Proof by Contradiction

Q.

Prove that $\sqrt{2}+\sqrt{3}$ is irrational.

Q.

Prove that $\surd 7$ is an irrational number.

Q.

Prove that $(5-2\sqrt{3})$ is an irrational number.

Q.

Question 14
Prove that $\sqrt{p}+\sqrt{q}$ is irrational, where p and q are primes.

Q. Show that $3\sqrt{7}$ is an irrational number

Q. Prove that $\sqrt{5}$ is an irrational number. Hence show that $3+2\sqrt{5}$ is also an irrational number.

Q.

Prove that $(2\sqrt{3}-1)$ is an irrational number.

Q.

Prove that $\frac{1}{\sqrt{3}}$ is irrational.

Q.

Prove that $\sqrt{5}+\sqrt{3}$ is irrational.

Q.

Which of the following is irrational?

1. $\frac{\sqrt{4}}{\sqrt{9}}$

2. $\frac{\sqrt{12}}{\sqrt{3}}$

3. $\sqrt{7}$

4. $\sqrt{81}$

Q. Question 1
Let x and y be rational and irrational numbers respectively. Is x + y necessarily an irrational number? Give an example in support of your answer.

Q. If p, q are prime positive integers, prove that $\sqrt{p}+\sqrt{q}$ is an irrational number.

Q. Prove that the following are irrational:
(i) $\frac{1}{\sqrt{2}}$ (ii) $7\sqrt{5}$ (iii) $6+\sqrt{2}$

Q.

Prove that for any prime positive integer p, $\sqrt{p}$ is an irrational number.

Q.

Show that $5-\sqrt{3}$ is an irrational number.

Q.

Question 10
Prove that $\sqrt{3}+\sqrt{5}$ is irrational.

Q. Show that $2-\sqrt{3}$ is an irrational number.

Q.

If $\frac{3+\sqrt{5}}{2\sqrt{5}+3}=a+b\sqrt{5}$, then the values of irrational numbers $a$ and $b$ are

1. 
2. 
3. 
4. 

Q.

Surds are the_____ roots of any equation with ___ coefficients.

1. Irrational, Irrational

2. Irrational, Rational

3. Rational, Irrational

4. Rational, Rational

Q.

Prove that $3-sqrt5$ is irrational

Q. Question 10
Prove that $\sqrt{3}+\sqrt{5}$ irrational

Q. If $\frac{3+\sqrt{5}}{2\sqrt{5}+3}=a+b\sqrt{5}$, then the values of irrational numbers $a$ and $b$ are

1. 
2. 
3. 
4. 

Q. Show that $3+\sqrt{2}$ is an irrational number.

Q. Let x and y be rational and irrational numbers, respectively, then x + y necessarily an irrational number.

1. True
2. False

Q. State whether the following statement is TRUE or FALSE.

1. True
2. False

Q.

Prove that the following is irrational : $6+\sqrt{2}$ [3 MARKS]

Q. Given that $\sqrt{2}$is irrational, prove that $(5+3\sqrt{2})$ is an irrational number.

Q.

Prove that $2\sqrt{3}-1$ is an irrational number.

Q. Prove that $4-5\sqrt{2}$ is an irrational number.

Q. is an irrational number.

1. $\sqrt{5}$
2. $5^2$
3. $\frac{1}{2}$