Inverse Trigonometric FunctionsmediumFree

Inverse Trigonometric Functions: Roots Equation 10x Find Value

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

Question
If the roots of the equation x3−10x+11=0x^{3}-10x+11=0 are u,v,wu,v,w, then find the value of 3csc⁡2(tan⁡−1u+tan⁡−1v+tan⁡−1w)3\csc^{2}(\tan^{-1}u+\tan^{-1}v+\tan^{-1}w).
Solution
Answer: 6
Step 1: By Vieta's formulas for x3+0⋅x2−10x+11=0x^{3}+0\cdot x^{2}-10x+11=0:
S1=u+v+w=0S_{1}=u+v+w=0
S2=uv+vw+wu=−10S_{2}=uv+vw+wu=-10
S3=uvw=−11S_{3}=uvw=-11
 Step 2: Let α=tan⁡−1u\alpha=\tan^{-1}u, β=tan⁡−1v\beta=\tan^{-1}v, γ=tan⁡−1w\gamma=\tan^{-1}w, so tan⁡α=u\tan\alpha=u, tan⁡β=v\tan\beta=v, tan⁡γ=w\tan\gamma=w. Step 3: Use the identity:
tan⁡(α+β+γ)=S1−S31−S2=0−(−11)1−(−10)=1111=1\tan(\alpha+\beta+\gamma)=\dfrac{S_{1}-S_{3}}{1-S_{2}}=\dfrac{0-(-11)}{1-(-10)}=\dfrac{11}{11}=1
 Step 4: Therefore:
α+β+γ=tan⁡−11=π4\alpha+\beta+\gamma=\tan^{-1}1=\dfrac{\pi}{4}
 Step 5: Hence:
3csc⁡2(α+β+γ)=3csc⁡2π4=3⋅(2)2=3⋅2=63\csc^{2}(\alpha+\beta+\gamma)=3\csc^{2}\dfrac{\pi}{4}=3\cdot(\sqrt{2})^{2}=3\cdot 2=6
 Answer: 66
Still stuck on this question?Ask your doubt on WhatsApp
Similar questions
Inverse Trigonometric Functions · medium
Let SS be the set of domain of f(x)=π2−tan⁡−1−x2+5x−6f(x)=\sqrt{\dfrac{\pi}{2}-\tan^{-1}\sqrt{-x^{2}+5x-6}}. If λ=α+1α\lambda=\alpha+\dfrac{1}{\alpha} where α∈S\alpha\in S and λ\lambda is an integer, then find the value of λ2\lambda^{2}.
Inverse Trigonometric Functions · medium
If x=sin⁡−1(a6+1)+cos⁡−1(a4+1)−tan⁡−1(a2+1)x=\sin^{-1}(a^{6}+1)+\cos^{-1}(a^{4}+1)-\tan^{-1}(a^{2}+1), a∈Ra\in\mathbb{R}, then find the value of sec⁡2x\sec^{2}x.
Inverse Trigonometric Functions · medium
If the domain of the function f(x)=3cos⁡−1(4x)−πf(x)=\sqrt{3\cos^{-1}(4x)-\pi} is [a,b][a,b], then find the value of (4a+64b)(4a+64b).
Inverse Trigonometric Functions · medium
If cot⁡−1(4+24)+cot⁡−1(4+64)+cot⁡−1(4+124)+…∞=tan⁡−1(ab)\cot^{-1}\left(4+\dfrac{2}{4}\right)+\cot^{-1}\left(4+\dfrac{6}{4}\right)+\cot^{-1}\left(4+\dfrac{12}{4}\right)+\ldots\infty=\tan^{-1}\left(\dfrac{a}{b}\right) where aa and bb are coprime, then find the value of (a3+b3)(a^{3}+b^{3}).

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