JEE Mains · Maths · STD 12 - 1. relation and function
Let \(f, \mathrm{~g}:(1, \infty) \rightarrow \mathbb{R}\) be defined as \(f(\mathrm{x})=\frac{2 x+3}{5 x+2}\) and \(g(x)=\frac{2-3 x}{1-x}\). If the range of the function \(f \circ g:[2,4] \rightarrow \mathbb{R}\) is \([\alpha, \beta]\), then \(\frac{1}{\beta-\alpha}\) is equal to
- A 68
- B 29
- C 2
- D 56
Answer & Solution
Correct Answer
(D) 56
Step-by-step Solution
Detailed explanation
\begin{aligned} & \operatorname{fog}(x)=f(g(x)) \\ & =f\left(\frac{2-3 x}{1-x}\right)=\frac{2\left(\frac{2-3 x}{1-x}\right)+3}{5\left(\frac{2-3 x}{1-x}\right)+2} \\ & =\frac{4-6 x+3-3 x}{10-15 x+2-2 x}=\left(\frac{7-9 x}{12-17 x}\right) \\ & \therefore\left[\begin{array}{c} 12-7…
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
- The area (in sq. units) bounded by the parabola \(y = x^2 -1\), the tangent at the point \((2, 3)\) into it and the \(y -\) axis isJEE Mains 2019 Hard
- If the plane \(2 x + y -5 z =0\) is rotated about its line of intersection with the plane \(3 x-y+4 z-7=0\) by an angle of \(\frac{\pi}{2}\), then the plane after the rotation passes through the pointJEE Mains 2022 Hard
- Let \(a>0, b>0\). Let \(e\) and \(\ell\) respectively be the eccentricity and length of the latus rectum of the hyperbola \(\frac{ x ^{2}}{ a ^{2}}-\frac{ y ^{2}}{ b ^{2}}=1\). Let \(e ^{\prime}\) and \(\ell^{\prime}\) respectively the eccentricity and length of the latus rectum of its conjugate hyperbola. If \(e ^{2}=\frac{11}{14} \ell\) and \(\left( e ^{\prime}\right)^{2}=\frac{11}{8} \ell^{\prime}\), then the value of \(77 a+44 b\) is equal toJEE Mains 2022 Hard
- The value of \(36(4 \cos ^2 9^{\circ}-1)(4 \cos ^2 27^{\circ}-1) (4\cos ^2 81^{\circ}-1) (4 \cos ^2 243^{\circ}-1)\) isJEE Mains 2023 Hard
- If \(\beta \) is one of the angles between the normals to the ellipse, \(x^2 + 3y^2 = 9\) at the points \(\left( {3\cos \theta ,\sqrt {3\,} \sin \theta } \right)\) and \(\left( { - 3\sin \,\theta ,\sqrt 3 \,\cos \theta } \right); \in \left( {0,\frac{\pi }{2}} \right)\); then \(\frac{{2\,\cot \beta }}{{\sin \,2\theta }}\) is equal toJEE Mains 2018 Hard
- For \(p\,>\,0\), a vector \(\vec{v}_{2}=2 \hat{i}+(p+1) \hat{j}\) is obtained by rotating the vector \(\vec{v}_{1}=\sqrt{3} p \hat{i}+\hat{j}\) by an angle \(\theta\) about origin in counter clockwise direction. If \(\tan \theta=\frac{(\alpha \sqrt{3}-2)}{4 \sqrt{3}+3}\), then the value of \(\alpha\) is equal to \(....\)JEE Mains 2021 Hard
More PYQs from JEE Mains
- Let \(f: R-\left\{\frac{\alpha}{6}\right\} \rightarrow R\) be defined by \(f(x)=\frac{5 x+3}{6 x-\alpha} .\) Then the value of \(\alpha\) for which \((fof)(x)=x\), for all \(x \in R-\left\{\frac{\alpha}{6}\right\}\), is:JEE Mains 2021 Medium
- The product \(2^{\frac{1}{4}} \cdot 4^{\frac{1}{16}} \cdot 8^{\frac{1}{48}} \cdot 16^{\frac{1}{128}} \cdot \ldots .\) to \(\infty\) is equal toJEE Mains 2020 Hard
- Let \(f: R \rightarrow R\) be defined as \(f(x)=\left\{\begin{array}{ll}\frac{x^{3}}{(1-\cos 2 x)^{2}} \log _{e}\left(\frac{1+2 x e^{-2 x}}{\left(1-x e^{-x}\right)^{2}}\right), & x \neq 0 \\ \,\alpha & , x=0\end{array}\right.\) If \(\mathrm{f}\) is continuous at \(\mathrm{x}=0\), then \(\alpha\) is equal to :JEE Mains 2021 Hard
- The area(in sq. units) of the region bounded by the curves \(y = 2^x\) and \(y = |x +1|\) in the first quadrant isJEE Mains 2019 Hard
- If \(S = \left\{\theta \in [-\pi, \pi] : \cos\theta \cos\dfrac{5\theta}{2} = \cos 7\theta \cos\dfrac{7\theta}{2}\right\}\), then \(n(S)\) is equal to _______.JEE Mains 2026 Hard
- The value of the integral \(\int \limits_{-\pi / 2}^{\pi / 2} \frac{d x}{\left(1+e^{x}\right)\left(\sin ^{6} x+\cos ^{6} x\right)}\) is equal toJEE Mains 2022 Hard