Download App

JEE Main 04-Apr-2024 Paper - Shift 2 — JEE STD 11 Science — Question

Gujarat BoardEnglish MediumSTD 11 ScienceJEEJEE Main 04-Apr-2024 Paper - Shift 24 Marks
MCQ
Let $f(x)=3 \sqrt{x-2}+\sqrt{4-x}$ be a real valued function. If  $\alpha $ and $\beta$ are respectively the minimum and the maximum values of $f,$ then $\alpha^2+2 \beta^2$ is equal to
  • A
    44
  • B
    42
  • C
    24
  • D
    38
✓

Answer

$ f ( x )=3 \sqrt{ x -2}+\sqrt{4- x }$
$x -2 \geq 0 \ 4- x \geq 0$
$\therefore x \in[2,4]$
Let $ x =2 \sin ^2 \theta+4 \cos ^2 \theta$
$\therefore f ( x )=3 \sqrt{2}|\cos \theta|+\sqrt{2}|\sin \theta|$
$\therefore \sqrt{2} \leq 3 \sqrt{2}|\cos \theta|+\sqrt{2}|\sin \theta| \leq \sqrt{9 \times 2+2}$
$\sqrt{2} \leq 3 \sqrt{2}|\cos \theta|+\sqrt{2}|\sin \theta| \leq \sqrt{20}$
$\therefore \alpha=\sqrt{2} \beta=\sqrt{20}$
$\alpha^2+2 \beta^2$
$=2+40=42$

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

JEE STD 11 Science question papers→Gujarat Board STD 11 Science question papers→Gujarat Board question paper generator→

Similar questions

If odds against solving a question by three students are $2 : 1 ,  5:2$ and $5:3$ respectively, then probability that the question is solved only by one student is
→
Length of the line segment joining the points $ - 1 - i$ and $2 + 3i$ is
→
If $arg\,z < 0$ then $arg\,( - z) - arg\,(z)$ is equal to
→
Let $F: R \rightarrow R$ be a thrice differentiable function. Suppose that $F (1)=0, F (3)=-4$ and $F^{\prime}( x )<0$ for all $x \in$ $(1 / 2,3)$. Let $f(x)=x F(x)$ for all $x \in R$.

$1.$ The correct statement$(s)$ is(are)

$(A)$ $f^{\prime}(1) < 0$

$(B)$ $f(2) < 0$

$(C)$ $f^{\prime}(x) \neq 0$ for any $x \in(1,3)$

$(D)$ $f^{\prime}(x)=0$ for some $x \in(1,3)$

$2.$ If $\int_1^3 x^2 F^{\prime}(x) d x=-12$ and $\int_1^3 x^3 F^{\prime \prime}(x) d x=40$, then the correct expression$(s)$ is(are)

$(A)$ $9 f^{\prime}(3)+f^{\prime}(1)-32=0$

$(B)$ $\int_1^3 f(x) d x=12$

$(C)$ $9 f^{\prime}(3)-f^{\prime}(1)+32=0$

$(D)$ $\int_1^3 f(x) d x=-12$

Give the answer question $1$ and $2.$

→
The value of the integral $\int {\frac{{\sin \,\left( {\ln \,(2 + 2x)} \right)}}{{x + 1}}dx} $ is
→
A coin is tossed three times in succession. If $E$ is the event that there are at least two heads and $F$ is the event in which first throw is a head, then $P\,\left( {\frac{E}{F}} \right) = $
→
If $f(x) = {{{x^2} - 1} \over {{x^2} + 1}}$, for every real number $x,$ then the minimum value of $f$
→
$\lim _{n \rightarrow \infty}\left(1+\frac{1+\frac{1}{2}+\ldots \ldots .+\frac{1}{n}}{n^{2}}\right)^{n}=.........$
→
Let $f$ be a function defined on $R$ (the set of all real numbers) such that $f^{\prime}(x)=2010(x-2009)(x-2010)^2$ $(x-2011)^3(x-2012)^4$, for all $x \in R$. If $g$ is a function defined on $R$ with values in the interval $(0, \infty)$ such that $f(x)=\ln (g(x))$, for all $x \in R$, then the number of points in $R$ at which $g$ has a local maximum is
→
The number of integers, greater than $7000$ that can be formed, using the digits $3,5,6,7,8$ without repetition, is
→