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: Equation from Point
Passes through $(1, 2) \implies 2 = a(1) + b(1) \implies a + b = 2$.
Step 2: Area Integral
$\int_0^4 (a\sqrt{x} + bx) dx = 8$.
$[a \frac{x^{3/2}}{3/2} + \frac{bx^2}{2}]_0^4 = 8 \implies \frac{2a}{3}(8) + \frac{b}{2}(16) = 8$.
$\frac{16a}{3} + 8b = 8 \implies 2a + 3b = 3$.
Step 3: Solve
From $a+b=2 \implies b = 2-a$.
$2a + 3(2-a) = 3 \implies 2a + 6 - 3a = 3 \implies a = 3$.
$b = 2-3 = -1$.
$a - b = 3 - (-1) = 4$.
Final Answer: (D)