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For my answer I got 4.5 and when you sketch -1-4i on the Argand diagram, it is in third quadrant. So when I did arctan(4), I got 1.3 and then I did pi+ 1.3 which gave me 4.467.. However, in the answer it says that it is 1.3. I am unsure whether there is a mistake in the mark-scheme or I did something wrong.

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The conjugate is $\bar{w}=-1+4i$, so the product is $w\bar{w}=1+16=17$; the modulus is $\sqrt{17}\approx 4.1231$.

The argument should certainly be an angle in the third quadrant. You have correctly done $$ \arctan 4\approx 1.3258 $$ and the argument is $$ \pi+\arctan 4\approx 4.4674 $$

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