Multiple choice

The least value of | a | for which sin θ and cosec θ are the roots of the equation x2 + ax + b = 0 is

  1. 2

  2. 1

  3. 1/2

  4. 0

  5. None of these
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A Correct answer
Explanation

Let the roots be sin(theta) and cosec(theta). Their product is sin(theta) * cosec(theta) = 1, so b = 1. Their sum is -a = sin(theta) + cosec(theta). Since |sin(theta) + cosec(theta)| >= 2 for all real theta, |a| must be at least 2.

AI explanation

If sin and cosec are the roots of x2 + ax + b = 0, their product is sin times cosec, which equals 1. This makes the constant term b equal to 1, simplifying the equation to x2 + ax + 1 = 0. The sum of the roots equals -a, so a = -(sin + cosec). The absolute value is |a| = |sin + 1/sin|. When sin is positive, the expression becomes sin + 1/sin, which is always greater than or equal to 2 by the AM-GM inequality. When sin is negative, it becomes -(sin + 1/sin), which is also always greater than or equal to 2. The result is 2.

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