\(B_2O_3\)
\(CO(BO_2)_2\)
\(COB_4O_7\)
\(CO[B_4O_5(OH)_4]\)
The question involves the borax bead test, a qualitative analysis technique used to identify certain metal ions based on the color of the bead formed when a metal compound is fused with borax in a flame. Let's go through the process to identify why the blue-colored bead is formed when borax is heated with cobalt oxide (CoO).
When borax \((\text{Na}_2\text{B}_4\text{O}_7 \cdot 10\text{H}_2\text{O})\) is heated, it loses water and forms sodium metaborate \((\text{NaBO}_2)\) and boric oxide \((\text{B}_2\text{O}_3)\):
\(\text{Na}_2\text{B}_4\text{O}_7 \cdot 10\text{H}_2\text{O} \rightarrow 2\text{NaBO}_2 + \text{B}_2\text{O}_3 + 10\text{H}_2\text{O}\)
The boric oxide \((\text{B}_2\text{O}_3)\) then reacts with the cobalt oxide \((\text{CoO})\) present on the platinum loop to form cobalt metaborate \((\text{Co(BO}_2)_2)\). This compound is responsible for the distinctive blue color of the bead:
\(\text{CoO} + \text{B}_2\text{O}_3 \rightarrow \text{Co(BO}_2)_2\)
Now, let's analyze why the correct answer is \(\text{Co(BO}_2)_2\):
Thus, the blue-colored bead is indeed due to the formation of cobalt metaborate, \(\text{Co(BO}_2)_2\).
Method used for separation of mixture of products (B and C) obtained in the following reaction is: 
In the following \(p\text{–}V\) diagram, the equation of state along the curved path is given by \[ (V-2)^2 = 4ap, \] where \(a\) is a constant. The total work done in the closed path is: 
Let \( ABC \) be a triangle. Consider four points \( p_1, p_2, p_3, p_4 \) on the side \( AB \), five points \( p_5, p_6, p_7, p_8, p_9 \) on the side \( BC \), and four points \( p_{10}, p_{11}, p_{12}, p_{13} \) on the side \( AC \). None of these points is a vertex of the triangle \( ABC \). Then the total number of pentagons that can be formed by taking all the vertices from the points \( p_1, p_2, \ldots, p_{13} \) is ___________.