Translate the following statement into symbolic into symbolic from.
(i) Rahul passed in Hindi and English.
(ii) x and y are even integers.
(iii) 2, 3 and 6 are factors of 12.
(iv) Either x or x + 1 is an odd integer.
(v) A number is either divisible by 2 or 3.
(vi) Either x= 2 or x = 3 is a root of `3x^(2)` - x - 10= 0
(vii) Students can take Hindi or English as an optional paper.

Explanation

To translate the given statements into symbolic form, we will define propositions for each statement and use logical symbols to represent the relationships. Here is the step-by-step solution:

### Step 1: Define Propositions
1. **(i)** Let \( P \): "Rahul passed in Hindi"
Let \( Q \): "Rahul passed in English"
Symbolic form: \( P \land Q \) (where \( \land \) represents "and")

2. **(ii)** Let \( P \): "x is an even integer"
Let \( Q \): "y is an even integer"
Symbolic form: \( P \land Q \)

3. **(iii)** Let \( P \): "2 is a factor of 12"
Let \( Q \): "3 is a factor of 12"
Let \( R \): "6 is a factor of 12"
Symbolic form: \( P \land Q \land R \)

4. **(iv)** Let \( P \): "x is an odd integer"
Let \( Q \): "x + 1 is an odd integer"
Symbolic form: \( P \lor Q \) (where \( \lor \) represents "or")

5. **(v)** Let \( P \): "A number is divisible by 2"
Let \( Q \): "A number is divisible by 3"
Symbolic form: \( P \lor Q \)

6. **(vi)** Let \( P \): "x = 2 is a root of \( 3x^2 - x - 10 = 0 \)"
Let \( Q \): "x = 3 is a root of \( 3x^2 - x - 10 = 0 \)"
Symbolic form: \( P \lor Q \)

7. **(vii)** Let \( P \): "Students can take Hindi as an optional paper"
Let \( Q \): "Students can take English as an optional paper"
Symbolic form: \( P \lor Q \)

### Final Symbolic Forms
1. \( P \land Q \) (Rahul passed in Hindi and English)
2. \( P \land Q \) (x and y are even integers)
3. \( P \land Q \land R \) (2, 3, and 6 are factors of 12)
4. \( P \lor Q \) (Either x or x + 1 is an odd integer)
5. \( P \lor Q \) (A number is either divisible by 2 or 3)
6. \( P \lor Q \) (Either x = 2 or x = 3 is a root of \( 3x^2 - x - 10 = 0 \))
7. \( P \lor Q \) (Students can take Hindi or English as an optional paper)