NTA JEE Main 6th September 2020 Shift 1

The angle of elevation of the top of a hill from a point on the horizontal plane passing through the foot of the hill is found to be $$45^\circ$$. After walking a distance of $$80$$ meters towards the top, up a slope inclined at angle of $$30^\circ$$ to the horizontal plane the angle of elevation of the top of the hill becomes $$75^\circ$$. Then the height of the hill (in meters) is_____.

Set $$A$$ has $$m$$ elements and set $$B$$ has $$n$$ elements. If the total number of subsets of $$A$$ is 112 more than the total number of subsets of $$B$$, then the value of $$m \cdot n$$ is___.

Let $$f : \mathbb{R} \to \mathbb{R}$$ be defined as $$f(x) = \begin{cases} x^5\sin\left(\frac{1}{x}\right) + 5x^2, & x < 0 \\ 0, & x = 0 \\ x^5\cos\left(\frac{1}{x}\right) + \lambda x^2, & x > 0 \end{cases}$$. The value of $$\lambda$$ for which $$f''(0)$$ exists, is___.

Let $$AD$$ and $$BC$$ be two vertical poles at $$A$$ and $$B$$ respectively on a horizontal ground. If $$AD = 8\,\text{m}$$, $$BC = 11\,\text{m}$$, $$AB = 10\,\text{m}$$; then the distance (in meters) of a point M lying in between AB from the point A such that $$MD^2 + MC^2$$ is minimum, is___.

If $$\vec{a}$$ and $$\vec{b}$$ are unit vectors, then the greatest value of $$\sqrt{3}|\vec{a} + \vec{b}| + |\vec{a} - \vec{b}|$$ is___.