Rational Numbers

Q.

Write a rational number between $\sqrt{2}$ and $\sqrt{3}$.

Q. prove that root 2 + root 5 is irrational

Q.

Find five rational numbers between $\frac{2}{3}$ and $\frac{4}{5}$.

Q.

Find five Rational numbers between $1$ and $2$.

Q.

Find five rational numbers between $\frac{1}{2}$ and $\frac{1}{3}$.

Q.

Is 2/root3 is rational or not.

Q.

Is zero a rational number? Can you write it in the form $\frac{p}{q}$ where $p$ and $q$ are integers and $q\ne 0$ ?

Q.

Explain why $3.\overline{1416}$ is a rational number.

Q. Which of the following rational numbers is in the standard form?

1. $-\frac{9}{28}$
2. $-\frac{26}{78}$
3. $-\frac{14}{16}$
4. $\frac{48}{96}$

Q.

All surds are irrational but not all irrational numbers are surds. True/False.

1. False

2. True

Q.

Find five rational numbers between $\frac{1}{4}$ and $\frac{1}{2}$.

Q.

If $\frac{p}{q}$ is a rational number with terminating decimal expansion where $p$ and $q$ are coprimes, then $q$ can be represented as:
(Here, n and m are non-negative integers.)

1. 2 or 5 or both

2. 3 or 5 or both

3. 2 or 3 or both

4. 2, 3 or 5

Q.

Division of rational number by a rational number.

Q.

The reciprocal of an irrational number is :

1. An integer

2. Rational

3. Irrational

4. None of these

Q. Given that HCF $(306$ and $657)=9$, find LCM of $(306$ and $657)$

Q.

In the standard form of a rational number, the denominator is always a

1. $0$

2. negative integer

3. positive integer

4. $1$

Q.

Insert4 irrational numbers between 3×root 2 and 2× root3

Q.

Decide whether the real number 0.120 1200 12000 120000.... is rational or not. If rational, and of the form $\frac{p}{q}$, what can you say about the prime factors of q?

Q.

show that the reciprocal of 3+2 root 2 is an irrational number

Q.

Prove that 7 root 5 is irrational

Q. A rational number (of form p/q) has a terminating decimal expansion if prime factorisation of q has only:

1. 3 or 4 or both
2. 4 or 5 or both
3. 6 or 7 or both
4. 2 or 5 or both

Q. Find the LCM and HCF of the following pairs of integers and verify that LCM $\times$HCF = product of the two numbers.
(i) $336$ and $54$

Q.

Prove that √x + √y is irrational, where x and y are primes.

Q.

Decide whether the real number with decimal expansion as $125.\overline{987654321}$ is rational or not. If rational, and of the form $\frac{p}{q}$, what can you say about the prime factors of q? :

1. Rational Number, Prime factors of q will be of form $2^n{.5}^m$ , $m$ and $n$ are whole numbers.

2. Not rational Number

3. Rational Number, prime factor cannot be represented in the form $2^n{.5}^m$, $m$ and $n$ are whole numbers.

4. Rational Number, Prime factor of q will be 3.

Q. All of the following are rational numbers between 1 and 2 except 11/10. 193/100. 1714/1000. 903/1000

Q.

A number ‘s’ is called irrational if it cannot be written in the form of _____ , where p and q are integers and ______.

1. $\frac{p}{q}$ , $q=0$

2. $\frac{p}{q}$ , $q\ne 0$

3. $pq$ , $p\ne 0$

4. $\frac{p}{q}$ , $p=0$

Q. Subtraction of a rational number and an irrational number gives always.

1. a rational number
2. an irrational number
3. zero

Q. State whether the real number 52.0521 is rational or not. If it is rational express it in the form $\frac{p}{q},$ where p, q are co - prime, integers and $q\ne 0.$ What can you say about prime factorisation of q?

Q. Draw the graphs of the equations $x-y+1=0$ and $3x+2y-12=0$. Determine the coordinates of the vertices of the triangle formed by these lines and the $x$-axis, and shade the triangular region.

Q. The combined set of rational and irrational numbers forms the set of real numbers.

1. True
2. False