Can you solve IIT-JEE 2026 Question?
A math educator solves an IIT-JEE 2026 trigonometry problem involving a quadratic equation whose roots are tan(a) and tan(b). Using Vieta's formulas and trigonometric identities, the presenter derives the value of 20·sin²((a+b)/2), arriving at the answer 10 - 3√10. The video is framed around the presenter's personal connection to IIT-JEE results from 9 years ago.
Summary
The video opens with the presenter acknowledging that IIT-JEE results were announced the previous day, and reflecting that their own JEE result 9 years ago changed their life completely. This personal anecdote sets a motivational tone before diving into the mathematical problem.
The problem presented is the quadratic equation x² - 2x - 5 = 0, where tan(a) and tan(b) are given as its roots. Using Vieta's formulas, the presenter identifies that tan(a) + tan(b) = 2 (i.e., -b/a = -(-2)/1) and tan(a)·tan(b) = -5 (i.e., c/a = -5/1).
With these values, the presenter applies the tangent addition formula: tan(a+b) = (tan a + tan b) / (1 - tan a·tan b) = 2 / (1 - (-5)) = 2/6 = 1/3.
A right triangle is then constructed with tan(a+b) = 1/3, giving opposite side = 1, adjacent side = 3, and hypotenuse = √10. This yields cos(a+b) = 3/√10.
Finally, the half-angle identity sin²(θ/2) = (1 - cosθ)/2 is applied with θ = a+b, giving sin²((a+b)/2) = (1 - 3/√10)/2 = (√10 - 3)/(2√10). Multiplying by 20 (as implied by the problem's expression 20·sin²((a+b)/2)) gives the final answer: 10 - 3√10, which corresponds to option D.
Key Insights
- The presenter uses Vieta's formulas to extract tan(a) + tan(b) = 2 and tan(a)·tan(b) = -5 directly from the quadratic x² - 2x - 5 = 0 without solving for individual roots.
- The presenter applies the tangent addition formula tan(a+b) = (tan a + tan b)/(1 - tan a·tan b), substituting the Vieta values to get tan(a+b) = 2/6 = 1/3.
- The presenter constructs a right triangle with tan(a+b) = 1/3 to geometrically determine that cos(a+b) = 3/√10, avoiding abstract algebraic manipulation.
- The half-angle identity sin²(θ/2) = (1 - cosθ)/2 is identified as a key formula that must be memorized to solve this problem, according to the presenter.
- After multiplying sin²((a+b)/2) by 20 as required by the problem, the presenter arrives at the final answer 10 - 3√10, identifying option D as correct.
Topics
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