cos$$\theta) 2 2$$\omega1$$\$omega2 = 0 + i$(65 sin$$\$theta + 120$sin$$\theta cos$$\$theta + 65$cos$$\theta)$$\alpha +$$\$beta = 65 + 60$sin 2q$$\alpha +$$\beta$| = 125 max$\alpha +$$\beta$| = 5 min Ans. =$125 + 5 = 130$option$(2)$
Correct Answer: 2