Prove that  

Show that  

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.

two non-singular 2×2 matrices C and D, with 

two non-zero, 3×3 matrices A and B , with

Solve by substitution, the system you have obtained in 5(a).

Apply the Gaussian elimination process to determine values of A for which the following linear system is consistent:

The annual bonus given to the employees of a company is 10% of their taxable incomes, after the state and central taxes are deducted. The state tax is 15% of taxable income. The central tax is 15% of taxable income after deducting the state tax. Formulate this situation for determining the bonus, as a linear system.

Solve x²-8x²+21x2-20x+5=0 given that the sum of two of its roots is 4.

Obtain the discriminant of the equation  Hence provide the nature of its roots

Find the polynomial over R of least degree which has  as its roots.  as its roots.

Find the sum of the 5^th roots of unity 

Apply De Moivre’s theorem to write  ( 5 + )i in the form a + ib , with a . b∈ R

Obtain the polar and exponential representations of  where    

Express   in standard (algebraic) form. Further, give an Argand diagram

in which  and are plotted

Find x and y , given that  

Draw a Venn diagram of sets B,A and C where 

What is the universal set you have chosen? Justify your choice of sets in the diagram.

For any two sets A and B , in a universal set U , 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.

ii)   for any three sets B,A and C .

iii)  

iv)   , has a root in R only if m is an odd number.

v) 

vi) Any system of three linear equations in two variables has no solution

vii) If a matrix has 2 n entries, where n ∈ N , then it is a square matrix.

viii) For n > 4 , the AM of the first n natural numbers is greater than n +1

Explain the natural drivers of climate change