JEE Main 2004 — Calculus

A function $y=f(x)$ has a second order derivative $f^{\prime \prime}(x)=6(x-1)$. If its graph passes through the point $(2,1)$ and at that point the tangent to the graph is $y=3 x-5$, then the function is

A point on the parabola $y^2=18 x$ at which the ordinate increases at twice the rate of the abscissa is

If $f(x)=\frac{e^x}{1+e^x}, l_1=\int_{f(-a)}^{f(a)} x g\{x(1-x)\} d x$ and $I_2=\int_{f(-a)}^{f(a)} g\{x(1-x)\} d x$ then the value of $\frac{l_2}{l_1}$ is

If $2 a+3 b+6 c=0$, then at least one root of the equation $a x^2+b x+c=0$ lies in the interval

If $x=e^{y+e^{y+. .10 \infty}}, x>0$, then $\frac{d y}{d x}$ is

If $\int_0^\pi x f(\sin x) d x=A \int_0^{\pi / 2} f(\sin x) d x$, then $A$ is

If $\lim _{x \rightarrow \infty}\left(1+\frac{a}{x}+\frac{b}{x^2}\right)^{2 x}=e^2$, then the values of $a$ and $b$, are

If $\int \frac{\sin x}{\sin (x-\alpha)} d x=A x+B \log \sin (x-\alpha)+C$, then value of $(A, B)$ is

$\lim _{n \rightarrow \infty} \sum_{r=1}^n \frac{1}{n} e^{\frac{-}{n}}$ is

$\int \frac{d x}{\cos x-\sin x}$ is equal to

Let $f(x)=\frac{1-\tan x}{4 x-\pi}, x \neq \frac{\pi}{4}, x \in\left[0, \frac{\pi}{2}\right]$. If $f(x)$ is continuous in $\left[0, \frac{\pi}{2}\right]$, then $f\left(\frac{\pi}{4}\right)$ is

The area of the region bounded by the curves $y=|x-2|, x=1, x=3$ and the $x$-axis is

The solution of the differential equation $y d x+\left(x+x^2 y\right) d y=0$ is

The value of $\int_{-2}^3\left|1-x^2\right| d x$ is

The value of $I=\int_0^{\pi / 2} \frac{(\sin x+\cos x)^2}{\sqrt{1+\sin 2 x}} d x$ is