enEnglishguગુજરાતી
JEE Mains · Maths · STD 12 - 6. Application of derivatives
If \(\theta_{1}\) and \(\theta_{2}\) be respectively the smallest and the largest values of \(\theta\) in \((0,2 \pi)-\{\pi\}\) which satisfy the equation, \(\quad 2 \cot ^{2} \theta-\frac{5}{\sin \theta}+4=0,\) then \(\int\limits_{\theta_{1}}^{\theta_{2}} \cos ^{2} 3 \theta \mathrm{d} \theta \) is equal to
- A \(\frac{2\pi}{3}\)
- B \(\frac{\pi}{3}+\frac{1}{6}\)
- C \(\frac{\pi}{9}\)
- D \(\frac{\pi}{3}\)
Answer & Solution
Correct Answer
(D) \(\frac{\pi}{3}\)
Step-by-step Solution
Detailed explanation
\(2 \cos ^{2} \theta-5 \sin \theta+4 \sin ^{2} \theta=0\) \(3 \sin ^{2} \theta-5 \sin \theta+2=0\) \(\sin \theta=\frac{1}{2}, 2(\text { Rejected })\)…
See the Complete Solution
Get step-by-step explanations for this and 2.5 Lakh+ more JEE, NEET & CET questions.
- Unlock all solutions
- Practice the full chapter
- Track accuracy across PYQs
4.8 rated on Google Play · 14,000+ reviews
More questions from Maths
- If \(\cot ^{-1}(\alpha)=\cot ^{-1} 2+\cot ^{-1} 8+\cot ^{-1} 18\) \(+\cot ^{-1} 32+\ldots . .\) upto \(100\) terms, then \(\alpha\) isJEE Mains 2021 Hard
- Let the sum of the first \(n\) terms of a non-constant \(A.P., a_1, a_2, a_3, ……\) be \(50\,n\, + \,\frac{{n\,(n\, - 7)}}{2}A,\) where \(A\) is a constant. If \(d\) is the common difference of this \(A.P.,\) then the ordered pair \((d,a_{50})\) is equal toJEE Mains 2019 Hard
- If \(\int\limits_0^x {f\left( t \right)} dt = {x^2} + \int\limits_x^1 {{t^2}f\left( t \right)dt} \), then \(f'(1/2)\) isJEE Mains 2019 Hard
- Mean of \(5\) observations is \(7.\) If four of these observations are \(6, 7, 8, 10\) and one is missing then the variance of all the five observations isJEE Mains 2013 Medium
- The integral \(\int_{1 / 4}^{3 / 4} \cos \left(2 \cot ^{-1} \sqrt{\frac{1-\mathrm{x}}{1+\mathrm{x}}}\right) \mathrm{dx}\) is equal to :JEE Mains 2024 Medium
- Let \(\alpha\) and \(\beta\) be the roots of \(x^{2}-3 x+p=0\) and \(\gamma\) and \(\delta\) be the roots of \(x^{2}-6 x+q=0 .\) If \(\alpha\) \(\beta, \gamma, \delta\) form a geometric progression. Then ratio \((2 q+p):(2 q-p)\) isJEE Mains 2020 Hard
More PYQs from JEE Mains
- If the two lines \(l_{1}: \frac{ x -2}{3}=\frac{ y +1}{-2}, z =2\) and \(l_{2}: \frac{x-1}{1}=\frac{2 y+3}{\alpha}=\frac{z+5}{2}\) perpendicular, then an angle between the lines \(l_{2}\) and \(l_{3}: \frac{1- x }{3}=\frac{2 y -1}{-4}=\frac{ z }{4}\) isJEE Mains 2022 Medium
- Let \(S\) be the set of all functions \(f:[0,1] \rightarrow \mathrm{R}\) which are continuous on \([0,1]\) and differentiable on \((0,1) .\) Then for every \(f\) in \(\mathrm{S},\) there exists a \(\mathrm{c} \in(0,1),\) depending on \(f,\) such thatJEE Mains 2020 Hard
- If \(f(x)=\sin \left(\cos ^{-1}\left(\frac{1-2^{2 x}}{1+2^{2 x}}\right)\right)\) and its first derivative with respect to \(x\) is \(-\frac{ b }{ a } \log _{ e } 2\) when \(x =1,\) where \(a\) and \(b\) are integers, then the minimum value of \(\left| a ^{2}- b ^{2}\right|\) is.........JEE Mains 2021 Hard
- If \(x\, = \,{\sin ^{ - 1}}(\sin \,10)\) and \(y = \,{\cos ^{ - 1}}\,(\cos \,10)\) , then \(y -x\) is equal toJEE Mains 2019 Hard
- If the coefficients of \(x^2\) and \(x^3\) are both zero, in the expansion of the expression \((1 + ax + bx^2) (1 -3x)^{t5}\) in powers of \(x\), then the ordered pair \((a, b)\) is equal toJEE Mains 2019 Hard
- Let \(A (1,4)\) and \(B (1,-5)\) be two points. Let \(P\) be a point on the circle \((x-1)^{2}+(y-1)^{2}=1\) such that \(( PA )^{2}+( PB )^{2}\) have maximum value, then the points \(P , A\) and \(B\) lie on ...... .JEE Mains 2021 Hard