Question

On the basis of the given polynomial equation

$x^4 - (\sin\theta + \cos\theta)x^3 + (\sin\theta\cos\theta - 1)x^2 + (\sin\theta + \cos\theta)x - \sin\theta\cos\theta = 0$.

Answer the following:

6. Sum of any 2 roots of the equation can never exceed -

7. Product of the roots of the equation will always be more than -

8. Number of real roots of the equation is/are -

• A. -1
• B. 0
• C. 1
• D. 2

Answer 1

Answer: (D)

2

Answer 2

The roots of the equation are $1, -1, \sin\theta, \cos\theta$. The largest possible sum of any two roots is obtained by maximizing $1 + \sin\theta$ or $1 + \cos\theta$, which occurs when $\sin\theta = 1$ or $\cos\theta = 1$. In either case, the maximum sum is $1 + 1 = 2$.