Real Numbers Real Numbers Set-1

Q1. The value of $\dfrac{\sqrt{7}+\sqrt{3}}{\sqrt{7}-\sqrt{3}}$ is equal to:

Q2. If $x>0$ satisfies $\sqrt{x}-\dfrac{1}{\sqrt{x}}=\sqrt{2}$, then $x$ equals:

Q3. Assertion (A): $x=\sqrt{5-2\sqrt{6}}$ is irrational.
Reason (R): For any two irrational numbers $p$ and $q$, the difference $p-q$ is necessarily irrational.
Which option is correct?

Q4. Let $M=\sqrt{5+2\sqrt{6}}\cdot \sqrt{5-2\sqrt{6}}$. Then $M$ equals:

Q5. The value of $x$ satisfying $\sqrt{x+2}-\sqrt{x-2}=2$ (where the expression is defined) is:

Q6. Find the value of $\sqrt{50}-\sqrt{18}$.

Q7. If $x$ is a rational number such that $\sqrt{x}+\sqrt{3x}$ is also rational, then the value of $x$ is:

Q8. Evaluate $\sqrt{7+4\sqrt{3}}-\sqrt{7-4\sqrt{3}}$.

Q9. Assertion (A): For all $a,b\ge 0$, $\sqrt{a}-\sqrt{b}=\sqrt{a+b-2\sqrt{ab}}$.
Reason (R): $(\sqrt{a}-\sqrt{b})^2=a+b-2\sqrt{ab}$.
Choose the correct option.

Q10. Let $n$ be a non-negative integer such that $\sqrt{n}+\sqrt{n+1}$ is rational. Then $n$ equals:

Q11. Find the value of $\sqrt{18}-\sqrt{8}$.

Q12. If $\sqrt{x+3}-\sqrt{x-1}=1$ for real $x$, then $x$ equals:

Q13. If $\sqrt{x+1}+\sqrt{x-2}=3$ for real $x$, then $x$ equals:

Q14. Consider the following statements for the equation $\sqrt{2x+1}=\sqrt{x+1}-2$.


Assertion (A): The equation has no real solution.
Reason (R): After squaring, we get $x=0$ or $x=24$, and for both values the right-hand side becomes negative, so the original equation cannot hold.

Q15. If $\sqrt{2x-3}-\sqrt{x-1}=1$ for real $x$, then $x$ equals: