Algebra Answers

Let A = x{ ∈Z x is a multiple of 5} and B = x{ ∈Z x is a divisor of 20}.
Represent B,A and A B
c ∩ by the listing method and in a Venn diagram

Apply Cardano’s method for finding the roots of .0 2x 3x

Give a direct proof, as well as a proof by contradiction, of the following statement:
‘ B A ∩ B ⊆ A ∪ for any two sets A and B .’

Solve the linear system
2 x + 2y = ,4 3x − y = …………….. (I)
by substitution.
If 3 4x + y = ax,c + by = is a system having the same solution set as (I),
find c,b,a .

Apply Cramer’s rule to solve the following system of equations:

2x x x 4 1 + 2 + 3 =
x x 2x 2 1 − 2 + 3 =
3x 2x x 0 1 − 2 − 3 =

Consider the equation 3 E ≡ 5x − 2y = .
Write down equations E , respectively so that 1 E,
2 E,
3
i) E and E are inconsistent; 1
ii) E and E have a unique solution; 2
iii) E and E have infinitely many solutions.

3 major projects P are being funded by 3 voluntary agencies 1 P,
2 P,
3 1 2 V3 V . ,V ,
1 2 V3 V are willing to pay Rs. 8,000/-, Rs. 4,000/- and R ,V , s. 2, 000/-, respectively per
person on the project P ; Rs. 4,000/-, Rs. 3,000/- and Rs. 4, 000/- respec 1
tively per
person on P ; and Rs. 3,000/-, Rs. 5,000/-, Rs. 8,000/- respec 2
tively on the project P . 3
Further, the amount that 1 2 V3 V have kept aside for paying people on these projec ,V , ts
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?

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) For any two sets A and B, A B
c A ∩ = B .
ii) The matrix 





0 0
1 1
is singular.
iii) The contrapositive of ‘ ∃ y such that )y(P is true’ is ‘ ∈Z ∃ x such that )x(P ∈Z
is true’.
iv) The system 1 2x − 3y = and 0 6y − 4x + 2 = has a unique solution.
v) If y,x such that y ∈C x
2
= and x y
2
= , then 1 x = y = .
vi) a ≥ b ⇔ −a ≤ −b is an absolute inequality.
vii) If }2 A = ϕ,B = },2,1{ C = {− ,1 − , then A× B×C has 4 elements.
viii) The argument of 1+ i3 is
3
π
.
ix) A linear equation over R can have at most one root in C R .
x) x1 − x2 = x1 − x2 ∀x1 x,
2 ∈R .

Suppose that a ball is thrown straight up into the air and its height after t second is 4 + 48t - 16t^2 meters. Determine how long it will take for the ball to reach its maximum height and determined the maximum height.

Prove that 2^n > 1+n√2^n-1, for every n>2, using linear inequalities