Statement 1: The sum of the series $1+\left(1+2+4\right)+\left(4+6+9\right)+\left(9+12+16\right)+......$$+\left(361+380+400\right)is8000$.Statement 2: $\sum _{k=1}^n\left(k^3-{\left(k-1\right)}^3\right)=n^3$for any natural number n.

Let $\hat{a}$and $\hat{b}$be two unit vectors. If the vectors $\vec{c}=\hat{a}+2\hat{b}and\vec{d}=5\hat{a}-4\hat{b}$are perpendicular to each other, then the angle between $\hat{a}$and $\hat{b}$is

A spherical balloon is filled with 4500p cubic meters of helium gas. If a leak in the balloon causes the gas to escape at the rate of $72\pi$cubic meters per minute, then the rate (in meters per minute) at which the radius of the balloon decreases 49 minutes after the leakage began is

Let $x_1,x_2,......,x_n$be n observations, and let $\overline{x}$be their arithematic mean and ${\sigma }^2$be their variance.Statement 1: Variance of $2x_1,2x_2,......,2x_{n}is4{\sigma }^2$.Statement 2: Arithmetic mean of $2x_1,2x_2,......,2x_{n}is4x$ .

Let a, b R be such that the function f given by $f\left(x\right)=\ln \left|x\right|+b{x}^2+ax,x\ne 0$has extreme values at $x=1$and $x=2$.Statement 1: f has local maximum at $x=1$and at $x=2$.Statement 2: $a=\frac{1}{2}\text{and}b=\frac{-1}{4}$

If the integral $\int \frac{5\tan x}{\tan x-2}dx=x+a\text{ln}\left|\sin x-2\cos x\right|+k$ then a is equal to

If n is a positive integer, then ${\left(\sqrt{3}+1\right)}^{2n}-{\left(\sqrt{3}-1\right)}^{2n}$is

In a $\Delta PQR$, if $3s\in P+4\cos Q=6$ and $4s\in Q+3\cos P=1$, then the angle R is equal to

The population p(t) at time t of a certain mouse species satisfies the differential equation $\frac{dp\left(t\right)}{dt}=0.5p\left(t\right)-450$If $p\left(0\right)=850$, then the time at which the population becomes zero is

Three numbers are chosen at random without replacement from {1, 2, 3, ...... 8}. The probability that their minimum is 3, given that their maximum is 6, is