Let 𝑄∗ be the set of non-zero rational numbers and * be the binary operation on Q definedby 𝑎 ∗ 𝑏 = 𝑎𝑏/3. Prove that 𝑄∗ is group under *.

If 𝑎 + 𝑏 + 𝑐 = 5 𝑎𝑛𝑑 𝑎𝑏 + 𝑏𝑐 + 𝑐𝑎 = 10, then prove that 𝑎3 + 𝑏3 + 𝑐3 − 3𝑎𝑏𝑐 = −25.**************************

Let a⃗ =2i^+2j^+k^𝑎→=2𝑖^+2𝑗^+𝑘^ and b⃗ 𝑏→ be another vector such that a⃗ .b⃗ =14𝑎→.𝑏→=14 and a⃗ ×b⃗ =3i^+j^−8k^𝑎→×𝑏→=3𝑖^+𝑗^−8𝑘^ the vector b⃗ 𝑏→ =

Suppose that 𝑎+1𝑏a+ b1​  and 𝑏+1𝑎b+ a1​  are the roots of the equation 𝑥2−𝑝𝑥+𝑞=0x 2 −px+q=0. If 𝑎𝑏=1ab=1, what is the value of 𝑞q?

If 𝑓(𝑥) = 𝑠𝑖𝑛−1𝑥, 0 < 𝑎 < 𝑏 < 1, use Mean value theorem to prove that𝑏 − 𝑎√1 − 𝑎2 < 𝑠𝑖𝑛−1𝑏 − 𝑠𝑖𝑛−1𝑎 < 𝑏 − 𝑎√1 − 𝑏2

Let p ∈ Q be a rational number such that p > 1 and letq := pp − 1 .Prove that for all a, b ≥ 0,ab ≤ app + bqq