Let's go through the questions one by one and solve them:

1. Which of the following is rational?

   • (a) $\sqrt{3}$
   • (b) $\frac{4}{0}$
   • (c) $\frac{0}{4}$
   • (d) $\pi$

   Answer: The rational numbers are numbers that can be expressed as the ratio of two integers, where the denominator is not zero.

   • $\sqrt{3}$ is an irrational number.
   • $\frac{4}{0}$ is undefined, as division by zero is not allowed.
   • $\frac{0}{4}$ is $0$, which is rational.
   • $\pi$ is irrational.

   Therefore, the answer is (c) $\frac{0}{4}$.

2. If $x + \frac{1}{x} = 5$, then $x^2 + \frac{1}{x^2} =$ ____________.

   • (a) 23
   • (b) 27
   • (c) 25
   • (d) 10

   Solution: Start with the given equation $x + \frac{1}{x} = 5$.

   To find $x^2 + \frac{1}{x^2}$, square both sides: $\left( x + \frac{1}{x} \right)^2 = 5^2$ Expanding the left side: $x^2 + 2 \cdot x \cdot \frac{1}{x} + \frac{1}{x^2} = 25$ Simplifying: $x^2 + 2 + \frac{1}{x^2} = 25$ $x^2 + \frac{1}{x^2} = 25 - 2 = 23$

   Therefore, the answer is (a) 23.

3. Two points having same abscissa but different ordinate lie on __________.

   • (a) Y-axis
   • (b) X-axis
   • (c) a line parallel to Y-axis
   • (d) a line parallel to X-axis

   Answer: The abscissa refers to the x-coordinate, so two points with the same abscissa will have the same x-coordinate. The points will lie on a vertical line, which is parallel to the Y-axis. Therefore, the correct answer is (c) a line parallel to Y-axis.

4. The root of the equation $2x + 5 = 0$ is __________.

   • (a) 5
   • (b) 2
   • (c) $\frac{5}{2}$
   • (d) $-\frac{5}{2}$

   Solution: Solve the equation: $2x + 5 = 0$ Subtract 5 from both sides: $2x = -5$ Divide both sides by 2: $x = -\frac{5}{2}$

   Therefore, the answer is (d) $-\frac{5}{2}$.

5. The equation $x = 7$ in two variables can be written as __________.

   • (a) $1 \cdot x + 1 \cdot y = 7$
   • (b) $1 \cdot x + 0 \cdot y = 7$
   • (c) $0 \cdot x + 1 \cdot y = 7$
   • (d) $0 \cdot x + 0 \cdot y = 7$

   Answer: The equation $x = 7$ is in the form $x + 0 \cdot y = 7$, which can be written as (b) $1 \cdot x + 0 \cdot y = 7$.

Would you like further explanations or clarifications on any of these answers?

Here are some related questions:

1. What is the difference between rational and irrational numbers?
2. How do you solve equations involving squares of variables, like $x^2 + \frac{1}{x^2}$?
3. What is the geometric interpretation of having two points with the same x-coordinate?
4. How do you solve linear equations with one variable?
5. What is the general form of a linear equation in two variables?

Tip: When dealing with equations, try to simplify them step-by-step. Avoid skipping steps as it can help you avoid mistakes!