CBSE Important Questions

Q. 3. Based on LCM and HCF:

Formula: LCM x HCF = product of numbers  Or product of numbers = LCM x HCF

Hint: If LCM or HCF is to be found then use the first formula. IF value of any of the numbers is to found use second formula. That is, always keep the unknown variable on the LHS to avoid confusion.

Solved example:

1. Find HCF (26,91) if LCM(26,91) is 182

Sol: We know that LCM x HCF = Product of numbers.
         or 182 x HCF = 26 x 91
         or HCF = 26 x 91 = 13
                             182
Hence HCF (26, 91) = 13.

2. LCM and HCF of two numbers are 3024 and 6 respectively. If one of the number is 336 find the other number.

Sol: We know that

Product of numbers = LCM x HCF
Or Number = LCM x HCF
                             Given number
Or number = 3024 x 6 = 54
                                   336
Hence the other number is 54.
Solve similar questions from your text book.

Q. 4. Based on irrational numbers.

By heart the following jingles.

1. Let us assume on the contrary the √__ is rational. That is we can find co-primes a and b ( b≠0) such that √__ = a/b.
2. ____ divides a². Hence it follows that ____ divides a.
3. So we can write a= ____c.
4. ____ divides b². Hence it follows that ____ divides b.
5. Now a and b have at least ____ as a common factor.
6. But this contradicts the fact that a and b are co-primes.
7. This contradiction has arisen because of our incorrect assumption that √__ is irrational.
8. Hence √__ is rational.

Sample question: prove that √5 is irrational.

Solution:

let us assume on the contrary that √5 is rational. That is we can find co-primes a and b b (≠0) such that √5 = a/b. (1^st jingle)
Or √5b = a.
Squaring both sides we get
5b² = a².
This means 5 divides a². Hence it follows that 5 divides a. (2^nd jingle)
So we can write a = 5c for some integer c. (3^rd jingle)
Putting this value of a we get

5b² = (5c)²
Or 5b² = 25c²
Or b² = 5b².

It follows that 5 divides b2. Hence 5 divides b. (4^th jingle)
Now a and b have at least 5 as a common factor. (5^th jingle)
But this contradicts the fact that a and b are co-primes. (6^th jingle)
This contradiction has arisen because of our incorrect assumption that √5 is rational. (7^th jingle)
Hence it follows that √5 is irrational. (8^th jingle)