NCERT Class 11-Math's: Chapter –14 Mathematical Reasoning Part 11
Glide to success with Doorsteptutor material for NCO Class-2: fully solved questions with step-by-step explanation- practice your way to success.
Question 12:
Prove by direct method that for any integer , is always even.
[Hint: Two cases is even, is odd.]
Answer:
We have given,
Let us Assume, is even
Let , where k is natural number
Therefore, is even.
Now, Let us Assume n is odd
Let , where k is natural number
Therefore, is even.
Hence, is always even
Question 13:
Check the validity of the following statement.
(i) is divisible by and .
(ii) is a multiple of or .
Answer:
(i) p: 125 is divisible by 5 and 7
Let,
is divisible by .
is divisible .
Here, q is true and r is false.
Therefore, is False
Hence, is not valid.
(ii) is a multiple of or
Let,
is a multiple of .
is a multiple of .
Here, is false and is False
Therefore, is False
Hence, is not valid
Question 14:
Prove the following statement by contradiction method.
The sum of an irrational number and a rational number is irrational.
Answer:
Let p is false, as the sum of an irrational number and a rational number is irrational.
Let is irrational and n is rational number
But, we know that is irrational where as is rational which is contradiction.
Here, Our Assumption is False
Hence, P is true.
Question 15:
Prove by direct method that for any real numbers , if , then .
Answer:
We have Given for any real number if
To Find:
Explanation: Let us Assume
where x and y are real number
On squaring both sides we get
(Assumption)
Therefore,
Hence, Proved