Assertion (A) : The remainder of `x^(3)+2x^(2)-5x-3` which is divided by `x-2` is 3.
Reason (R) : The remainder of the polynomial `f(x)` when divided by `x-a` is `f(a)`

Options

A. Both A & R are true and R is correct explanation of A
B. Both A & R are true and R is not correct explanation of A
C. A is true but R is false
D. A is false but R is true

Answer

To solve the problem, we need to verify both the assertion (A) and the reason (R) provided in the question.

### Step 1: Verify the Assertion (A)

The assertion states that the remainder of the polynomial \( f(x) = x^3 + 2x^2 - 5x - 3 \) when divided by \( x - 2 \) is 3.

According to the Remainder Theorem, the remainder of a polynomial \( f(x) \) when divided by \( x - a \) is given by \( f(a) \). Here, \( a = 2 \).

So, we need to calculate \( f(2) \):

\[ f(2) = (2)^3 + 2(2)^2 - 5(2) - 3 \]

Calculating each term:

\[ = 8 + 2(4) - 10 - 3 \]
\[ = 8 + 8 - 10 - 3 \]
\[ = 16 - 10 - 3 \]
\[ = 6 - 3 \]
\[ = 3 \]

Thus, the remainder when \( f(x) \) is divided by \( x - 2 \) is indeed 3. Therefore, assertion (A) is **true**.

### Step 2: Verify the Reason (R)

The reason states that the remainder of the polynomial \( f(x) \) when divided by \( x - a \) is \( f(a) \). This is a statement of the Remainder Theorem.

Since we have already applied this theorem to find \( f(2) \) and confirmed that it equals 3, we can conclude that the reason (R) is also **true**.

### Conclusion

Both the assertion (A) and the reason (R) are true, and the reason correctly explains the assertion. Therefore, the correct option is:

**Both A and R are true, and R is a correct explanation of A.**

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