AIEEE 2012 PYQ Mock Test

The equation $e^{\sin x}-e^{-\sin x}-4=0$ has:

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 $4500\pi$ 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:

Statement 1: The sum of the series $1+(1+2+4)+(4+6+9)+(9+12+16)+\cdots +(361+380+400)$ is $8000$.Statement 2: ${\sum }_{k=1}^n(k^3-(k-1{)}^3)=n^3$ for any natural number $n$.

The negation of the statement “If I become a teacher, then I will open a school” is: