Question

The slope of a function y = x3 + kx at x= 2 is equal to the area under the curve z = a2 + a between points a = 0 and a = 3 Then the value of k is

• A. 1.5
• B. 5.5
• C. 6.5
• D. Cannot be determined

Answer 1

Answer: (A)

1.5

Answer 2

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.