Let a curve y=f(x) pass through the point (2, ( _ e 2 )^ 2 ) and … - Vidyadip

MCQ

Let a curve $y=f(x)$ pass through the point $\left(2,\left(\log _{e} 2\right)^{2}\right)$ and have slope $\frac{2 y}{x \log _{e} x}$ for all positive real value of $x$. Then the value of $f(e)$ is equal to $.....$

✓
$1$
B
$2$
C
$3$
D
$4$

✓

Answer

Correct option: A.

$1$

a
$y^{\prime}=\frac{2 y}{x \ell n x} \Rightarrow \frac{d y}{d x}=\frac{2 y}{x \ln x}$

$\Rightarrow \frac{d y}{y}=\frac{2 d x}{x \ell n x}$

$\Rightarrow \ell n|y|=2 \ell n|\ell n x|+C$

$\text { Put } x=2, y=(\ell n 2)^{2}$

$\Rightarrow \ln \left|(\ln 2)^{2}\right|=\ln \left|(\ln 2)^{2}\right|+c$

$\Rightarrow c=0$

$\Rightarrow y=(\text { en } x)^{2}$

$\Rightarrow f(e)=1$

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 "M.C.Q (1 Marks)" from STD 12 - 9. differential equations→STD 12 - 9. differential equations — all question types→Mathematics STD 12 Science question papers→CBSE Board STD 12 Science question papers→CBSE Board question paper generator→

Similar questions

The differential equation satisfied by the function $y = \sqrt {\sin x + \sqrt {\sin x + \sqrt {\sin x + .....\infty } } } $, is

The area bounded by the y-axis, $\text{y}=\cos\text{x}$ and $\text{y}=\sin\text{x}$ when $0\leq\text{x}\leq\frac{\pi}{2}$ is:

If ${\sin ^{ - 1}}a + {\sin ^{ - 1}}b + {\sin ^{ - 1}}c = \pi ,$ then the value of $a\sqrt {(1 - {a^2})} + b\sqrt {(1 - {b^2})} + c\sqrt {(1 - {c^2})} $ will be

If $a,b,c$ are in $A.P$., then the value of $\left| {\,\begin{array}{*{20}{c}}{x + 2}&{x + 3}&{x + a}\\{x + 4}&{x + 5}&{x + b}\\{x + 6}&{x + 7}&{x + c}\end{array}\,} \right|$ is

$3 x^2-6 x+5$ is an increasing function, if

Let $A$ be a $3 \times 3$ matrix such that $A^2 -5A+ 7I = 0$.

Statement $-I$ : ${A^{ - 1}} = \frac{1}{7}\left( {5I - A} \right).$

Statement $II$ : the polynomial $A^3 - 2A^2 - 3A + I$ can be reduced to $5\, (A - 4I)$.

The equation of the plane through the intersection of the planes $x + 2y + 3z = 4$ and $2x + y - z = -5$ and perpendicular to the plane $5x + 3y + 6z + 8 = 0$ is:

The area of the region above the $x-$ axis bounded by the curve $y\, = tan\, x$, $0 \leq x \leq \frac{\pi }{2}$ and the tangent to the curve at $x\, = \frac{\pi}{4}$ is

If A is skew symmetric matrix of order 3, then the value of |A| is

The value of $\int\frac{\text{d}(\text{x}^2+1)}{\sqrt{\text{x}^2+2}}$ is: