Let the six numbers $${\mathrm{a}}_1,{\mathrm{a}}_2,{\mathrm{a}}_3,{\mathrm{a}}_4,{\mathrm{a}}_5,{\mathrm{a}}_6$$, be in A.P. and $${\mathrm{a}}_1+{\mathrm{a}}_3=10$$. If the mean of these six numbers is $$\frac{19}{2}$$ and their variance is $${\sigma }^2$$, then 8$${\sigma }^2$$ is equal to :

The mean and standard deviation of 10 observations are 20 and 8 respectively. Later on, it was observed that one observation was recorded as 50 instead of 40. Then the correct variance is :

Let the mean of 6 observations $$1,2,4,5,\mathrm{x}$$ and $$\mathrm{y}$$ be 5 and their variance be 10 . Then their mean deviation about the mean is equal to :

Let sets A and B have 5 elements each. Let the mean of the elements in sets A and B be 5 and 8 respectively and the variance of the elements in sets A and B be 12 and 20 respectively. A new set C of 10 elements is formed by subtracting 3 from each element of $$\mathrm{A}$$ and adding 2 to each element of $$\mathrm{B}$$. Then the sum of the mean and variance of the elements of $$\mathrm{C}$$ is ___________.

Let $$\mu$$ be the mean and $$\sigma$$ be the standard deviation of the distribution .tg {border-collapse:collapse;border-spacing:0;} .tg td{border-color:black;border-style:solid;border-width:1px;font-family:Arial, sans-serif;font-size:14px; overflow:hidden;padding:10px 5px;word-break:normal;} .tg th{border-color:black;border-style:solid;border-width:1px;font-family:Arial, sans-serif;font-size:14px; font-weight:normal;overflow:hidden;padding:10px 5px;word-break:normal;} .tg .tg-baqh{text-align:center;vertical-align:top} $$x_i$$ 0 1 2 3 4 5 $$f_i$$ $$k+2$$ $$2k$$ $$k^2-1$$ $$k^2-1$$ $$k^2+1$$ $$k-3$$ where $$\sum f_i=62$$. If $$[x]$$ denotes the greatest integer $$\le x$$, then $$\left[{\mu }^2+{\sigma }^2\right]$$ is equal to :

Let the mean and variance of 12 observations be $$\frac{9}{2}$$ and 4 respectively. Later on, it was observed that two observations were considered as 9 and 10 instead of 7 and 14 respectively. If the correct variance is $$\frac{m}{n}$$, where $$\mathrm{m}$$ and $$\mathrm{n}$$ are coprime, then $$\mathrm{m}+\mathrm{n}$$ is equal to :

The mean and variance of a set of 15 numbers are 12 and 14 respectively. The mean and variance of another set of 15 numbers are 14 and $${\sigma }^2$$ respectively. If the variance of all the 30 numbers in the two sets is 13 , then $${\sigma }^2$$ is equal to :

The sum \sum\limits_{n = 1}^\infty {{{2{n^2} + 3n + 4} \over {(2n)!}}} is equal to :

Let $a_1,a_2,a_3,\dots$ be an A.P. If $a_7=3$, the product $a_1{a}_4$ is minimum and the sum of its first $n$ terms is zero, then $n!-4a_{n(n+2)}$ is equal to :

The sum of 10 terms of the series {1 \over {1 + {1^2} + {1^4}}} + {2 \over {1 + {2^2} + {2^4}}} + {3 \over {1 + {3^2} + {3^4}}}\, + \,.... is

If the sum and product of four positive consecutive terms of a G.P., are 126 and 1296 , respectively, then the sum of common ratios of all such GPs is

Let $a,b,c>1,a^3,b^3$ and $c^3$ be in A.P., and ${\log }_{a}b,{\log }_{c}a$ and ${\log }_{b}c$ be in G.P. If the sum of first 20 terms of an A.P., whose first term is $\frac{a+4b+c}{3}$ and the common difference is $\frac{a-8b+c}{10}$ is $-444$, then $a b c$ is equal to :

If {a_n} = {{ - 2} \over {4{n^2} - 16n + 15}}, then $$a_1+a_2+....+a_{25}$$ is equal to :

For three positive integers p, q, r, $$x^{p{q}^2}=y^{qr}=z^{p^{2}r}$$ and r = pq + 1 such that 3, 3 log$${}_{y}x$$, 3 log$${}_{z}y$$, 7 log$${}_{x}z$$ are in A.P. with common difference $$\frac{1}{2}$$. Then r-p-q is equal to

Let $A_1$ and $A_2$ be two arithmetic means and $G_1,G_2,G_3$ be three geometric means of two distinct positive numbers. Then $G_1^4+G_2^4+G_3^4+G_1^2{G}_3^2$ is equal to :

Let a$${}_1$$, a$${}_2$$, a$${}_3$$, .... be a G.P. of increasing positive numbers. Let the sum of its 6th and 8th terms be 2 and the product of its 3rd and 5th terms be $$\frac{1}{9}$$. Then $$6(a_2+a_4)(a_4+a_6)$$ is equal to

Let $$s_1,s_2,s_3,\dots ,s_{10}$$ respectively be the sum to 12 terms of 10 A.P. s whose first terms are $$1,2,3,\dots .10$$ and the common differences are $$1,3,5,\dots \dots ,19$$ respectively. Then \sum_\limits{i=1}^{10} s_{i} is equal to :

Let $$ $$ be a sequence such that $$a_1+a_2+\dots +a_n=\frac{n^2+3n}{(n+1)(n+2)}$$. If 28 \sum_\limits{k=1}^{10} \frac{1}{a_{k}}=p_{1} p_{2} p_{3} \ldots p_{m}, where $${\mathrm{p}}_1,{\mathrm{p}}_2,\dots .,{\mathrm{p}}_{\mathrm{m}}$$ are the first $$\mathrm{m}$$ prime numbers, then $$\mathrm{m}$$ is equal to

Let $$a,b,c$$ and $$d$$ be positive real numbers such that $$a+b+c+d=11$$. If the maximum value of $$a^5{b}^3{c}^{2}d$$ is $$3750\beta$$, then the value of $$\beta$$ is

Let $$x_1,x_2,\dots ,x_{100}$$ be in an arithmetic progression, with $$x_1=2$$ and their mean equal to 200 . If $$y_i=i\left(x_i-i\right),1\le i\le 100$$, then the mean of $$y_1,y_2,\dots ,y_{100}$$ is :