JEE Main 2003 — Calculus

The real number $x$ when added to its inverse gives the minimum value of the sum at $x$ equal to

If $f(x)= \begin{cases}x e^{-\left(\frac{1}{|x|}+\frac{1}{x}\right)}, & x \neq 0 \text { then } f(x) \text { is } \\ 0 & , x=0\end{cases}$

$$ \lim _{x \rightarrow \frac{\pi}{2}} \frac{\left[1-\tan \left(\frac{x}{2}\right)\right][1-\sin x]}{\left[1+\tan \left(\frac{x}{2}\right)\right]\left[\pi-2 x^3\right]} $$ is

If $\mathrm{f}(\mathrm{y})=\mathrm{e}^{\mathrm{y}}, \mathrm{g}(\mathrm{y})=\mathrm{y} ; \mathrm{y}>0$ and $F(t)=\int_0^{\mathrm{t}} \mathrm{f}(\mathrm{t}-\mathrm{y}) \mathrm{g}(\mathrm{y})$, then

Let $f(x)$ be a function satisfying $f^{\prime}(x)=f(x)$ with $f(0)=1$ and $g(x)$ be a function that satisfies $f(x)+g(x)=x^2$. Then the value of the integral $\int_0^1 f(x) g(x) d x$, is

$\lim _{n \rightarrow \infty} \frac{1+2^4+3^4+\ldots n^4}{n^5}-\lim _{n \rightarrow \infty} \frac{1+2^3+3^3+\ldots n^3}{n^5}$

If $\lim _{x \rightarrow 0} \frac{\log (3+x)-\log (3-x)}{x}=k$, the value of $k$ is

Let $\mathrm{f}(\mathrm{a})=\mathrm{g}(\mathrm{a})=\mathrm{k}$ and their $n$th derivatives $\mathrm{f}^{\mathrm{n}}(\mathrm{a}), \mathrm{g}^{\mathrm{n}}(\mathrm{a})$ exist and are not equal for some $\mathrm{n}$. Further if $\lim _{x \rightarrow a} \frac{f(a) g(x)-f(a)-g(a) f(x)+f(a)}{g(x)-f(x)}=4$ then the value of $k$ is

The degree and order of the differential equation of the family of all parabolas whose axis is X-axis, are respectively.

The value of the integral $I=\int_0^1 x(1-x)^n d x$ is

If $\mathrm{f}(\mathrm{a}+\mathrm{b}-\mathrm{x})=\mathrm{f}(\mathrm{x})$ then $\int_a^b \mathrm{xf}(\mathrm{x}) \mathrm{dx}$ is equal to

Let $\frac{d}{d x} F(x)=\left(\frac{e^{\text {sinx }}}{x}\right) x>0$. If $\int_1^4 \frac{3}{x} e^{\text {sin }^3} d x=F(k)-F(1)$ then one of the possible values of $k$, is

The solution of the differential equation $\left(1+y^2\right)+\left(x-e^{\tan ^{-1} y}\right) \frac{d y}{d x}=0$, is

The value of $\lim _{x \rightarrow 0} \frac{\int_0^{x^2} \sec ^2 t d t}{x \sin x}$ is

If the function $\mathrm{f}(\mathrm{x})=2 \mathrm{x}^2-9 a \mathrm{x}^2+12 \mathrm{a}^2 \mathrm{x}+1$, where $\mathrm{a}>0$, attains its maximum and minimum at $\mathrm{p}$ and $\mathrm{q}$ respectively such that $\mathrm{p}^2=\mathrm{q}$, then a equals

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