Let f, ~g:(1, ) R be defined as f(x)=2 x+3 5 x+2 and g(x)=2-3 x 1… - Vidyadip

MCQ

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 $fo g:[2,4] \rightarrow \mathbb{R}$ is $[\alpha, \beta]$, then $\frac{1}{\beta-\alpha}$ is equal to

A
68
B
29
C
2
✓
56

✓

Answer

Correct option: D.

56

(D) 56
fog $(x)=f(g(x))$
$=f\left(\frac{2-3 x}{1-x}\right)=\frac{2\left(\frac{2-3 x}{1-x}\right)+3}{5\left(\frac{2-3 x}{1-x}\right)+2}$
$=\frac{4-6 x+3-3 x}{10-15 x+2-2 x}=\left(\frac{7-9 x}{12-17 x}\right)$
$\therefore\left[\begin{array}{c}12-7 \mathrm{x} \neq 0 \\ \mathrm{x} \neq \frac{12}{17}\end{array}\right.$
$\left[\begin{array}{l}\operatorname{fog}(2)=\frac{7-9(2)}{12-17(2)}=\frac{-11}{-22}=\frac{1}{2} \\ \operatorname{fog}(4)=\frac{7-9(4)}{12-17(4)}=\frac{-29}{-56}=\frac{29}{56}\end{array}\right.$
Range of fog : $[\alpha, \beta]=\left[\frac{1}{2}, \frac{29}{56}\right]$
$\therefore(\beta-\alpha)=\frac{29}{56}-\frac{1}{2}=\frac{29-28}{56}=\frac{1}{56}$
$\frac{1}{(\beta-\alpha)}=56$

Need a full question paper?

Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.

Start Generating Free

Explore more

More "SECTION - A [MATHS - MCQ]" from JEE Main 4-April-2025 Paper - Shift 1→JEE Main 4-April-2025 Paper - Shift 1 — all question types→JEE STD 11 Science question papers→Gujarat Board STD 11 Science question papers→Gujarat Board question paper generator→

Similar questions

The curve represented by $x = 3(\cos t + \sin t)$, $y = 4(\cos t - \sin t)$ is

In a football championship, there were played $153$ matches. Every team played one match with each other. The number of teams participating in the championship is

$a.(b × c)$ is equal to

The maximum value of the function $f\,(x)\, = 3{x^3} - 18{x^2} + 27x\,\, - 40$ on the set $S = \{ x\, \in \,R\,:\,{x^2}\, + \,30\, \le \,11x\} $ is

The solution of the differential equation $\frac{{dy}}{{dx}} + \frac{y}{x} = {x^2}$is

If $a$ and $b$ are the odd integers, then the roots of the equation $2a{x^2} + (2a + b)x + b = 0,\;a \ne 0,$ will be

The differential equations of all circles passing through origin and having their centres on the $x$ - axis is

The height of a right circular cylinder of maximum volume inscribed in a sphere of radius $3$ is

Let a rectangle $A B C D$ of sides $2$ and $4$ be inscribed in another rectangle $P Q R S$ such that the vertices of the rectangle $A B C D$ lie on the sides of the rectangle $P Q R S$. Let $a$ and $b$ be the sides of the rectangle $P Q R S$ when its area is maximum. Then $(a+b)^2$ is equal to :

Let $z \in C$ with $Im(z) = 10$ and it satisfies $\frac{{2z - n}}{{2z + n}} = 2i - 1$ for some natural number $n$. Then