Question:

Let y=f(x) y = f(x) be a thrice differentiable function in (5,5) (-5, 5) . Let the tangents to the curve y=f(x) y = f(x) at (1,f(1)) (1, f(1)) and (3,f(3)) (3, f(3)) make angles π6 \frac{\pi}{6} and π4 \frac{\pi}{4} , respectively, with the positive x-axis. If 2131((f(t))2+1)f(t)dt=α+β32 \int_{\frac{1}{\sqrt{3}}}^{1} \left( \left( f'(t) \right)^2 + 1 \right) f''(t) \, dt = \alpha + \beta \sqrt{3} where α \alpha , β \beta are integers, then the value of α+β \alpha + \beta equals

Updated On: Mar 20, 2025
  • -14
  • 26
  • -16
  • 36
Hide Solution
collegedunia
Verified By Collegedunia

The Correct Option is B

Solution and Explanation

From the tangents, we find:

f(1)=13,f(3)=1. f'(1) = \frac{1}{\sqrt{3}}, \quad f'(3) = 1.

Assume f(t)=t f'(t) = t (consistent with the slopes), then f(t)=1 f''(t) = 1 .

Substitute into the integral:

2131(t2+1)dt=α+β3. 2 \int_{\frac{1}{\sqrt{3}}}^{1} \left( t^2 + 1 \right) dt = \alpha + \beta \sqrt{3}.

α+β3=27(4310273)=36103. \alpha + \beta \sqrt{3} = 27 \left( \frac{4}{3} - \frac{10}{27} \sqrt{3} \right) = 36 - 10\sqrt{3}.

Here α=36,β=10 \alpha = 36, \beta = -10 .

α+β=3610=26 \alpha + \beta = 36 - 10 = 26

Compute the integral to find α+β=26 \alpha + \beta = 26 .

Was this answer helpful?
2
0