Multiple choice

The least value of $|a|$ for which $\tan q$ and $\cot q$ are the roots of the equation $x^2 + ax + b = 0$ is

  1. $2$
  2. $1$
  3. $1/2$
  4. $0$
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A Correct answer
Explanation

Roots are tan(q) and cot(q). Their product is tan(q) * cot(q) = 1. From the equation x^2 + ax + b = 0, the product of roots is b = 1. The sum of roots is tan(q) + cot(q) = -a. Since tan(q) + cot(q) = 2/sin(2q), its range is (-inf, -2] U [2, inf). Thus |a| >= 2. The least value is 2.

AI explanation

For the roots tan q and cot q, the product of the roots is (tan q)(cot q)=1. In the quadratic equation x^2+ax+b=0, the product of the roots equals b, so b=1. The sum of the roots is tan q + cot q, which can be written as (sin^2 q + cos^2 q)/(sin q cos q), simplifying to 1/(sin q cos q) or 2/sin 2q. This sum equals -a, making |a|=|2/sin 2q|. The minimum value of |sin 2q| is 1, so the least possible value of |a| is 2/1=2.

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