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

Updated On: Sep 25, 2024
  • 1.5
  • 5.5
  • 6.5
  • Cannot be determined
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The Correct Option is A

Solution and Explanation

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.
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