P(A ∩ B) और P(A ∪ B) का मान ज्ञात कीजिए। ay² = x³ के बिंदु (am², am³) पर प्रवणता ज्ञात कीजिए। बिंदु A(2, 3, 4), B(-1, -2, 1) तथा C(1, 2, 3) का उपयोग कीजिए। P(A ∩ B) और P(A ∪ B) का मान ज्ञात कीजिए। ay² = x³ के बिंदु (am², am³) पर प्रवणता ज्ञात कीजिए। बिंदु A(2, 3, 4), B(-1, -2, 1) तथा C(1, 2, 3) का उपयोग कीजिए। Read more

Steps to Solve

1. Probability Calculation for A and B To find $P(A \cap B)$ and $P(A \cup B)$, we need to know the definitions:

   • $P(A \cap B) = P(A) + P(B) - P(A \cup B)$
   • $P(A \cup B) = P(A) + P(B) - P(A \cap B)$

   However, since we don't know the probabilities of A and B yet, we will need to find them or assume values to use this formula.

2. Finding the Derivative for Slope Calculation We have the equation $ay^2 = x^3$. First, we need to differentiate it implicitly to find the slope.

   • Rearranging gives: $y^2 = \frac{x^3}{a}$
   • Differentiate both sides with respect to $x$ using implicit differentiation.

   The differentiation yields: $$ 2y \frac{dy}{dx} = \frac{3x^2}{a} $$

   Now, solving for $\frac{dy}{dx}$ we get: $$ \frac{dy}{dx} = \frac{3x^2}{2ay} $$

3. Evaluate the Slope at the Point (am², am³) Substitute the values $(am^2, am^3)$ into the derivative to find the slope at this point:

   • Here, $x = am^2$ and $y = am^3$.
   • Now substitute these into the slope formula: $$ \frac{dy}{dx} = \frac{3(am^2)^2}{2a(am^3)} = \frac{3a^2m^4}{2a^2m^3} = \frac{3m}{2} $$

4. Final Slope Interpretation Thus, the slope of the tangent line at the point $(am^2, am^3)$ is: $$ \text{slope} = \frac{3m}{2} $$