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

x^{3}-10x+11=0

are

u,v,w

, then find the value of

3\csc^{2}(\tan^{-1}u+\tan^{-1}v+\tan^{-1}w)

Solution

Answer: 6

Step 1: By Vieta's formulas for

x^{3}+0\cdot x^{2}-10x+11=0

S_{1}=u+v+w=0

S_{2}=uv+vw+wu=-10

S_{3}=uvw=-11

Step 2: Let

\alpha=\tan^{-1}u

\beta=\tan^{-1}v

\gamma=\tan^{-1}w

, so

\tan\alpha=u

\tan\beta=v

\tan\gamma=w

. Step 3: Use the identity:

\tan(\alpha+\beta+\gamma)=\dfrac{S_{1}-S_{3}}{1-S_{2}}=\dfrac{0-(-11)}{1-(-10)}=\dfrac{11}{11}=1

Step 4: Therefore:

\alpha+\beta+\gamma=\tan^{-1}1=\dfrac{\pi}{4}

Step 5: Hence:

3\csc^{2}(\alpha+\beta+\gamma)=3\csc^{2}\dfrac{\pi}{4}=3\cdot(\sqrt{2})^{2}=3\cdot 2=6

Answer:

6

Still stuck on this question?Ask your doubt on WhatsApp

Solve more, learn faster

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