Consider the following group under matrix multiplication:
\[ H = \left\{ \begin{bmatrix} 1 & p & q \\ 0 & 1 & r \\ 0 & 0 & 1 \end{bmatrix} : p, q, r \in \mathbb{R} \right\}. \]
Then the center of the group is isomorphic to
The radius of convergence of the power series \( \displaystyle \sum_{n=1}^{\infty} \left( \dfrac{n+2}{n} \right)^{n^2} x^n \) is
If \( u = x^3 \) and \( v = y^2 \) transform the differential equation \( 3x^5 dx - y(y^2 - x^3) dy = 0 \) to \( \dfrac{dv}{du} = \dfrac{\alpha u}{2(u - v)} \), then \( \alpha \) is
Let \( s_n = 1 + \dfrac{(-1)^n}{n}, \, n \in \mathbb{N}. \) Then the sequence \(\{s_n\}\) is
Identify the type of fault in the given figure.
Match the primary sedimentary structure in Group I with the corresponding process of formation in Group II. \[\begin{array}{|l|l|} \hline \textbf{Group I} & \textbf{Group II} \\ \hline \text{P. Asymmetric ripples} & \text{1. Water escape} \\ \hline \text{Q. Dish and pillar} & \text{2. Bed load transportation of sediments} \\ \hline \text{R. Flute cast} & \text{3. Deposition from alternate traction and suspension load} \\ \hline \text{S. Wavy bedding} & \text{4. Scouring by turbulent eddy} \\ \hline \end{array}\]
Match the mineral/metal deposit in Group I with the corresponding Indian occurrence in Group II. \[\begin{array}{|l|l|} \hline \textbf{Group I} & \textbf{Group II} \\ \hline \text{P. Diamond} & \text{1. Malanjkhand} \\ \hline \text{Q. Iron} & \text{2. Wajarkaru} \\ \hline \text{R. Fluorite} & \text{3. Dalli Rajhara} \\ \hline \text{S. Copper} & \text{4. Ambadongar} \\ \hline \end{array}\]
Match the name of granitoid in Group I with the corresponding Craton in Group II. \[\begin{array}{|l|l|} \hline \textbf{Group I} & \textbf{Group II} \\ \hline \text{P. Closepet} & \text{1. Singhbum} \\ \hline \text{Q. Berach} & \text{2. Bastar} \\ \hline \text{R. Dongargarh} & \text{3. Dharwar} \\ \hline \text{S. Mayurbhanj} & \text{4. Aravalli} \\ \hline \end{array}\]
A northerly dipping fault (F-F) has displaced beds P, Q and R. The thickness of the beds across the fault is same. Identify the fault type(s).
A bedding plane, pictorially represented at X, will be plotted in stereonet as
A schematic deformation (D)-metamorphism (M)-time map of a metamorphic belt is shown below. The belt recorded two phases of deformation, D₁-D₂ and attendant schistosity, S₁-S₂ and metamorphism, M₁-M₂. The metamorphic minerals stable during these events are shown by solid lines. On the basis of this information, which of the following statement(s) is/are correct?
