Is Cramer’s Rule applicable for solving the linear system below? If yes, apply it.
Otherwise, alter the last equation in the system so that the solution can be obtained
by applying the Rule.
x + y + z = π
− πx + πy +sqrt(2z) = 0
π2 x + π2 y + 2z = 0 .
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Apply De Moivre’s theorem to write (sqrt(3 + 1)5 in the form a + ib , with a, b ∈ R
Solve x4 − 8x3 + 21x2 − 20x + 5 = 0 given that the sum of two of its roots is 4.
Find the roots of the equation 2x3- x2 − 22x − 24 = 0 , if two of them are in the ratio 3 : 4 .
The annual bonus given to the employees of a company is 5% of their taxable incomes, after the state and central taxes are deducted. The state tax is 10% of taxable income. The central tax is 20% of taxable income after deducting the state tax. Formulate this situation for determining the bonus, as a linear system.
Prove that 
Which of the following statements are true? Justify your answers. (This means that if you think a statement is false, give a short proof or an example that shows it is false. If it is true, give a short proof for saying so. For instance, to show that ‘{1, padma, blue} is a set’ is true, you need to say that this is true because it is a well-defined collection of 3 objects.)
i) The collection of Venn diagrams is a set.
iii) If z ∈ R , then | z | = z .
vi) Any system of three linear equations in two variables has no solution.
vii) If a matrix has n2 entries, where n ∈ N , then it is a square matrix.
viii) For n > 2 , the AM of the first n natural numbers is greater than n + 1.
Solve by substitution, the system you have obtained in 5(a) .