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Which of the following statements are true? Please justify the answers.

1. a≥b⇔-a≤-b is an absolute inequality.

2. If A=φ,B={1,2},C={-1,-2}, then A×B×C has 4 elements.

3. The argument of 1+√(3)i is π/3.

4. A linear equation over R can have at most one root in C\R.

5. |x₁-x₂|=|x₁|-|x₂|∀x₁,x₂∈R

1. a≥b⇔-a≤-b is an absolute inequality.

2. If A=φ,B={1,2},C={-1,-2}, then A×B×C has 4 elements.

3. The argument of 1+√(3)i is π/3.

4. A linear equation over R can have at most one root in C\R.

5. |x₁-x₂|=|x₁|-|x₂|∀x₁,x₂∈R

Which of the following statements are true? Please justify the answers.

1. For any two sets A and B, A∩Bᶜ=A\B.

2. The matrix [1 1 is singular.

0 0]

3. The contrapositive of '∃ y∈Z such that P(y) is true' is '∃ x∈Z such that P(x) is true'.

4. The system 2x-3y=1 and 6y-4x+2=0 has a unique solution.

5. If x, y∈C such that x²=y and y²=x, then x=y=1.

1. For any two sets A and B, A∩Bᶜ=A\B.

2. The matrix [1 1 is singular.

0 0]

3. The contrapositive of '∃ y∈Z such that P(y) is true' is '∃ x∈Z such that P(x) is true'.

4. The system 2x-3y=1 and 6y-4x+2=0 has a unique solution.

5. If x, y∈C such that x²=y and y²=x, then x=y=1.

To prove that 2ⁿ>1+n√(2ⁿ⁻¹)∀n>2, using:

Cauchy-Schwarz Inequality;

Weierstrass Inequalities;

Tchebychev's Inequalities.

Cauchy-Schwarz Inequality;

Weierstrass Inequalities;

Tchebychev's Inequalities.

3 major projects P₁, P₂, P₃, are being funded by 3 voluntary agencies V₁, V₂, V₃. V₁, V₂, V₃, are willing to pay Rs. 8,000/-, Rs. 4,000/- and Rs. 2, 000/-, respectively per person on the project P₁; Rs. 4,000/-, Rs. 3,000/- and Rs. 4, 000/- respectively per person on P₂; and Rs. 3,000/-, Rs. 5,000/-, Rs. 8,000/- respectively on the project P₃. Further, the amount that V₁, V₂, V₃, have kept aside for paying people on these projects is Rs. 2,17,000/-, Rs. 1,42,000/- and Rs. 1,32,000/- respectively. How many people should each project employ so that the total money available is utilised?

Consider the equation E ≡ 5x-2y=3.

Write down equations E₁, E₂, E₃, respectively so that

1.) E and E₁ are inconsistent;

2.) E and E₂ have a unique solution;

3.) E and E₃ have infinitely many solutions.

Write down equations E₁, E₂, E₃, respectively so that

1.) E and E₁ are inconsistent;

2.) E and E₂ have a unique solution;

3.) E and E₃ have infinitely many solutions.

To apply Cramer's rule to solve the following system of equations:

2x₁+x₂+x₃=4

x₁-x₂+2x₃=2

3x₁-2x₂-x₃=0

2x₁+x₂+x₃=4

x₁-x₂+2x₃=2

3x₁-2x₂-x₃=0

To solve the linear system:

x+2y=4, 3x-y=2 .......(1)

by substitution.

If 4x+y=c, ax+by=3 is a system having the same solution set as (1), find a,b,c.

x+2y=4, 3x-y=2 .......(1)

by substitution.

If 4x+y=c, ax+by=3 is a system having the same solution set as (1), find a,b,c.

How to write the converse of the statement:

If p and q are the only roots of a polynomial f over C, then deg f = 2 .

If p and q are the only roots of a polynomial f over C, then deg f = 2 .

A prefilled bag of infusion fluid contains a concentration of 0.5% of the medication and you are asked to administer 50mgs.How many mls do you administer to give the required dose?

For a₁,....,aₙ∈R, a₁<a₂<....<aₙ, show that :

n/a₁-a₀+n-1/a₂-a₁+....+

1/aₙ-aₙ₋₁≥∑^n_k=1(k²/a_k)

n/a₁-a₀+n-1/a₂-a₁+....+

1/aₙ-aₙ₋₁≥∑^n_k=1(k²/a_k)