State T for and F for false.
(i) `a ^(3) - b ^(3) = (a - b) ^(3)` if ab = 0
(ii) `(x + (1)/(x)) ^( 2) - (x - (1)/(x)) ^(2) = 2`
(iii) The remainder when `15 ^(2) - 26 x + 8` is divided by `3x - 4` is 1.

Options

A. `{:(""(i), (ii), (iii)),(T, F, T):}`
B. `{:(""(i), (ii), (iii)),(T,T,F):}`
C. `{:(""(i), (ii), (iii)),(F,T,T):}`
D. `{:(""(i), (ii), (iii)),(T,F,F):}`

Answer

Let's solve each part of the question step by step.

### (i) \( a^3 - b^3 = (a - b)^3 \) if \( ab = 0 \)

**Step 1:** Recall the identity for the difference of cubes:
\[ a^3 - b^3 = (a - b)(a^2 + ab + b^2) \]

**Step 2:** If \( ab = 0 \), then either \( a = 0 \) or \( b = 0 \).

**Step 3:** Substitute \( ab = 0 \) into the identity:
- If \( a = 0 \), then \( a^3 - b^3 = -b^3 \) and \( (a - b)^3 = (-b)^3 = -b^3 \).
- If \( b = 0 \), then \( a^3 - b^3 = a^3 \) and \( (a - b)^3 = a^3 \).

**Conclusion:** In both cases, \( a^3 - b^3 = (a - b)^3 \) holds true when \( ab = 0 \).

**Final Answer:** True (T)

### (ii) \( (x + \frac{1}{x})^2 - (x - \frac{1}{x})^2 = 2 \)

**Step 1:** Expand both squares:
\[ (x + \frac{1}{x})^2 = x^2 + 2 + \frac{1}{x^2} \]
\[ (x - \frac{1}{x})^2 = x^2 - 2 + \frac{1}{x^2} \]

**Step 2:** Now subtract the second expression from the first:
\[ (x^2 + 2 + \frac{1}{x^2}) - (x^2 - 2 + \frac{1}{x^2}) = 2 + 2 = 4 \]

**Conclusion:** The left-hand side simplifies to 4, not 2.

**Final Answer:** False (F)

### (iii) The remainder when \( 15^2 - 26x + 8 \) is divided by \( 3x - 4 \) is 1.

**Step 1:** First, calculate \( 15^2 - 26x + 8 \):
\[ 15^2 = 225 \quad \text{so} \quad 225 - 26x + 8 = 233 - 26x \]

**Step 2:** To find the remainder when dividing by \( 3x - 4 \), we can use the Remainder Theorem. Set \( 3x - 4 = 0 \) to find \( x \):
\[ 3x = 4 \quad \Rightarrow \quad x = \frac{4}{3} \]

**Step 3:** Substitute \( x = \frac{4}{3} \) into \( 233 - 26x \):
\[ 233 - 26 \left(\frac{4}{3}\right) = 233 - \frac{104}{3} \]
Convert 233 to a fraction:
\[ 233 = \frac{699}{3} \quad \Rightarrow \quad \frac{699}{3} - \frac{104}{3} = \frac{595}{3} \]

**Step 4:** Check if \( \frac{595}{3} = 1 \):
\[ \frac{595}{3} \neq 1 \]

**Conclusion:** The remainder is not equal to 1.

**Final Answer:** False (F)

### Summary of Answers:
(i) True (T)
(ii) False (F)
(iii) False (F)