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Answer:
Section - A
1. The expression \(\left(\frac{5}{3}\right)^{-3} \times\left(\frac{1}{4}\right)^{-11}=\left(\frac{9}{5}\right)^{2 \pi}\) can be simplified as follows:
\(\left(\frac{5}{3}\right)^{-3} \times\left(\frac{1}{4}\right)^{-11} = \left(\frac{3}{5}\right)^{3} \times \left(\frac{4}{1}\right)^{11} = \left(\frac{3}{5}\right)^{3} \times 4^{11} = \frac{27}{125} \times 4194304 = 14680064\).
Therefore, the value of \(x\) is 14680064.
2. To find the number by which \((-12)^{-1}\) should be divided to get \(\left(\frac{2}{5}\right)^{-1}\), we can rewrite the division as multiplication by the reciprocal:
\((-12)^{-1} \div \left(\frac{2}{5}\right)^{-1} = (-12)^{-1} \times \left(\frac{5}{2}\right)^{1} = \frac{1}{-12} \times \frac{5}{2} = \frac{-5}{24}\).
Therefore, the number by which \((-12)^{-1}\) should be divided is \(-\frac{5}{24}\).
3. To express \(\left\{\left(\frac{3}{4}\right)^{-1}-\left(\frac{1}{4}\right)^{-1}\right\}^{-1}\) as a rational number, we can simplify the expression inside the curly brackets first:
\(\left(\frac{3}{4}\right)^{-1}-\left(\frac{1}{4}\right)^{-1} = \frac{4}{3} - 4 = \frac{4}{3} - \frac{12}{3} = \frac{-8}{3}\).
Taking the reciprocal of \(\frac{-8}{3}\), we get \(-\frac{3}{8}\).
Therefore, the expression is equal to \(-\frac{3}{8}\).
4. Given \(x=\left(\frac{3}{3}\right)^{2} \times\left(\frac{z}{3}\right)^{-1}\), we can simplify it as follows:
\(x=\left(\frac{3}{3}\right)^{2} \times\left(\frac{z}{3}\right)^{-1} = 1^{2} \times \frac{3}{z} = \frac{3}{z}\).
To find the value of \(x^{-2}\), we take the reciprocal and square it:
\(x^{-2} = \left(\frac{3}{z}\right)^{-2} = \left(\frac{z}{3}\right)^{2} = \frac{z^{2}}{9}\).
Therefore, the value of \(x^{-2}\) is \(\frac{z^{2}}{9}\).
5. The volume of a rectangular box is given by the formula \(V = l \times b \times h\), where \(l\) is the length, \(b\) is the breadth, and \(h\) is the height. In this case, the length is \(2ax\), the breadth is \(3by\), and the height is \(scz\). Therefore, the volume is:
\(V = (2ax) \times (3by) \times (scz) = 6abcsxyz\).
Therefore, the volume of the rectangular box is \(6abcsxyz\).
Section - C
6. Using Euler's formula, we can find the unknown values in the given table:
- Faces: \(?\), 5, 20
- Vertices: 6, \(?\), 12
- Edges: 12, 9, \(?\)
According to Euler's formula, \(F + V - E = 2\), where \(F\) is the number of faces, \(V\) is the number of vertices, and \(E\) is the number of edges.
From the table, we can substitute the known values into the formula:
\(?\ + 6 - 12 = 2\)
\(?\ - 6 = 2\)
\(?\ = 8\)
Therefore, the unknown value in the table is 8.
7. To find the value of \(p\) in the equation \(\left(\frac{2}{5}\right)^{3} \times\left(\frac{2}{4}\right)^{-6}=\left(\frac{1}{