NTA JEE Main 7th January 2020 Shift 1

If the sum of the coefficients of all even powers of $$x$$ in the product $$(1 + x + x^2 + \ldots + x^{2n})(1 - x + x^2 - x^3 + \ldots + x^{2n})$$ is 61, then $$n$$ is equal to

Let $$A(1, 0)$$, $$B(6, 2)$$ and $$C\left(\frac{3}{2}, 6\right)$$ be the vertices of a triangle ABC. If P is a point inside the triangle ABC such that the triangles APC, APB and BPC have equal areas, then the length of the line segment PQ, where Q is the point $$\left(-\frac{7}{6}, -\frac{1}{3}\right)$$, is

$$\lim_{x \to 2} \frac{3^x + 3^{3-x} - 12}{3^{-x/2} - 3^{1-x}}$$ is equal to

If the variance of the first $$n$$ natural numbers is 10 and the variance of the first $$m$$ even natural numbers is 16, then the value of $$m + n$$ is equal to

Let S be the set of points where the function $$f(x) = |2 - |x - 3||$$, $$x \in R$$, is not differentiable. Then $$\sum_{x \in S} f(f(x))$$ is equal to