If A = \sum\limits_{n = 1}^\infty {{1 \over {{{\left( {3 + {{( - 1)}^n}} \right)}^n}}}} and B = \sum\limits_{n = 1}^\infty {{{{{( - 1)}^n}} \over {{{\left( {3 + {{( - 1)}^n}} \right)}^n}}}}, then {A \over B} is equal to :

The sum 1 + 2 . 3 + 3 . 32 + ......... + 10 . 39 is equal to :

Let x, y > 0. If x3y2 = 215, then the least value of 3x + 2y is

If $$\{a_i{\}}_{i=1}^n$$, where n is an even integer, is an arithmetic progression with common difference 1, and $$\sum _{i=1}^n{a}_i=192,\sum _{i=1}^{n/2}{a}_{2i}=120$$, then n is equal to :

The value of 1 + {1 \over {1 + 2}} + {1 \over {1 + 2 + 3}} + \,\,....\,\, + \,\,{1 \over {1 + 2 + 3 + \,\,.....\,\, + \,\,11}} is equal to:

The sum \sum\limits_{n = 1}^{21} {{3 \over {(4n - 1)(4n + 3)}}} is equal to

Consider two G.Ps. 2, 22, 23, ..... and 4, 42, 43, .... of 60 and n terms respectively. If the geometric mean of all the 60 + n terms is {(2)^{{{225} \over 8}}}, then $$\sum _{k=1}^{n}k(n-k)$$ is equal to :

Suppose $$a_1,a_2,\dots ,a_n$$, .. be an arithmetic progression of natural numbers. If the ratio of the sum of first five terms to the sum of first nine terms of the progression is $$5:17$$ and , $$110

Let the sum of an infinite G.P., whose first term is a and the common ratio is r, be 5 . Let the sum of its first five terms be $$\frac{98}{25}$$. Then the sum of the first 21 terms of an AP, whose first term is $$10\mathrm{ar},{\mathrm{n}}^{\text{th }}$$ term is $${\mathrm{a}}_{\mathrm{n}}$$ and the common difference is $$10{\mathrm{ar}}^2$$, is equal to :

Consider the sequence $$a_1,a_2,a_3,\dots$$ such that $$a_1=1,a_2=2$$ and $$a_{n+2}=\frac{2}{a_{n+1}}+a_n$$ for $$\mathrm{n}=1,2,3,\dots .$$ If $$\left(\frac{{\mathrm{a}}_1+\frac{1}{{\mathrm{a}}_2}}{{\mathrm{a}}_3}\right)\cdot \left(\frac{{\mathrm{a}}_2+\frac{1}{{\mathrm{a}}_3}}{{\mathrm{a}}_4}\right)\cdot \left(\frac{{\mathrm{a}}_3+\frac{1}{{\mathrm{a}}_4}}{{\mathrm{a}}_5}\right)\dots \left(\frac{{\mathrm{a}}_{30}+\frac{1}{{\mathrm{a}}_{31}}}{{\mathrm{a}}_{32}}\right)=2^{\alpha }\left({}^{61}{\mathrm{C}}_{31}\right)$$, then $$\alpha$$ is equal to :

$$\begin{array}{rl}& \text{ Let }{\left\{a_n\right\}}_{n=0}^{\infty }\text{ be a sequence such that }{a}_0=a_1=0\text{ and }\\ & \\ & a_{n+2}=3a_{n+1}-2a_n+1,\forall n\ge 0.\end{array}$$ Then $$a_{25}{a}_{23}-2a_{25}{a}_{22}-2a_{23}{a}_{24}+4a_{22}{a}_{24}$$ is equal to

A wire of length 22 m is to be cut into two pieces. One of the pieces is to be made into a square and the other into an equilateral triangle. Then, the length of the side of the equilateral triangle, so that the combined area of the square and the equilateral triangle is minimum, is :

Let f : R $$\to$$ R be a function defined by f(x) = (x $$-$$ 3)n1 (x $$-$$ 5)n2, n1, n2 $$\in$$ N. Then, which of the following is NOT true?

The number of real solutions of $$x^7+5x^3+3x+1=0$$ is equal to ____________.

The sum of the absolute minimum and the absolute maximum values of the function f(x) = |3x $$-$$ x2 + 2| $$-$$ x in the interval [$$-$$1, 2] is :

Let S be the set of all the natural numbers, for which the line {x \over a} + {y \over b} = 2 is a tangent to the curve {\left( {{x \over a}} \right)^n} + {\left( {{y \over b}} \right)^n} = 2 at the point (a, b), ab $$\ne$$ 0. Then :

Let $$f(x)=2{\cos }^{-1}x+4{\cot }^{-1}x-3x^2-2x+10$$, $$x\in [-1,1]$$. If [a, b] is the range of the function f, then 4a $$-$$ b is equal to :

Consider a cuboid of sides 2x, 4x and 5x and a closed hemisphere of radius r. If the sum of their surface areas is a constant k, then the ratio x : r, for which the sum of their volumes is maximum, is :

Water is being filled at the rate of 1 cm3 / sec in a right circular conical vessel (vertex downwards) of height 35 cm and diameter 14 cm. When the height of the water level is 10 cm, the rate (in cm2 / sec) at which the wet conical surface area of the vessel increases is

If the angle made by the tangent at the point (x0, y0) on the curve $$x=12(t+\sin t\cos t)$$, $$y=12(1+\sin t{)}^2$$, $$0 0 is equal to: