Find x and y , given that (5x + y, 3x − y) = (2, 2x) ∈ Q × Q .
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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 .
Write an odd natural number as a sum of two integers m1 and m2 in a way that m1m 2 is maximum.
Obtain the polar and exponential representations of z1 , z2 and z1z 2 , where zi = 1/2 - 2i and z2 = 3 + i
Express z = 1/-5 − i in standard (algebraic) form. Further, give an Argand diagram in which z, –z and − z are plotted.
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 
For any two sets A and B , in a universal set U , prove that
A ⊆ B ⇔ A ∪ B = B .