Mathematics Test - 9

y = x is tangent to the parabola
y=ax² +c
if a= then c=?
y ′=2ax
y ’= 2(7/2)x  =1
x = 1/7
1/7 = 2(1/7)² + c
c = 1/7 * 2/49
c = 7/2

Given the equation is,
2x² +3y² −8x −18y+35=K
Or, 2{x² −4x+4} + 3{y² −6y+9}=K
Or, 2(x −2)² + 3(y −3)² =K.
From the above equation it is clear that if K >0 then the given equation will represent an ellipse and for K <0, no geometrical interpretation.
Also if K=0 then the given equation will be reduced to a point and the point will be (2,3).

We have ∗is defined on the set {1,2,3,4,5}  by a ∗b = LCM of a and b
Now, 3 ∗4 = LCM of 3 and 4=12
But, 12 does not belong to {1,2,3,4,5}.
Hence, ′∗′is not a binary operation.

Given that οis a binary operation on Q –{ –1} defined by a οb = a + b –ab for all a,b ∈Q –{ –1}.
We know that commutative property is p οq = q οp, where οis a binary operation.
Let ’s check the commutativity of given binary operation:
⇒a οb = a + b –ab
⇒b οa = b + a –ba = a + b –ab
⇒b*a = a*b
∴Commutative property holds for given binary operation ‘ο’on ‘Q –[ –1]’.
We know that associative property is (p οq)οr = p ο(q οr)
Let ’s check the associativity of given binary operation:
⇒(a οb)οc = (a + b –ab)οc
⇒(a οb)οc = a + b –ab + c –((a + b –ab)×c)
⇒(a οb)οc = a + b + c –ab –ac –ab + abc ...... (1)
⇒a ο(b οc) = a ο(b + c –bc)
⇒a ο(b οc) = a + b + c –bc –(a ×(b + c –bc))
⇒a*(b*c) = a + b + c –ab –bc –ac + abc ...... (2)
From (1) and (2)
⇒(aob)oc = ao(boc)
Hence,we can clearly say that associativity hold for the given binary operation on ‘Q –{ –1}’.
Given that οis a binary operation on Q –{ –1} defined by a οb = a + b –ab for all a,b ∈Q –{ –1}.
We know that commutative property is p οq = q οp, where οis a binary operation.
Let ’s check the commutativity of given binary operation:
⇒a οb = a + b –ab
⇒b οa = b + a –ba = a + b –ab
⇒b*a = a*b
∴Commutative property holds for given binary operation ‘ο’on ‘Q –[ –1]’.
We know that associative property is (p οq)οr = p ο(q οr)
Let ’s check the associativity of given binary operation:
⇒(a οb)οc = (a + b –ab)οc
⇒(a οb)οc = a + b –ab + c –((a + b –ab)×c)
⇒(a οb)οc = a + b + c –ab –ac –ab + abc ...... (1)
⇒a ο(b οc) = a ο(b + c –bc)
⇒a ο(b οc) = a + b + c –bc –(a ×(b + c –bc))
⇒a*(b*c) = a + b + c –ab –bc –ac + abc ...... (2)
From (1) and (2)
⇒(aob)oc = ao(boc)
Hence,we can clearly say that associativity hold for the given binary operation on ‘Q –{ –1}’.

A  ∩ B = {4, 3}

Probability of event A or B

The probability that Events A and B both occur is the probability of the intersection of A and B. The probability of the intersection of Events A and B is denoted by P(A ∩B). If Events A and B are mutually exclusive, P(A ∩B) = 0. The probability that Events A or B occur is the probability of the union of A and B.