Theory of EquationsmediumPYQ · JEE Main · 5 Apr 2026 · Shift 1 (Morning)Free

20 sin²((A+B)/2) = 10 - 3√10 | JEE Main 2026

JEE Maths question with a full step-by-step solution.

Question
Let tan⁡A,tan⁡B\tan A,\tan B, where A,B∈(−π2,π2)A,B\in\left(-\dfrac{\pi}{2},\dfrac{\pi}{2}\right), be the roots of x2−2x−5=0x^2-2x-5=0. Then 20sin⁡2(A+B2)20\sin^2\left(\dfrac{A+B}{2}\right) is equal to
A10+1010+\sqrt{10}
B10−21010-2\sqrt{10}
C10−31010-3\sqrt{10}correct
D10−1010-\sqrt{10}
Solution
Step 1: By Vieta on x2−2x−5=0x^2-2x-5=0:
tan⁡A+tan⁡B=2,tan⁡Atan⁡B=−5.\tan A+\tan B=2,\qquad \tan A\tan B=-5.
 Step 2: tan⁡(A+B)=tan⁡A+tan⁡B1−tan⁡Atan⁡B=21−(−5)=26=13\tan(A+B)=\dfrac{\tan A+\tan B}{1-\tan A\tan B}=\dfrac{2}{1-(-5)}=\dfrac{2}{6}=\dfrac13.
∴ cos⁡(A+B)=312+32=310.\therefore\ \cos(A+B)=\frac{3}{\sqrt{1^2+3^2}}=\frac{3}{\sqrt{10}}.
 Step 3: sin⁡2 ⁣(A+B2)=1−cos⁡(A+B)2\sin^2\!\left(\dfrac{A+B}{2}\right)=\dfrac{1-\cos(A+B)}{2}.
20sin⁡2 ⁣(A+B2)=10(1−cos⁡(A+B))=10(1−310).20\sin^2\!\left(\frac{A+B}{2}\right)=10\big(1-\cos(A+B)\big)=10\left(1-\frac{3}{\sqrt{10}}\right).
 Step 4: 1010=10\dfrac{10}{\sqrt{10}}=\sqrt{10}:
10−3010=10−310.10-\frac{30}{\sqrt{10}}=10-3\sqrt{10}.
 Correct answer: (3)
Still stuck on this question?Ask your doubt on WhatsApp
Similar questions
Theory of Equations · expert
If roots of the equation a(x−ℓ)(x−m)+n=0a(x-\ell)(x-m) + n = 0 are α,β\alpha, \beta, then roots of the equation a ⁣(xx−k−α) ⁣(xx−k−β)−n=0a\!\left(\dfrac{x}{x-k} - \alpha\right)\!\left(\dfrac{x}{x-k} - \beta\right) - n = 0 are
Theory of Equations · medium
If α\alpha and β\beta are the roots of the equation x2−2x−5=0x^2 - 2x - 5 = 0, then the equation whose roots are αα−1\dfrac{\alpha}{\alpha-1} and ββ−1\dfrac{\beta}{\beta-1} is
Theory of Equations · hard
Consider ax2−bx+c=0ax^2-bx+c=0 with a,b,c∈Na, b, c\in\mathbb{N}. If it has two distinct real roots in (1,2)(1,2), then
Theory of Equations · medium
Find the only integer which does not lie in the range of the function f(x)=x2−3x+2x2+x−6.f(x) = \frac{x^{2}-3x+2}{x^{2}+x-6} .

Solve more, learn faster

Sign up free to solve more JEE Maths questions and explore doMath — timed drills, mastery sprints, bookmarks, and chapter-wise progress tracking.

Sign up freeExplore doMath