The correct option is (A): 1.5
Explanation: To find the value of \( k \), we need to calculate the slope of the function \( y = x^3 + kx \) at \( x = 2 \) and the area under the curve \( z = a^2 + a \) from \( a = 0 \) to \( a = 3 \).
1. Calculate the slope at \( x = 2 \):
\[\text{Slope} = \frac{dy}{dx} = 3x^2 + k \]
At \( x = 2 \):
\[\text{Slope} = 3(2^2) + k = 3(4) + k = 12 + k\]
2. Calculate the area under the curve \( z = a^2 + a \):
\[\text{Area} = \int_{0}^{3} (a^2 + a) \, da\]
First, we find the integral:
\[\int (a^2 + a) \, da = \frac{a^3}{3} + \frac{a^2}{2}\]
Now, evaluate from \( 0 \) to \( 3 \):
\[ \left[\frac{3^3}{3} + \frac{3^2}{2}\right] - \left[0\right] = \left[9 + 4.5\right] = 13.5 \]
3. Set the slope equal to the area:
\[12 + k = 13.5\]
Solving for \( k \):
\[k = 13.5 - 12 = 1.5\]
Therefore, the value of \( k \) is \( 1.5 \), so the answer is A: 1.5.