Question

If a curve $y = a\sqrt{x} + bx$ passes through the point $(1, 2)$ and the area bounded by this curve, line $x = 4$ and the X -axis is 8 sq . units, then the value of $a - b$ is

• A. -2
• B. 2
• C. -4
• D. 4

Hint:

Area under curve $= \int y dx$.

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:
Point $(1,2)$ satisfies the curve equation.
Area calculation involves definite integration from $x=0$ to $x=4$.
Step 2: Key Formula or Approach:
At $(1,2): 2 = a(1) + b(1) \implies a + b = 2$.
Area $A = \int_0^4 (a\sqrt{x} + bx) dx = 8$.
Step 3: Detailed Explanation:
$[ a \frac{2}{3}x^{3/2} + b \frac{x^2}{2} ]_0^4 = 8 \implies \frac{16a}{3} + 8b = 8 \implies \frac{2a}{3} + b = 1$.
Solve $a+b=2$ and $2a/3+b=1$.
$a - 2a/3 = 2 - 1 \implies a/3 = 1 \implies a = 3, b = -1$.
$a - b = 3 - (-1) = 4$.
Step 4: Final Answer:
The value of $a - b$ is 4.