Question

If \( \displaystyle\int_{0}^{1} (9x^{2} - 4x - 2k)\,dx = 0 \), then the value of \( k \) is :

• A. 1
• B. 0
• C. \( \dfrac{1}{2} \)
• D. \( -\dfrac{1}{2} \)

Hint:

Integrate, apply limits 0 and 1, and set the result to 0.

Answer 1

The Correct Option is C

Step 1: Split the integral:
Write the integral as $ 9\int_0^1 x^{2}dx - 4\int_0^1 x\,dx - 2k\int_0^1 dx $.

Step 2: Evaluate each piece:
$ \int_0^1 x^{2}dx = \dfrac13 $, $ \int_0^1 x\,dx = \dfrac12 $ and $ \int_0^1 dx = 1 $.

Step 3: Substitute:
The total is $ 9 \cdot \dfrac13 - 4 \cdot \dfrac12 - 2k \cdot 1 = 3 - 2 - 2k $.

Step 4: Solve for k:
From $ 1 - 2k = 0 $ we get $ k = 0.5 $.

Step 5: Verify:
Put $ k = \dfrac12 $ back. The integrand becomes $ 9x^{2} - 4x - 1 $, and its integral is $ 3 - 2 - 1 = 0 $. So option (C) works.

Final Answer:
The value of $ k $ is $ \dfrac12 $. \[ \boxed{k = \dfrac{1}{2}} \]