Question

If \( \csc\theta \) and \( \cot\theta \) are the roots of \( cx^2 + bx + a = 0 \) where \( bc \neq 0 \), then evaluate \( b^2(b^2 - 4ac) \):

• A. \( -2c^4 \)
• B. \( 2c^4 \)
• C. \( -c^4 \)
• D. \( c^4 \)

Hint:

Try relating trigonometric roots to algebraic identities and use Vieta’s formulas with discriminants.

Answer 1

The Correct Option is D

Step 1: Use Vieta's formulas
Let the roots be \( \alpha = \csc\theta \), \( \beta = \cot\theta \)
Sum of roots: \( \alpha + \beta = -\frac{b}{c} \)
Product of roots: \( \alpha\beta = \frac{a}{c} \)

Step 2: Use identity
We know: \[ \csc^2\theta - \cot^2\theta = 1 \Rightarrow (\csc\theta + \cot\theta)(\csc\theta - \cot\theta) = 1 \] This helps relate terms but a key insight is to assume identities and solve using discriminant \( D = b^2 - 4ac \), then compute: \[ b^2(b^2 - 4ac) = c^4 \]