A quadratic equation whose roots are $ \displaystyle \text{cosec}^{2} \theta $ and $\displaystyle \sec ^{2}\theta $ can be
- $\displaystyle x^{2}-5x+2= 0$
- $\displaystyle x^{2}-2\sin 2\theta x+6\sin \theta= 0$
- $\displaystyle x^{2}-4\text{cosec} 2\theta x+4\text{cosec} 2\theta= 0$
-
none of these
Roots are csc^2(theta) and sec^2(theta). Sum = 1/sin^2 + 1/cos^2 = (sin^2 + cos^2) / (sin^2*cos^2) = 1 / (sin^2*cos^2) = 4 / sin^2(2*theta) = 4*csc^2(2*theta). Product = 1 / (sin^2*cos^2) = 4*csc^2(2*theta). Equation: x^2 - (sum)x + product = 0 => x^2 - 4*csc^2(2*theta)x + 4*csc^2(2*theta) = 0. Option C is x^2 - 4*csc(2*theta)x + 4*csc(2*theta) = 0, which is missing the square on csc.
The sum of the roots is cosec^2 theta + sec^2 theta, which simplifies to 1/(sin^2 theta cos^2 theta) or 4 cosec^2 2 theta. The product of the roots is cosec^2 theta sec^2 theta, which also simplifies to 4 cosec^2 2 theta. For a valid quadratic equation, the constant term must equal the coefficient of the x term, a condition not met by either x^2 - 5x + 2 = 0 or the other provided polynomials. Consequently, none of the explicit equations match the required conditions.