Question : Which of the following is a potash alum?
Option 1: $\mathrm{Na}_2 \mathrm{SO}_4 \mathrm{Al}_2\left(\mathrm{SO}_4\right)_3 \cdot 24 \mathrm{H}_2 \mathrm{O}$
Option 2: $\mathrm{K}_2 \mathrm{SO}_4 \mathrm{Al}_2\left(\mathrm{SO}_4\right)_3 .24 \mathrm{H}_2 \mathrm{O}$
Option 3: $\mathrm{K}_2 \mathrm{SO}_4 \mathrm{Al}_2\left(\mathrm{SO}_4\right)_3 .12 \mathrm{H}_2 \mathrm{O}$
Option 4: $\mathrm{Na}_2 \mathrm{SO}_4 \mathrm{Al}_2\left(\mathrm{SO}_4\right)_3 \cdot 20 \mathrm{H}_2 \mathrm{O}$
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Correct Answer: $\mathrm{K}_2 \mathrm{SO}_4 \mathrm{Al}_2\left(\mathrm{SO}_4\right)_3 .24 \mathrm{H}_2 \mathrm{O}$
Solution : The correct answer is K2SO4.Al2(SO4)3.24H2O.
Potash alum, sometimes known as potassium aluminium sulphate, is a chemical substance that is frequently found in the dodecahydrate form. It is a double salt that is frequently used to purify water and in medicine. Potash alum has the chemical formula K2SO4.Al2(SO4)3.24H2O.
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Question : What is the value of $ \left(\mathrm{k}-\frac{1}{\mathrm{k}}\right)\left(\mathrm{k}^2+\frac{1}{\mathrm{k}^2}\right)\left(\mathrm{k}^4+\frac{1}{\mathrm{k}^4}\right)\left(\mathrm{k}^8+\frac{1}{\mathrm{k}^8}\right)\left(\mathrm{k}^{16}+\frac{1}{\mathrm{k}^{16}}\right)\left(\mathrm{k}^{32}+\frac{1}{\mathrm{k}^{32}}\right)? $
Option 1: $\frac{\mathrm{k}^{64}-\frac{1}{\mathrm{k}^{64}}}{\mathrm{k}+\frac{1}{\mathrm{k}}}$
Option 2: $\frac{\mathrm{k}^{32}-\frac{1}{\mathrm{k}^{32}}}{\mathrm{k}-\frac{1}{\mathrm{k}}}\\$
Option 3: $\frac{\mathrm{k}^{32}-\frac{1}{\mathrm{k}^{32}}}{\mathrm{k}+\frac{1}{\mathrm{k}}}\\$
Option 4: $\frac{\mathrm{k}^{32}+\frac{1}{\mathrm{k}^{32}}}{\mathrm{k}+\frac{1}{\mathrm{k}}}$
Question : Which of the following statements is correct? I. The value of $100^2-99^2+98^2-97^2+96^2-95^2+94^2$ $- 93^2+\ldots \ldots+22^2-21^2$ is 4840. II. The value of $\left(\mathrm{k}^2+\frac{1}{\mathrm{k}^2}\right)\left(\mathrm{k}-\frac{1}{\mathrm{k}}\right)\left(\mathrm{k}^4+\frac{1}{\mathrm{k}^4}\right)\left(\mathrm{k}+\frac{1}{\mathrm{k}}\right)\left(\mathrm{k}^4-\frac{1}{\mathrm{k}^4}\right) \text { is } \mathrm{k}^{16}-\frac{1}{\mathrm{k}^{16}}$.
Option 1: Neither I nor II
Option 2: Both I and II
Option 3: Only II
Option 4: Only I
Question : Which of the following statements is correct? I. The value of $100^2-99^2+98^2-97^2+96^2-95^2+$ $94^2-93^2+\ldots \ldots+22^2-21^2$ is 4840. II. The value of $\left(\mathrm{k}^2+\frac{1}{\mathrm{k}^2}\right)\left(\mathrm{k}-\frac{1}{\mathrm{k}}\right)\left(\mathrm{k}^4+\frac{1}{\mathrm{k}^4}\right)\left(\mathrm{k}+\frac{1}{\mathrm{k}}\right)\left(\mathrm{k}^4-\frac{1}{\mathrm{k}^4}\right)$ is $\mathrm{k}^{16} - \frac{1}{\mathrm{k}^{16}}$.
Option 2: Only II
Option 3: Only I
Option 4: Both I and II
Question : If $\frac{(17)^3+(7)^3}{\left(17^2+7^2-\mathrm{k}\right)}=24$, then what is the value of $\mathrm{k}?$
Option 1: 119
Option 2: 128
Option 3: 24
Option 4: 109
Question : Evaluate the following: $\cos \left(36^{\circ}+A\right) \cdot \cos \left(36^{\circ}-A\right)+\cos \left(54^{\circ}+A\right) \cdot \cos \left(54^{\circ}-A\right)$
Option 1: $\sin 2A$
Option 2: $\cos A$
Option 3: $\sin A$
Option 4: $\cos 2A$
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