Which of the following statements are true:
(i) The rational number `29/(23)` lies to the left zero on the number line.
(ii) The rational number `(-12)/(-17)` lies to the right zero on the number line.
(iii) The rational numbers `(-12)/(5)and(-7)/(17)` are on th opposite side of zero on the number line.
(iv) The rational numbers `(-21)/(5)and7/(-31)` are on th opposite side of zero on the number line.

Options

A. Only (i)
B. (i) & (ii)
C. Only (iii)
D. (i), (ii) & (iv)

Answer

To determine which statements are true, we will evaluate each statement one by one.

### Step 1: Evaluate Statement (i)
**Statement:** The rational number \( \frac{29}{23} \) lies to the left of zero on the number line.

**Calculation:**
- \( \frac{29}{23} \) is a positive number because both the numerator and denominator are positive.
- The value of \( \frac{29}{23} \) is approximately \( 1.26 \), which is greater than zero.

**Conclusion:** This statement is **false** because \( \frac{29}{23} \) lies to the right of zero.

### Step 2: Evaluate Statement (ii)
**Statement:** The rational number \( \frac{-12}{-17} \) lies to the right of zero on the number line.

**Calculation:**
- The negative signs cancel out, so \( \frac{-12}{-17} = \frac{12}{17} \), which is positive.
- The value of \( \frac{12}{17} \) is approximately \( 0.705 \), which is greater than zero.

**Conclusion:** This statement is **true** because \( \frac{-12}{-17} \) lies to the right of zero.

### Step 3: Evaluate Statement (iii)
**Statement:** The rational numbers \( \frac{-12}{5} \) and \( \frac{-7}{17} \) are on the opposite side of zero on the number line.

**Calculation:**
- \( \frac{-12}{5} \) is negative, and its value is approximately \( -2.4 \).
- \( \frac{-7}{17} \) is also negative, and its value is approximately \( -0.412 \).
- Both numbers are negative, so they lie on the same side of zero.

**Conclusion:** This statement is **false** because both numbers are on the left side of zero.

### Step 4: Evaluate Statement (iv)
**Statement:** The rational numbers \( \frac{-21}{5} \) and \( \frac{7}{-31} \) are on the opposite side of zero on the number line.

**Calculation:**
- \( \frac{-21}{5} \) is negative, and its value is approximately \( -4.2 \).
- \( \frac{7}{-31} \) is also negative, as it can be rewritten as \( \frac{-7}{31} \), which is approximately \( -0.226 \).
- Both numbers are negative, so they lie on the same side of zero.

**Conclusion:** This statement is **false** because both numbers are on the left side of zero.

### Final Summary of Statements:
- (i) False
- (ii) True
- (iii) False
- (iv) False