Download App

JEE Main 7-April-2025 Paper - Shift 2 — JEE STD 11 Science — Question

Gujarat BoardEnglish MediumSTD 11 ScienceJEEJEE Main 7-April-2025 Paper - Shift 24 Marks
MCQ
Let $f: \mathbf{R} \rightarrow \mathbf{R}$ be a polynomial function of degree four having extreme values at $\mathrm{x}=4$ and $\mathrm{x}=5$.
If $\lim _{x \rightarrow 0} \frac{f(\mathrm{x})}{\mathrm{x}^{2}}=5$, then $f(2)$ is equal to :
  • A
    12
  • B
    10
  • C
    8
  • D
    14
✓

Answer

B. 10
$\lim _{x \rightarrow 0} \frac{f(x)}{x^{2}}=5$
$\lim _{x \rightarrow 0} \frac{\left.a x^{4}+b x^{3}+c x^{2}+d x+e\right)}{x^{2}}=5$
$\mathrm{c}=5$ and $\mathrm{d}=\mathrm{e}=0$
$f(x)=a x^{4}+b x^{3}+5 x^{2}$
$f^{\prime}(x)=4 a x^{3}+3 b x^{2}+10 x$
$=x\left(4 a x^{2}+3 b x+10\right)$
has extremes at 4 and so $f^{\prime}(4)=0 \& f^{\prime}(5)=0$
so $\mathrm{a}=\frac{1}{8} \& \mathrm{~b}=\frac{-3}{2}$
so $f(2)=\frac{1}{8} \times 2^{4}-\frac{3}{2} \times 2^{3}+5 \times 2^{2}$
$=2-12+20=10$

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 7-April-2025 Paper - Shift 2→JEE Main 7-April-2025 Paper - Shift 2 — all question types→JEE STD 11 Science question papers→Gujarat Board STD 11 Science question papers→Gujarat Board question paper generator→

Similar questions

Let $m, n$ be two distinct integers chosen randomly from the set $\{0,1,2, \ldots, 99\}$. Then, the probability that $4^m+4^n+3$ is divisible by $5$ lies in the interval
→
If the $n^{th}$ term of geometric progression $5, - \frac{5}{2},\frac{5}{4}, - \frac{5}{8},...$ is $\frac{5}{{1024}}$, then the value of $n$ is
→
Suppose A and B are the coefficients of $30^{\text {th }}$ and $12^{\text {th }}$ terms respectively in the binomial expansion of $(1+x)^{2 n-1}$. If $2 A=5 B$, then $n$ is equal to:
→
Order of the differential equation of the family of all concentric circles centered at $(h, k)$ is
→
If $^n{P_r}$=$ 720$.$^n{C_r},$ then $r$ is equal to
→
Let $R$ be the set of real numbers and $f: R \rightarrow R$ be given by $f(x)=\sqrt{|x|}-\log (1+|x|)$. We now make the following assertions:

$I.$ There exists a real number $A$ such that $f(x) \leq A$ for all $x$.

$II.$ There exists a real number $B$ such that $f(x) \geq B$ for all $x$.

→
If the function $f(x) = \,\left\{ {\begin{array}{*{20}{c}}{5x - 4}&,&{{\rm{if}}}&{0 < x \le 1}\\{4{x^2} + 3bx}&,&{{\rm{if}}}&{1 < x < 2}\end{array}} \right.$ is continuous at every point of its domain, then the value of $b$ is
→
If the sum of the coefficients in the expansion of ${(x + y)^n}$ is $1024$, then the value of the greatest coefficient in the expansion is
→
$\int_{}^{} {\frac{{{e^{ - x}}}}{{1 + {e^x}}}\;dx = } $
→
Through which of the following pairs of points does the circle ${x^2} + {y^2} - 12x + 1 = 0$ pass
→