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5. Define a relation on Q-{0} as follows: In -= 2for some keZ. Prove that this is an equivalence relation.
[1] 1. (a) Solve 2 β 31 < 7. Solution: We have β 7<-3 < 7 which gives -4 0 on (-0, 1] U [2,). Next, we need to see where Vr2 β 3r +2 -1 > 0. If r2 - 3.c + 2 β 1 > 0, then V.22 β 3.0 +2 > . Observe that this is definitely true if r < 0. If r > 0, then squar- ing both sides we get 22 - 3.+2 > 1 2 > 3. V A VICO Thus, the domain of f is (-00, ). [2] (c) Find the exact value of tan(sec-14). Solution: Let y = sec-14. Then, sec y = 4. From the diagram we get that tan(sec-? 4) = tan y = 15 [2] (d) Find all r such that sin(2.x) = COS I. Solution: We have 2 sin r cos r = COS I, SO 0 = 2 sin Cos I -Cos I = cos .r (2 sin c - 1). Hence, sin(2x) = cost when cos x = 0, or when sin r = Thus, r= +ik, kez, I= +2nk, ke Z, or r= +2nk, kez [2] (e) State the precise mathematical definition of lim f(x) = 0. Solution: For every e > 0 there exists a 8 >0 such that if 0 < x-al <, then f(c) >
[4] (c) lim *+ 5 2 - 5x + 6 -5
