$Let a_1, a_2, a_3, .., a_n be n positive consecutive terms of an arithmetic progression. If d>0 is its common difference, then$

$\underset{n\to \infty }{\lim }\sqrt{\frac{d}{n}}(\frac{1}{\sqrt{a_1}+\sqrt{a_2}}+\frac{1}{\sqrt{a_2}+\sqrt{a_3}}+...+\frac{1}{\sqrt{a_{n-1}}+\sqrt{a_n}}) is$

Among

$\left(S1\right): \underset{n\to \infty }{\lim }\frac{1}{n^2}(2+4+6+...+2n)=1$

$\left(S2\right): \underset{n\to \infty }{\lim }\frac{1}{n^{16}}(1^{15}+2^{15}+3^{15}+....+n^{15})=\frac{1}{16}$

$\underset{n\to \infty }{\lim }\left\{\right(2^{\frac{1}{2}}-2^{\frac{1}{3}}\left)\right(2^{\frac{1}{2}}-2^{\frac{1}{5}})....(2^{\frac{1}{2}}-2^{\frac{1}{2n+1}}\left)\right\} is equal to$

$\underset{x\to 0}{\lim }\left(\right(\frac{1-{\cos }^2\left(3x\right)}{{\cos }^3\left(4x\right)}\left)\right(\frac{{\sin }^3\left(4x\right)}{({\log }_e{(2x+1))}^5}\left)\right) is equal to \_\_\_\_\_ .$

$If \alpha >\beta >0 are the roots of the equation a{x}^2+bx+1=0, and \underset{x\to \frac{1}{\alpha }}{\lim }{\left(\frac{1-\cos (x^2+bx+a)}{2{(1-\alpha x)}^2}\right)}^{\frac{1}{2}}=\frac{1}{k}(\frac{1}{\beta }-\frac{1}{\alpha }), then k is equal to$

$If \underset{x\to 0}{\lim }\frac{e^{ax}-\cos \left(bx\right)-{\displaystyle \frac{cx{e}^{-cx}}{2}}}{1-\cos \left(2x\right)}=17, then 5a^2+b^2 is equal to$

$\underset{t\to 0}{\lim } {(1^{\frac{1}{{\sin }^{2}t}}+2^{\frac{1}{{\sin }^{2}t}}+...+n^{\frac{1}{{\sin }^{2}t}})}^{{\sin }^{2}t} is equal to$

$The set of all values of a for which \underset{x\to a}{\lim }\left(\right[x-5]-[2x+2\left]\right)=0, where \left[\alpha \right] denotes the greatest integer less than or equal to \alpha is equal to$

The value of

$\underset{x\to \infty }{\lim }\frac{1+2-3+4+5-6+...+(3n-2)+(3n-1)-3n}{\sqrt{2n^4+4n+3}-\sqrt{n^4+5n+4}} is$

$Let x=2 be a root of the equation x^2+px+q=0 and$

$f\left(x\right)=\{\begin{array}{ll}\frac{1-\cos (x^2-4px+q^2+8q+16)}{{(x-2p)}^4},& x\ne 2p\\ 0,& x=2p\end{array}$

$Then \underset{x\to 2p^{+}}{\lim }\left[f\right(x\left)\right], where [·] denotes greatest integer function, is$

$Let f,g and h be the real valued functions defined on R$

$f\left(x\right)=\{\begin{array}{ll}\frac{x}{\left|x\right|},& x\ne 0\\ 1,& x=0\end{array}, g(x)=\{\begin{array}{ll}\frac{\sin (x+1)}{(x+1)},& x\ne -1\\ 1,& x=-1\end{array}$

$and h\left(x\right)=2\left[x\right]-f\left(x\right), where \left[x\right] is the greatest integer\le x. Then the value of \underset{x\to 1}{\lim }g\left(h\right(x-1\left)\right) is$

$\underset{x\to \infty }{\lim }\frac{{(\sqrt{3x+1}+\sqrt{3x-1})}^6+{(\sqrt{3x+1}-\sqrt{3x-1})}^6}{{(x+\sqrt{x^2-1})}^6+{(x-\sqrt{x^2-1})}^6}{x}^3$

$If 2x^y+3y^x=20, then \frac{dy}{dx} at (2,2) is equal to$

$Let f\left(x\right)=\frac{\sin x+\cos x-\sqrt{2}}{\sin x-\cos x},x\in [0,\mathrm{\pi }]-\left\{\frac{\mathrm{\pi }}{4}\right\}. Then$

$f\left(\frac{7\mathrm{\pi }}{12}\right)f\text{'}\text{'}\left(\frac{7\mathrm{\pi }}{12}\right) is equal to$

Let $f\left(x\right)=[x^2-x]+|-x+[x\left]\right|,$ where $x\in \mathrm{ℝ}$ and [t]

denotes the greatest integer less than or equal to t. Then, $f$ is

Let $f$ and $g$ be two functions defined by

$f\left(x\right)=\{\begin{array}{ll}x+1,& x<0\\ |x-1|,& x\ge 0\end{array}$ and $g\left(x\right)=\{\begin{array}{ll}x+1,& x<0\\ 1,& x\ge 0\end{array}$

Then $\left(gof\right) \left(x\right)$ is

For the differentiable function $f:\mathrm{ℝ}-\left\{0\right\}\to \mathrm{ℝ},$

let $3f\left(x\right)+2f\left(\frac{1}{x}\right)=\frac{1}{x}-10,$then $\left|f\right(3)+f\text{'}(\frac{1}{4}\left)\right|$ is equal to

Let $\left[x\right]$ denote the greatest integer function and $f\left(x\right)=max\{1+x+[x],2+x,x+2[x\left]\right\}, 0\le x\le 2.$

Let m be the number of points in [0, 2], where $f$ is not continuous and n be the number of points in (0, 2), 

where $f$ is not differentiable. Then ${(m+n)}^2+2$ is equal to

If $y\left(x\right)=x^x,x>0,$ then $y\text{'}\text{'}\left(2\right)-2y\text{'}\left(2\right)$ is equal to

Let $f\left(x\right)=\{\begin{array}{ll}{x}^2\sin \left(\frac{1}{x}\right),& x\ne 0\\ 0,& x=0\end{array}$

Then at x = 0

If $f\left(x\right)=x^3-x^{2}f\text{'}\left(1\right)+xf\text{'}\text{'}\left(2\right)-f\text{'}\text{'}\text{'}\left(3\right),x\in R,$ then

Let $y\left(x\right)=(1+x)(1+x^2)(1+x^4)(1+x^8)(1+x^{16}).$

Then $y\text{'}-y\text{'}\text{'}$ at $x=-1$ is equal to:

If the function

$f\left(x\right)=\{\begin{array}{ll}(1+|\cos x\left|\right)\frac{\lambda }{\left|\cos x\right|},& 0<x<\frac{\mathrm{\pi }}{2}\\ \mu ,& x=\frac{\mathrm{\pi }}{2}\\ \frac{cot 6x}{e cot 4x},& \frac{\mathrm{\pi }}{2}<x<\mathrm{\pi }\end{array}$

is continuous at $x=\frac{\mathrm{\pi }}{2}$, then $9\lambda +6{\log }_e\mu +{\mu }^6-e^{6\lambda }$ is equal to

Let $f$ and $g$ be twice differentiable function on $\mathrm{ℝ}$ such that

$f\text{'}\text{'}\left(x\right)=g\text{'}\text{'}\left(x\right)+6x$

$f\text{'}\left(1\right)=4g\text{'}\left(1\right)-3=9$

$f\left(2\right)=3g\left(2\right)=12$

Then which of the following is NOT true?

Let $a\in \mathrm{\mathbb{Z}}$ and $\left[t\right]$ be the greatest integer $\le t$. Then the number of points, where the function 

$f\left(x\right)=[a+13\sin x]$, $x\in (0,\mathrm{\pi })$ is not differentiable, is _________ .

If $a_{\alpha }$ is the greatest term in the sequence $a_n=\frac{n^3}{n^4+147}, n=1,2,3,......,$ then $\alpha$ is equal to _______ .

Let $k$ and $m$ be positive real numbers such that the function

$f\left(x\right)=\{\begin{array}{ll}3x^2+k\sqrt{x+1},& 0<x<1\\ m{x}^2+k^2,& x\ge 1\end{array}$ is differentiable for all $x>0.$

Then $\frac{8f\text{'}\left(8\right)}{f\text{'}\left({\displaystyle \frac{1}{8}}\right)}$ is equal to _______________ .

Let $f:(-2,2)\to \mathit{R}$ be defined by $f\left(x\right)=\{\begin{array}{ll}x\left[x\right],& -2<x<0\\ (x-1)\left[x\right],& 0\le x<2\end{array}$ where [x] denotes the

greatest integer function. If m and n respectively are the number of points in

(-2, 2) at which $y=\left|f\right(x\left)\right|$ is not continuous and not differentiable, then m + n is equal to _______ .

Let [x] be the greatest integer $\le x$. Then the number of points in the interval (-2, 1),

where the function $f\left(x\right)=\left|\right[x\left]\right|+\sqrt{x-\left[x\right]}$ is discontinuous, is _____ .

Let $f\left(x\right)=\sum _{k=1}^{10}k{x}^k, x\in \mathrm{ℝ}.$ If $2f\left(2\right)+f\text{'}\left(2\right)=119 {\left(2\right)}^n+1$ then n is equal to ________ .

If $f\left(x\right)=x^2+g\text{'}\left(1\right)x+g\text{'}\text{'}\left(2\right)$ and $g\left(x\right)=f\left(1\right)x^2+xf\text{'}\left(x\right)+f\text{'}\text{'}\left(x\right),$

then the value of $f\left(4\right)-g\left(4\right)$ is equal to _______ .

Let $f:\mathrm{ℝ}\to \mathrm{ℝ}$ be a differentiable function that satisfies the relation

$f(x+y)=f\left(x\right)+f\left(y\right)-1,\forall x,y\in \mathrm{ℝ}.$ If $f\text{'}\left(0\right)=2,$ then $\left|f\right(-2\left)\right|$ is equal to _________ .

The distance of the point $(6,-2\sqrt{2})$ from the common tangent $y=mx+c,m>0,$

of the curves $x=2y^2$ and $x=1+y^2$ is

The number of points on the curve $y=54x^5-135x^4-70x^3+180x^2+210x$ at which the normal lines

are parallel to $x+90y+2=0$ is

Let a common tangent to the curves $y^2=4x$ and ${(x-4)}^2+y^2=16$ touch the curves at the points P and Q.

Then ${\left(PQ\right)}^2$ is equal to ________________ .

Let the quadratic curve passing through the point $(-1,0)$ and touching the line $y=x$ at (1, 1) be

$y=f\left(x\right)$. Then the x-intercept of the normal to the curve at the point $(\alpha ,\alpha +1)$ in the first quadrant is _______ .

If the equation of the normal to the curve $y=\frac{x-a}{(x+b)(x-2)}$ at the point $(1,-3)$ is

$x-4y=13,$ then the value of $a+b$ is equal to ________ .

Let $g\left(x\right)=f\left(x\right)+f(1-x)$ and $f\text{'}\text{'}\left(x\right)>0,x\in (0,1)$. If g is decreasing in the interval $(0,\alpha )$ and increasing in the interval $(\alpha ,1)$, then

${\tan }^{-1}\left(2\alpha \right)+{\tan }^{-1}\left(\frac{1}{\alpha }\right)+{\tan }^{-1}\left(\frac{\alpha +1}{\alpha }\right)$ is equal to

Let $f:[2,4]\to R$ be a differentiable function such that $\left(x{\log }_{e}x\right)f\text{'}\left(x\right)+\left({\log }_{e}x\right)f\left(x\right)+f\left(x\right)\ge 1, x\in [2, 4]$ with $f\left(2\right)=\frac{1}{2}$ and $f\left(4\right)=\frac{1}{4}.$

Consider the following two statements.

$\left(A\right):f\left(x\right)\le 1,$ for all $x\in [2,4]$

$\left(B\right):f\left(x\right)\ge \frac{1}{8},$ for all $x\in [2,4]$

Then,

Let $f:(0,1)\to \mathrm{ℝ}$ be a function defined by $f\left(x\right)=\frac{1}{1-e^{-x}}$ and $g\left(x\right)=\left(f\right(-x)-f(x\left)\right)$. Consider two statements

(I) g is an increasing function in (0, 1)

(II) g is one-one in (0, 1)

Then,

A square piece of tin of side 30 cm is to be made into a box without top by cutting a square from each corner and folding up the flaps to form a box. If the volume of the box is maximum, then its surface area ($in c{m}^2$) is equal to

If the local maximum value of the function $f\left(x\right)={\left(\frac{\sqrt{3e}}{2\sin x}\right)}^{{\sin }^{2}x}$, $x\in (0,\frac{\pi }{2})$ is $\frac{k}{e},$ then ${\left(\frac{k}{e}\right)}^8+\frac{k^8}{e^5}+k^8$ is equal to

$\underset{0\le x\le \pi }{max}\{x-2\sin x\cos x+\frac{1}{3}\sin 3x\}=$

Let $f\left(x\right)=2x+{\tan }^{-1}x$ and $g\left(x\right)={\log }_e(\sqrt{1+x^2}+x), x\in [0,3].$ Then

Let $f\left(x\right)=\left|\begin{array}{ccc}1+{\sin }^{2}x& {\cos }^{2}x& \sin 2x\\ {\sin }^{2}x& 1+{\cos }^{2}x& \sin 2x\\ {\sin }^{2}x& {\cos }^{2}x& 1+\sin 2x\end{array}\right|,x\in [\frac{\mathrm{\pi }}{6},\frac{\mathrm{\pi }}{3}].$

If $\alpha$ and $\beta$ respectively are the maximum and the minimum values of $f$, then

The sum of the absolute maximum and minimum values of the function $f\left(x\right)=|x^2-5x+6|-3x+2$ in the interval $[-1,3]$ is equal to

Let $x=2$ be a local minima of the function $f\left(x\right)=2x^4-18x^2+8x+12,x\in (-4,4).$ If M is local maximum value of the function

$f$ in (-4, 4), then M =

Let the function $f\left(x\right)=2x^3+(2p-7)x^2+3(2p-9)x-6$ have a maxima for some value of $x<0$ and a

minima for some value of $x>0.$Then, the set of all values of p is

If the functions $f\left(x\right)=\frac{x^3}{3}+2bx+\frac{a{x}^2}{2}$ and $g\left(x\right)=\frac{x^3}{3}+ax+b{x}^2,$

$a\ne 2b$ have a common extreme point, then $a+2b+7$ is equal to

A wire of length 20 m is to be cut into two pieces. A piece of length $l_1$ is bent to make a square of area $A_1$

and the other piece of length $l_2$ is made into a circle of area $A_2.$ If $2A_1+3A_2$ is minimum then $\left(\pi l_1\right):l_2$ is equal to:

The number of points, where the curve $y=x^5-20x^3+50x+2$ crosses the x-axis, is _______ .

Consider the triangles with vertices A(2, 1), B(0, 0) and C(t, 4), $t\in [0,4].$ If the maximum and the minimum perimeters of such triangles are obtained at $t=\alpha$ and $t=\beta$ respectively, then $6\alpha +21\beta$ is equal to ___________ .