Apply the Gaussian elimination process to determine values of λ for which the following linear system is consistent:
x − 3y + 4 = 0, 3x − 2y = λ, y = 6 − 2x .
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Find the polynomial over R of least degree which has i − 3 and
sqrt(7) + 5i as its roots.
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 .
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.
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 x4 − 8x3 + 21x2 − 20x + 5 = 0 given that the sum of two of its roots is 4.