JEE Main 2005 — Calculus

A function is matched below against an interval where it is supposed to be increasing. Which of the following pairs is incorrectly matched? $Interval \rightarrow Function$

A spherical iron ball $10 \mathrm{~cm}$ in radius is coated with a layer of ice of uniform thickness than melts at a rate of $50 \mathrm{~cm}^3 / \mathrm{min}$. When the thickness of ice is $5 \mathrm{~cm}$, then the rate at which the thickness of ice decreases, is

Area of the greatest rectangle that can be inscribed in the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ is

$\lim _{n \rightarrow \infty}\left[\frac{1}{n^2} \sec ^2 \frac{1}{n^2}+\frac{2}{n^2} \sec ^2 \frac{4}{n^2}+\ldots .+\frac{1}{n^2} \sec ^2 1\right]$ equals

If $\mathrm{I}_1=\int_0^1 2^{x^2} d x, I_2=\int_0^1 2^{x^3} d x, I_3=\int_1^2 2^{x^2} d x$ and $I_4=\int_1^2 2^{x^3} d x$ then

If $f$ is a real-valued differentiable function satisfying $|f(x)-f(y)| \leq(x-y)^2, x, y \in R$ and $f(0)=0$, then $f(1)$ equals

If $\mathrm{x}$ is so small that $\mathrm{x}^3$ and higher powers of $\mathrm{x}$ may be neglected, then $\frac{(1+x)^{3 / 2}-\left(1+\frac{1}{2} x\right)^3}{(1-x)^{1 / 2}}$

If the equation $a_n x^n+a_{n-1} x^{n-1}+\ldots \ldots+a_1 x=0, a_1 \neq 0, n \geq 2$, has a positive root $x=\alpha$, then the equation $n a_n x^{n-1}+(n-1) a_{n-1} x^{n-2}+\ldots . .+a_1=0$ has a positive root, which is

If $x \frac{d y}{d x}=y(\log y-\log x+1)$, then the solution of the equation is

$\int\left\{\frac{(\log x-1)}{\left(1+(\log x)^2\right.}\right\}^2 d x$ is equal to

Let $\alpha$ and $\beta$ be the distinct roots of $a x^2+b x+c=0$, then $\lim _{x \rightarrow \alpha} \frac{1-\cos \left(a x^2+b x+c\right)}{(x-\alpha)^2}$ is equal to

Let $f: R \rightarrow R$ be a differentiable function having $f(2)=6, f^{\prime}(2)=\left(\frac{1}{48}\right)$. Then $\lim _{x \rightarrow 2} \int_6^{f(x)} \frac{4 t^3}{x-2} d t$ equals

Let $f(x)$ be a non-negative continuous function such that the area bounded by the curve $y=f(x), x$-axis and the ordinates $x=\frac{\pi}{4}$ and $x=\beta>\frac{\pi}{4}$ is $\left(\beta \sin \beta+\frac{\pi}{4} \cos \beta+\sqrt{2} \beta\right)$. Then $f\left(\frac{\pi}{2}\right)$ is

Suppose $f(x)$ is differentiable $x=1$ and $\lim _{h \rightarrow 0} \frac{1}{h} f(1+h)=5$, then $f^{\prime}(1)$ equals

The area enclosed between the curve $y=\log _e(x+e)$ and the coordinate axes is

The differential equation representing the family of curves $y^2=2 c(x+\sqrt{c})$, where $c$ $>0$, is a parameter, is of order and degree as follows:

The parabolas $\mathrm{y}^2=4 \mathrm{x}$ and $\mathrm{x}^2=4 \mathrm{y}$ divide the square region bounded by the lines $\mathrm{x}=$ $4, y=4$ and the coordinate axes. If $S_1, S_2, S_3$ are respectively the areas of these parts numbered from top to bottom; then $\mathrm{S}_1: \mathrm{S}_2: \mathrm{S}_3$ is

The value of $\int_{-\pi}^\pi \frac{\cos ^2 x}{1+a^x} d x, a>0$, is