If $${\mathrm{S}}_n=4+11+21+34+50+\dots$$ to $$n$$ terms, then $$\frac{1}{60}\left({\mathrm{S}}_{29}-{\mathrm{S}}_9\right)$$ is equal to :

Let the first term $$\alpha$$ and the common ratio r of a geometric progression be positive integers. If the sum of squares of its first three terms is 33033, then the sum of these three terms is equal to

Let $${\mathrm{a}}_{\mathrm{n}}$$ be the $${\mathrm{n}}^{\text{th }}$$ term of the series $$5+8+14+23+35+50+\dots$$ and \mathrm{S}_{\mathrm{n}}=\sum_\limits{k=1}^{n} a_{k}. Then $${\mathrm{S}}_{30}-a_{40}$$ is equal to :

Let $$S_K=\frac{1+2+\dots +K}{K}$$ and \sum_\limits{j=1}^{n} S_{j}^{2}=\frac{n}{A}\left(B n^{2}+C n+D\right), where $$A,B,C,D\in \mathbb{N}$$ and $$A$$ has least value. Then

The sum of the first $$20$$ terms of the series $$5+11+19+29+41+\dots$$ is :

If $$\gcd (\mathrm{m},\mathrm{n})=1$$ and $$1^2-2^2+3^2-4^2+\dots ..+(2021{)}^2-(2022{)}^2+(2023{)}^2=1012m^{2}n$$ then $$m^2-n^2$$ is equal to :

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

A wire of length $$20\mathrm{m}$$ 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 :

If the functions $f(x)=\frac{x^3}{3}+2bx+\frac{a{x}^2}{2}$ and $g(x)=\frac{x^3}{3}+ax+b{x}^2,a\ne 2b$ have a common extreme point, then $a+2 b+7$ is equal to :

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 the function $$f(x)=2x^3+(2p-7)x^2+3(2p-9)x-6$$ have a maxima for some value of $$x0$$. Then, the set of all values of p is

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

Let $$f:(0,1)\to \mathbb{R}$$ be a function defined f(x) = {1 \over {1 - {e^{ - x}}}}, and $$g(x)=\left(f(-x)-f(x)\right)$$. Consider two statements (I) g is an increasing function in (0, 1) (II) g is one-one in (0, 1) Then,

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

Let $$f:[2,4]\to \mathbb{R}$$ be a differentiable function such that $$\left(x{\log }_{e}x\right)f^{\prime }(x)+\left({\log }_{e}x\right)f(x)+f(x)\ge 1,x\in [2,4]$$ with $$f(2)=\frac{1}{2}$$ and $$f(4)=\frac{1}{4}$$. Consider the following two statements : (A) : $$f(x)\le 1$$, for all $$x\in [2,4]$$ (B) : $$f(x)\ge \frac{1}{8}$$, for all $$x\in [2,4]$$ Then,

Let $$\mathrm{g}(x)=f(x)+f(1-x)$$ and $$f^{\prime \prime }(x)>0,x\in (0,1)$$. If $$\mathrm{g}$$ is decreasing in the interval $$(0,a)$$ and increasing in the interval $$(\alpha ,1)$$, then $${\tan }^{-1}(2\alpha )+{\tan }^{-1}\left(\frac{1}{\alpha }\right)+{\tan }^{-1}\left(\frac{\alpha +1}{\alpha }\right)$$ is equal to :

The slope of tangent at any point (x, y) on a curve $$y=y(x)$$ is {{{x^2} + {y^2}} \over {2xy}},x > 0. If $$y(2)=0$$, then a value of $$y(8)$$ is :

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 cm$${}^2$$) is equal to :

The area of the region given by $$\{(x,y):xy\le 8,1\le y\le x^2\}$$ is :