Solve by substitution, the system you have obtained in 5(a) .
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Find the polynomial over R of least degree which has i − 3 and
sqrt(7) + 5i as its roots.Solve x4 − 8x3 + 21x2 − 20x + 5 = 0 given that the sum of two of its roots is 4.
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.
Show that 1 + 1/sqrt(2)+ 1/sqrt(n) ≥ sqrt(2(n − 1)) , for n ∈ N, n > 1 .
Express z = 1/-5 − i in standard (algebraic) form. Further, give an Argand diagram in which z, –z and − z are plotted.
| Draw a Venn diagram of sets A, B and C where A ⊆ B, A ∩ C ≠ φ, B ∩ C = φ . |
| What is the universal set you have chosen? Justify your choice of sets in the diagram. |
Prove that 