CBSE Class 10-Mathematics: Chapter –1 Real Numbers Part 1
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1 Marks Questions
Question 1: Use Euclid Division lemma to show that the square of any positive integer is either of form for some integer
[Hint: Let x be any positive integer then it is of the form . Now square each of these and show that they can be rewritten in the form .]
Answer:
Let a be any positive integer and .
The for some integer
And because
Therefore, or
Or,
Where are some positive integers?
Hence, it can be said the square of any positive integer is either of the form .
Question 2:
Express each number as product of its prime factors:
(i)
(i)
(iv) 5005
(v)
Answer:
(i)
(ii)
(iii)
(iv)
Question 3:
Given that , find
Answer:
We know that Product of two numbers
Question 4:
Check whether can end with the digit for any natural number
Answer:
If any number ends with the digit , it should be divisible by .
In other words, it will also be divisible by and as
Prime Factorization of
It can be observed that is not in the prime factorisation of
Hence, for any value of will not be divisible by .
Therefore, cannot end with the digit for any natural number .
Question 5:
Prove that is Irrational
Answer:
Let us say is rational.
Then the co-prime x and y of the given rational number where is such that:
Rearranging, we get,
Since x and y are integers, thus, is a rational number.
Therefore, is also a rational number. But this confronts the fact that is irrational.
Hence, we get that is irrational.
Question 6:
is a
(a) Composite number
(b) Whole number
(c) Prime number
(d) None of these
Answer:
(a) and (b) both