If the volume of the solid in \( \mathbb{R}^3 \) bounded by the surfaces \[ x = -1, \ x = 1, \ y = -1, \ y = 1, \ z = 2, \ y^2 + z^2 = 2 \] is \( \alpha - \pi \), then \( \alpha = \text{.........}. \)
The value of the integral \[ \int_0^1 \int_x^1 y^4 e^{x y^2} \, dy \, dx\] is .........… (correct up to three decimal places).
If \( a = \int_{\pi/3}^{\pi/6} \dfrac{\sin t + \cos t}{\sqrt{\sin 2t}} \, dt, \) then the value of \( \left( 2 \sin \frac{\alpha}{2} + 1 \right)^2 \text{ is ............}. \)
\[ T(P) = QP. \]
The area of the parametrized surface \[ S = \left\{ \left( (2 + \cos u) \cos v, (2 + \cos u) \sin v, \sin u \right) \in \mathbb{R}^3 \mid 0 \leq u \leq \frac{\pi}{2}, 0 \leq v \leq \frac{\pi}{2} \right\} \] is .................. (correct up to two decimal places).
If \( x(t) \) is the solution to the differential equation \( \frac{dx}{dt} = x^2 t^3 + x t, \text{ for } t > 0, \text{ satisfying } x(0) = 1, \) then the value of \( x(\sqrt{2}) \) is .......... (correct up to two decimal places).
If \( y(x) = v(x) \sec x \) is the solution of \[ y'' - (2 \tan x) y' + 5y = 0, -\frac{\pi}{2} < x < \frac{\pi}{2}, \text{ satisfying } y(0) = 0 \text{ and } y'(0) = \sqrt{6}, \] then \( v \left( \frac{\pi}{6 \sqrt{6}} \right) \) is .............. (correct up to two decimal places).
The orientations of the fold axis and axial plane in the given figure indicate
Identify the rocks P and Q in the diagram as per the IUGS classification.
Which of the following statement(s) is/are correct for the upper hemisphere stereographic projection of a crystal given below?
