JEE Main 2026 Apr 8 Shift 2, Mathematics Q5: Arithmetic, Geometric And Harmonic Progressions

Split the $40$-term sum into two $20$-term arithmetic progressions:

Odd positions: $3+8+13+\dots$ ($a=3,d=5$) gives $\frac{20}{2}(6+19\cdot 5)=10\cdot 101=1010$.

Even positions: $4+9+14+\dots$ ($a=4,d=5$) gives $\frac{20}{2}(8+19\cdot 5)=10\cdot 103=1030$.

Total: $\alpha =2040$, so $\frac{\alpha }{1020}=2$.

The roots of $x^2+x-2=0$ are $x=1,-2$. Since $\beta \in (0,\pi /2)$, $\tan \beta >0$, so $(\tan \beta {)}^2=1\Rightarrow \tan \beta =1\Rightarrow \beta =\frac{\pi }{4}$.

Then ${\sin }^2\beta ={\cos }^2\beta =\frac{1}{2}$, giving ${\sin }^2\beta +3{\cos }^2\beta =\frac{1}{2}+\frac{3}{2}=2$.