a) Write the expansion of e(1 x) ; x <1 . [1] b) Write the total number of permutations of a set having n elements. [1] c) State De Moivre's theorem. [1] d) Write the sum of cubes of first n natural…

a) The expansion of ${\log }_e(1-x)$ for $|x|<1$ is given by the Taylor series expansion around $x=0$:

$${\log }_e(1-x)=-\sum _{k=1}^{\infty }\frac{x^k}{k}=-\left(x+\frac{x^2}{2}+\frac{x^3}{3}+\frac{x^4}{4}+\cdots \right)$$

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b) The total number of permutations of a set having $n$ distinct elements is given by the factorial of $n$, denoted as $n!$. Mathematically, it is expressed as:

$$n!=n\times (n-1)\times (n-2)\times \cdots \times 2\times 1$$

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c) De Moivre's Theorem states that for any integer $n$ and any real number $\theta$, the following holds:

$$(\cos \theta +i\sin \theta {)}^n=\cos (n\theta )+i\sin (n\theta )$$

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d) The sum of the cubes of the first $n$ natural numbers is given by the formula:

$$\sum _{k=1}^n{k}^3={\left(\frac{n(n+1)}{2}\right)}^2$$

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e) The given system of equations is: $$\{\begin{array}{l}3x+2y-1=0\\ 4x+y-3=0\end{array}$$

To write the augmented matrix, we first rewrite the equations in standard form:

$$\{\begin{array}{l}3x+2y=1\\ 4x+y=3\end{array}$$

The augmented matrix is:

$$\left[\begin{array}{cccc}3& 2& |& 1\\ 4& 1& |& 3\end{array}\right]$$