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\(\frac{(a - \frac{1}{a}) (a^{\frac{4}{3}} + \frac{1}{a^{\frac{2}{3}}})}{a^2 - \left(\frac{1}{a}\right)^2}\) \( = \frac{(a - \frac{1}{a}) \left(a^{\frac{4}{3}} + \frac{1}{a^{\frac{2}{3}}}\right)}{\left(a - \frac{1}{a}\right)\left(a + \frac{1}{a}\right)} \) \(= \frac{a^{\frac{4}{3}} + \frac{1}{a^{\frac{2}{3}}}}{a + \frac{1}{a}}\) \(= \frac{\frac{a^{2} + 1}{a^{\frac{2}{3}}}}{\frac{a^2 + 1}{a}}\) \(= \frac{1}{a^{\frac{2}{3}}} \cdot a\) \(= a^{1 - \frac{2}{3}} = a^{\frac{1}{3}}\) \(\boxed{a^{\frac{1}{3}}}\)

If k = -7 is put as l = -15, the equation x\(^4\) - kx\(^3\) + 10x\(^2\) + lx - 3 becomes x\(^4\) - (7x\(^3\)) + 10x\(^2\) + (15) -3 = x\(^4\) + 7x\(^3\) + 10x\(^2\) - 15x - 3 This equation is divisible by (x - 1) and (x + 2) with the remainder as 27 k = -7, l = -15

In < AFT: \(\frac{TF}{AF}\) = tan60º therefore, TF = xtan60º ..................1 In < BFT: \(\frac{TF}{BF}\) = tan45º TF = ( 2 + x )tan45º .......................2 equating equations 1 and 2 xtan60º = ( 2 + x )tan45º But tan60º = \(\sqrt3\) and tan45º = 1 x \(\sqrt3\) = 2 + x x \(\sqrt3\) - x = 2 x ( \(\sqrt3\) - 1) = 2 x = \(\frac{2}{(\sqrt3\)) - 1}\) rationalizing the denominator x = \(\frac{2(\sqrt3 + 2)}{3 - 1}\) = \(\frac{2(\sqrt3 + 1)}{2}\) = \(\sqrt3\) + 1

a = 2i - 3j - 3i + 6j = -i + 3j = \(\sqrt{5 \times 2}\) = \(\sqrt{10}\)

Let x be the no. \(\frac{3 + x}{

Possible multiples of 8 and 9 = 72, 144, 216, 288, 360, 432, 576, 648. \(\frac{387}{8}\) has remainder = 3; \(\frac{387}{9}\) has remainder 0. \(\frac{493}{8}\) has remainder 5; \(\frac{493}{9}\) has remainder 7. \(\frac{502}{8}\) has remainder 6; \(\frac{502}{9}\) has remainder 7. \(\frac{441}{8}\) has remainder 1; \(\frac{441}{9}\) has remainder 0. \(\frac{634}{8}\) has remainder 2; \(\frac{634}{9}\) has remainder 4. Hence 502 is the answer.

Angles ADE and EFC are similar, Therefore, = \(\frac{x}{3} = \frac{4}{2}\) = \(\frac{x}{3} = 2\) x = 3 x 2 = 6

x + y + z = 360^\circ (angles at a point)

The term "dark horse" is an idiomatic expression that refers to a little-known or less-favoured contender in a competition who unexpectedly achieves success. In the context of the sentence, describing the sub mariners as "the dark horse" suggests that they were not widely expected to win, but they surprised everyone by emerging victorious in the match against the uplanders team. Therefore, option C, "won unexpectedly," accurately conveys the meaning of the underlined portion in the sentence.

The expression "small fry" in this context is used as a metaphor to refer to unimportant or insignificant people. So, the sentence is saying that only the unimportant or insignificant individuals get punished for such social misdemeanors. Option B, "unimportant people," is the best fit because it conveys the idea that those who are not considered significant or influential in the situation are the ones facing punishment. The term "small fry" is often used to describe individuals of little importance or influence in a particular context.

Heart is in his/her mouth; If someone's heart is in their mouth, they are feeling extremely nervous feeling of being anxious and worried.

A leader in a newspaper is a piece of writing which gives the editor's opinion on an important news item.

A bore is generally an uninteresting person; a boring person. To talk or act in a way that makes someone lose interest:

The word "concurrently" means happening at the same time. In this context, it means that the sentences are served at the same time rather than one after the other.

The sentence suggests that the presentation has a sense of balance, harmony, and organization, which aligns with the meaning of "symmetry." Therefore, "orderliness" best captures this idea.