JEE Mathematics 2024 Question with Solution

Given: The tangents at $x=1$ and $x=3$ make angles $\pi/6$ and $\pi/4$ with the positive $x$-axis.

Find: The value of $\alpha+\beta$.

From slope of tangent,

$$f'(1)=\tan\left(\frac{\pi}{6}\right)=\frac{1}{\sqrt{3}}, \qquad f'(3)=\tan\left(\frac{\pi}{4}\right)=1$$

the solution evaluates the integral by substituting

$$u=f'(t), \qquad d\nu=f''(t) \, dt$$

so that

$$I=\int_{1/\sqrt{3}}^{1}(u^2+1) \, du$$

Then,

$$I=\left[\frac{u^3}{3}+u\right]_{1/\sqrt{3}}^{1}$$ $$=\left(\frac{1}{3}+1\right)-\left(\frac{1}{9\sqrt{3}}+\frac{1}{\sqrt{3}}\right)$$ $$=\frac{4}{3}-\frac{10}{9\sqrt{3}}=\frac{4}{3}-\frac{10\sqrt{3}}{27}$$

According to the extracted solution, the intended expression is scaled to give integer coefficients:

$$27I=27\left(\frac{4}{3}-\frac{10\sqrt{3}}{27}\right)=36-10\sqrt{3}$$

Hence,

$$\alpha=36, \qquad \beta=-10$$

and therefore

$$\alpha+\beta=26$$

So, the correct option is B.

Note: The solution itself flags a discrepancy in the printed integral and proceeds with the intended form to obtain the listed answer.

Given: The tangent slopes are determined by the given angles.

Identify principle: If a tangent makes angle $\theta$ with the positive $x$-axis, then its slope is $\tan\theta$.

Thus,

$$f'(1)=\frac{1}{\sqrt{3}}, \qquad f'(3)=1$$

Now use the substitution suggested by the integrand:

$$u=f'(t)$$

Then,

$$d\nu=f''(t) \, dt$$

Therefore,

$$\int \left((f'(t))^2+1\right)f''(t) \, dt=\int (u^2+1) \, du$$

With the corresponding limits used in the solution,

$$u=\frac{1}{\sqrt{3}} \text{ at the lower end, and } u=1 \text{ at the upper end}$$

So,

$$\int_{1/\sqrt{3}}^{1}(u^2+1) \, du$$ $$=\left[\frac{u^3}{3}+u\right]_{1/\sqrt{3}}^{1}$$ $$=\frac{4}{3}-\left(\frac{1}{9\sqrt{3}}+\frac{1}{\sqrt{3}}\right)$$ $$=\frac{4}{3}-\frac{10\sqrt{3}}{27}$$

The extracted solution then compares the scaled expression with $\alpha+\beta\sqrt{3}$:

$$36-10\sqrt{3}=\alpha+\beta\sqrt{3}$$

Hence,

$$\alpha=36, \qquad \beta=-10$$

Thus,

$$\alpha+\beta=26$$

Therefore, the correct option is B.