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.

Show 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) (A\B) UC=A\(B∩C) for any three sets A, B and C.

iii) If ze R, then |z|=z.

iv) x^m + a₁ x^{m -1}+ ....a_m-1 x + a_m = 0, a_i ∈ R∀ i = 1, ..., m, has a root in R only if m is an odd number.

v) x > 0 is necessary for x+3>1.

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 > 4, the AM of the first n natural numbers is greater than n +1.

Find the roots of the equation 2x³-x2-22x-24=0, if two of them are in the ratio 3:4.

Write an odd natural number as a sum of two integers m, and m₂, in a way that m₁m, is maximum.

किसी विषम प्राकृत संख्या को दो पूर्णांकों m₁, और m₂ के योग के रूप में इस प्रकार लिखिए कि ,m₁m₂ अधिकतम हो।

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.