Which of the following statements are true?
1. LCM of `10a^(2)b, 15abc, 20a^(2)b^(2)c^(2)` is `20a^(2)b^(2)c^(2)`
2. HCF of `(6x^(2)-7x-3) and (2x^(2)+11x-21)` is `2x-3`.
3. HCF of `(6x^(2)-7x-3) and (2x^(2)+11x-21)` is `2x-3`.

Options

A. 1, 2, 3
B. 1, 2
C. 1, 3
D. 2, 3

Answer

To determine which of the statements are true, we will analyze each statement one by one.

### Step 1: Analyze the first statement
**Statement 1:** LCM of `10a^(2)b`, `15abc`, `20a^(2)b^(2)c^(2)` is `20a^(2)b^(2)c^(2)`.

**Finding LCM:**
1. **Identify the factors:**
- `10a^(2)b` can be factored as `2 * 5 * a^(2) * b`
- `15abc` can be factored as `3 * 5 * a * b * c`
- `20a^(2)b^(2)c^(2)` can be factored as `2^2 * 5 * a^(2) * b^(2) * c^(2)`

2. **Determine the LCM:**
- For the coefficients: LCM of `10`, `15`, and `20` is `60` (as `60` is the smallest number that is a multiple of all three).
- For `a`: The maximum power is `a^(2)`.
- For `b`: The maximum power is `b^(2)`.
- For `c`: The maximum power is `c^(2)`.

Thus, the LCM is `60a^(2)b^(2)c^(2)`.

**Conclusion:** The statement is **false** because the correct LCM is `60a^(2)b^(2)c^(2)`.

### Step 2: Analyze the second statement
**Statement 2:** HCF of `(6x^(2)-7x-3)` and `(2x^(2)+11x-21)` is `2x-3`.

**Finding HCF:**
1. **Factor the polynomials:**
- For `6x^(2)-7x-3`, we can factor it as follows:
- The factors of `6x^2` are `(2x + 1)(3x - 3)`.
- For `2x^(2)+11x-21`, we can factor it as:
- The factors are `(2x + 3)(x - 7)`.

2. **Identify common factors:**
- The common factor from both factorizations is `2x - 3`.

**Conclusion:** The statement is **true**.

### Step 3: Analyze the third statement
**Statement 3:** HCF of `(6x^(2)-7x-3)` and `(2x^(2)+11x-21)` is `2x-3`.

**Conclusion:** This statement is **false** because the HCF is `2x - 3`, which is the same as in statement 2. However, since it is a repeat of the second statement, it is not a new statement.

### Final Conclusion
- **Statement 1** is **false**.
- **Statement 2** is **true**.
- **Statement 3** is **false** (as it repeats the second statement).

### Summary of Results
- True Statements: 2
- False Statements: 1, 3