Quantrex Quantrex AcademyJEE · NEET · NDA PYQs with solutions Open app
NTA Abhyas 2020
Exam
NTA Abhyas
Year
2020
Subject
Mathematics
Chapter
Continuity and Differentiability

NTA Abhyas 2020 · Mathematics · Continuity and Differentiability

If f(x) = \ cases (1 + | x|)(p)/(| x|) & : & - (π)/(6) < x < 0 q &...

If f\left(x\right) = \left\{\begin{cases}\left(1 + \left|\sin x\right|\right)\dfrac{p}{\left|\sin x\right|} & : & - \dfrac{\pi}{6} < x < 0 \\ q & : & x = 0 \\ {e}^{\tan 3x . \cot 5x} & : & 0 < x < \dfrac{\pi}{6}\end{cases}\right) is continuous atx = 0,thenthevalueof, then the value of2p + 10ln q is equal to

Correct answer

7.20

Step-by-step solution

lim⁡x→0+f(x)=q⇒lim⁡x→0+etan⁡3xtan⁡5x=q\lim_{x \to {0}^{+}}f\left(x\right) = q ⇒ \lim_{x \to {0}^{+}}{e}^{\dfrac{\tan 3x}{\tan 5x}} = q
=lim⁡x→0+etan⁡3x3x.5xtan⁡5x.3x5x⇒e35=q= \lim_{x \to {0}^{+}}{e}^{\dfrac{\tan 3x}{3x} . \dfrac{5x}{\tan 5x} . \dfrac{3x}{5x}} ⇒ {e}^{\dfrac{3}{5}} = q
Now, lim⁡x→0−f(x)=q⇒elim⁡x→0−(1−sinx)−psin⁡x=q\lim_{x \to {0}^{-}}f\left(x\right) = q ⇒ {e}^{\lim_{x \to {0}^{-}}\left(1 - sin x\right)\dfrac{- p}{\sin x}} = q
=elim⁡x→0−−psinx(1−sinx−1)=ep= {e}^{\lim_{x \to {0}^{-}}\dfrac{- p}{sin x}\left(1 - sin x - 1\right)} = {e}^{p}
∴ep=e35⇒p=35,q=e35∴ {e}^{p} = e\dfrac{3}{5} ⇒ p = \dfrac{3}{5}, q = e\dfrac{3}{5}
2p+10.lnq=65+10(35)=3652p + 10 . ln q = \dfrac{6}{5} + 10\left(\dfrac{3}{5}\right) = \dfrac{36}{5}
Practice Continuity and Differentiability on Quantrex Academy →

More from Continuity and Differentiability

Let ℝ denote the set of all real numbers. Let f : ℝ arrow ℝ be an arbitrary function and… JEE Advanced 2026Consider the function f : (-(π)/(2), (π)/(2)) arrow (-∞, ∞) defined by f(x) = (|x| + |x… JEE Advanced 2026For a real number α , let [α] denote the greatest integer less than or equal to α . For a… JEE Advanced 2026For the function f(x) = e^ |x| - |x| , x ∈ ℝ , consider the following statements… JEE Main 2026Let f(x) = cases (1)/(3), & x ≤ π/2 (b(1- x))/((π-2x)^2), & x > π/2 cases . If f is… JEE Main 2026Let f(x) = cases x^3 + 8 ; & x Then the number of points, where the function g ∘ f is… JEE Main 2026Let f(x)= cases e^ x-1 , & x<0 x^2-5x+6, & x ≥ 0 cases and g(x)=f(|x|)+|f(x)| . If the… JEE Main 2026The number of points, at which the function f(x) = \ 6x, 2 + 3x^2\ + |x - 1|| |x^2… JEE Main 2026The number of points in the interval [2, 4] , at which the function f(x) = [x^2 - x… JEE Main 2026Let f(x)= _ θ arrow 0 ( π x-x^ ((2)/(θ)) (x-1) 1+x^ ((2)/(θ)) (x-1) ), x ∈ R . Consider… JEE Main 2026Let [t] denote the greatest integer less than or equal to t . If the function f(x)=\… JEE Main 2026If the function f(x)= e^ x (e^ x-x -1)+ _ e ( x+ x)-x x-x is continuous at x=0 , then the… JEE Main 2026Let α, β ∈ ℝ be such that the function f(x)= cases 2 α(x^ 2 -2)+2 β x &, x be… JEE Main 2026If f(x)=\ array cc a|x|+x^ 2 -2( |x|)( |x|) x &, x ≠ 0 b &, x=0 array . is continuous at… JEE Main 2026Let f(x)= cases a x^ 2 +2 a x+3 4 x^ 2 +4 x-3 &, x ≠-(3)/(2), (1)/(2) ~b &, x=-(3)/(2)… JEE Main 2026Let [·] denote the greatest integer function, and let f(x)= \ 2 x, x^ 2 \ . Let S = \ x… JEE Main 2026 Full Continuity and Differentiability list Continuity and Differentiability JEE Maths PYQsJEE Mathematics PYQs All NTA Abhyas JEE Main PYQs