please do its urgent (1,3,4)
Answers
Step-by-step explanation:
Solutions:-
1) Given that:
7 √(1/3) - 2 1/3 √(1/3) +3√147
=>7(1/√3) - (7/3)(1/√3) +3√(3×7×7)
=> (7/√3) - (7/3√3) +3×7√3
=> (7/√3) - (7/3√3) + 21√3
On Rationalising the denominators then
=>[ 7√3/(√3×√3)]-[7√3/(3×√3×√3) +21√3
The Rationalising factor of√3 =√3
=> (7√3/3)-[7√3/(3×3)]+21√3
=> (7√3/3)-(7√3/9)+21√3
LCM of 3 and 9 = 9
=> [(3×7√3)-(7√3)+(9×21√3)]/9
=> (21√3-7√3+189√3)/9
=>[(21-7+189)√3]/9
=> [(210-7)√3]/9
=> 203√3/9
3)
Given that :
x = 5-2√6
1/x = 1/(5-2√6)
The denominator = 5-2√6
The Rationalising factor of 5-2√6 = 5+2√6
On Rationalising the denominator then
=> 1/x = [1/(5-2√6)]×[(5+2√6)/5+2√6)]
=> 1/x = (5+2√6)/[5-2√6)(5+2√6)]
=> 1/x = (5+2√6)/[5²-(2√6)²]
Since (a+b)(a-b) = a²-b²
Where , a = 5 and b = 2√6
=>1/x = (5+2√6)/(25-24)
=> 1/x = (5+2√6)/1
=> 1/x = 5+2√6
Now,
x +(1/x)
=> 5-2√6+5+2√6
=> 5+5
=> 10
Therefore, x+(1/x) = 10
On squaring both sides then
=> [x+(1/x)]² = 10²
=> x²+2(x)(1/x)+(1/x)² =100
=> x²+2+(1/x)² = 100
=> x²+(1/x²) = 100-2
=> x²+(1/x²) = 98
Therefore, x²+(1/x²) = 98
4)Given number = √5
Let us assume that √5 is a rational number.
It must be in the form of p/q
Where, p and q are integers and q≠0
Let √5 = a/b (a,b are co primes)
On squaring both sides
=> (√5)² = (a/b)²
=> 5 = a²/b²
=> 5b² = a² ---------(1)
=> b² = a²/5
=> 5 divides a²
=>5 divides a also
=> 5 is a multiple of a ------(2)
Put a = 5c in (1) then
5b² = (5c)²
=> 5b² = 25c²
=> b² = 5c²
=> c² = b²/5
=> 5 divides b²
=> 5 divides also b
=> 5 is a multiple of b --------(3)
From (2)&(3)
We have,
5 is a multiple of both a and b
=> 5 is a common multiple of a and b
but a and b are co primes
They have only one common factor that is 1
This contradicts to our assumption that is√5 is a rational number.
So, √5 is not a rational number.
√5 is an irrational number.
Hence, Proved.
and
Given number = 3√5
Let us assume that 3√5 is a rational number.
It must be in the form of p/q
Where, p and q are integers and q≠0
Let 3√5 = a/b (a,b are co primes)
=> √5 = (a/b)/3
=> √5 = a/3b
=> √5 is in the form of p/q
=>√5 is a rational number
But √5 is not a rational number.
It is an irrational number
This contradicts to our assumption that is 3√5 is a rational number.
So, 3√5 is not a rational number.
3√5 is an irrational number.
Hence, Proved.
Note :-
The product of a rational and an irrational is an irrational number.
Answer:-
1) 203√3/9
3)x+(1/x) = 10 and x²+(1/x²) = 98
4) √5 and 3√5 are irrational numbers
Used formulae:-
- The Rationalising factor of a-√b is a+√b
Used Method:-
- Method of Contradiction ( Indirect method)