5. Out of the following irrationals, which has sum and product both rationals?
(a) (3 + 5 and 13 - 5
(b) 13 + 15 and 13 - 15
(c) 5+ 73 and 5 - V3
(d) All of these
Answers
Answer:
ANSWER
(i)
2
1
Let us assume
2
1
is rational.
So we can write this number as
2
1
=
b
a
---- (1)
Here, a and b are two co-prime numbers and b is not equal to zero.
Simplify the equation (1) multiply by
2
both sides, we get
1=
b
a
2
Now, divide by b, we get
b=a
2
or
a
b
=
2
Here, a and b are integers so,
a
b
is a rational number,
so
2
should be a rational number.
But
2
is a irrational number, so it is contradictory.
Therefore,
2
1
is irrational number.
(ii) 7
5
Let us assume 7
5
is rational.
So, we can write this number as
7
5
=
b
a
---- (1)
Here, a and b are two co-prime numbers and b is not equal to zero.
Simplify the equation (1) divide by 7 both sides, we get
5
=
7b
a
Here, a and b are integers, so
7b
a
is a rational
number, so
5
should be a rational number.
But
5
is a irrational number, so it is contradictory.
Therefore, 7
5
is irrational number.
(iii) 6+
2
Let us assume 6+
2
is rational.
So we can write this number as
6+
2
=
b
a
---- (1)
Here, a and b are two co-prime number and b is not equal to zero.
Simplify the equation (1) subtract 6 on both sides, we get
2
=
b
a
−6
2
=
b
a−6b
Here, a and b are integers so,
b
a−6b
is a rational
number, so
2
should be a rational number.
But
2
is a irrational number, so it is contradictory.
Therefore, 6+
2
is irrational number.
Step-by-step explanation:
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Answer:
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