Consider the following statements :
1. `(dy)/(dx)` at a point on the curve gives slope of the tangent at the point.
2. If a(t) denotes acceleration of a particle. Then `inta(t)dt+c` gives velocity of the particle.
3. If s(t) gives displacement of a particle at time t, then `(ds)/(dt)` gives its acceleration at that instant.
Which of the above statements is/are correct ?

Options

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

Answer

To determine which of the statements are correct, let's analyze each statement step by step.

### Step 1: Analyze the first statement
**Statement 1:** `(dy)/(dx)` at a point on the curve gives the slope of the tangent at that point.

- **Explanation:** The derivative of a function, represented as `(dy)/(dx)`, measures the rate of change of `y` with respect to `x`. At any point on the curve, this derivative represents the slope of the tangent line to the curve at that point. Therefore, this statement is **correct**.

### Step 2: Analyze the second statement
**Statement 2:** If `a(t)` denotes the acceleration of a particle, then `∫a(t) dt + c` gives the velocity of the particle.

- **Explanation:** Acceleration is defined as the derivative of velocity with respect to time, i.e., `a(t) = (dv)/(dt)`. To find velocity, we integrate acceleration with respect to time:
\[ dv = a(t) dt \implies v(t) = \int a(t) dt + C \]
where `C` is the constant of integration. Thus, this statement is also **correct**.

### Step 3: Analyze the third statement
**Statement 3:** If `s(t)` gives the displacement of a particle at time `t`, then `(ds)/(dt)` gives its acceleration at that instant.

- **Explanation:** The derivative `(ds)/(dt)` represents the velocity of the particle, not the acceleration. Acceleration is the second derivative of displacement with respect to time, which is given by:
\[ a(t) = \frac{d^2s}{dt^2} \]
Therefore, this statement is **incorrect**.

### Conclusion
Based on the analysis:
- Statement 1 is correct.
- Statement 2 is correct.
- Statement 3 is incorrect.

Thus, the correct answer is that **only statements 1 and 2 are correct**.

### Final Answer
The correct statements are: **1 and 2**.

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