The top of a hill observed from the top and bottom of a building of height h is at angles of elevation \( \frac{\pi}{6} \) and \( \frac{\pi}{3} \) respectively. What is the height of the hill?
If \( \theta = \frac{\pi}{8} \), then what is the value of \( (2\cos\theta + 1)^{10}(2\cos 2\theta - 1)^{10}(2\cos\theta - 1)^{10}(2\cos 4\theta - 1)^{10} \)?
If \( \cos\alpha \) and \( \cos\beta \) \( (0 < \alpha < \beta < \pi) \) are the roots of the quadratic equation \( 4x^{2} - 3 = 0 \), then what is the value of \( \sec\alpha \times \sec\beta \)?
Consider the following values of x:
1. 8
2. −4
3. \( \frac{1}{6} \)
4. \( -\frac{1}{4} \)
Which of the above values of x is/are the solution(s) of the equation \( \tan^{-1}(2x) + \tan^{-1}(3x) = \frac{\pi}{4} \)?
The ratio of roots of the equations \( ax^{2} + bx + c = 0 \) and \( px^{2} + qx + r = 0 \) are equal. If \( D_{1} \) and \( D_{2} \) are respective discriminants, then what is \( \frac{D_{1}}{D_{2}} \) equal to?
