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#### CaptainBlack

##### Well-known member

- Jan 26, 2012

- 890

Just to show that not everything in a STEP paper in difficult, this is an easy question:

Which of the following are true and which false? Justify your answers

(i) \(a^{\ln(b)}=b^{\ln(a)}\), for all \(a,b \gt 0\).

(ii) \(\cos(\sin(\theta))=\sin(\cos(\theta))\), for all real \(\theta\).

(iii) There exists a polynomial \(P\) such that \(|P(\theta)-\cos(\theta)| \lt 10^{ -6 } \) for all real \(\theta\)

(iv) \(x^4+3+x^{-4} \ge 5\) for all \(x\gt 0\).

Which of the following are true and which false? Justify your answers

(i) \(a^{\ln(b)}=b^{\ln(a)}\), for all \(a,b \gt 0\).

(ii) \(\cos(\sin(\theta))=\sin(\cos(\theta))\), for all real \(\theta\).

(iii) There exists a polynomial \(P\) such that \(|P(\theta)-\cos(\theta)| \lt 10^{ -6 } \) for all real \(\theta\)

(iv) \(x^4+3+x^{-4} \ge 5\) for all \(x\gt 0\).

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