NTA JEE Main 1st September 2021 Shift 2

If for the complex numbers $$z$$ satisfying $$|z - 2 - 2i| \leq 1$$, the maximum value of $$|3iz + 6|$$ is attained at $$a + ib$$, then $$a + b$$ is equal to _________.

All the arrangements, with or without meaning, of the word FARMER are written excluding any word that has two R appearing together. The arrangements are listed serially in the alphabetic order as in the English dictionary. Then the serial number of the word FARMER in this list is _________.

If the sum of the coefficients in the expansion of $$(x + y)^n$$ is 4096, then the greatest coefficient in the expansion is _________.

Let the points of intersections of the lines $$x - y + 1 = 0$$, $$x - 2y + 3 = 0$$ and $$2x - 5y + 11 = 0$$ are the mid points of the sides of a triangle ABC. Then the area of the triangle ABC is _________.

A man starts walking from the point $$P(-3, 4)$$, touches the x-axis at $$R$$, and then turns to reach at the point $$Q(0, 2)$$. The man is walking at a constant speed. If the man reaches the point Q in the minimum time, then $$50[(PR)^2 + (RQ)^2]$$ is equal to _________.

Let $$f(x) = x^6 + 2x^4 + x^3 + 2x + 3$$, $$x \in R$$. Then the natural number $$n$$ for which $$\lim_{x \to 1} \frac{x^n f(1) - f(x)}{x - 1} = 44$$ is _________.

Let $$f(x)$$ be a polynomial of degree 3 such that $$f(k) = -\frac{2}{k}$$ for $$k = 2, 3, 4, 5$$. Then the value of $$52 - 10 f(10)$$ is equal to _________.

Let $$[t]$$ denote the greatest integer $$\leq t$$. The number of points where the function $$f(x) = [x]|x^2 - 1| + \sin\frac{\pi}{[x]+3} - [x+1]$$, $$x \in (-2, 2)$$ is not continuous is _________.

Let $$\vec{a} = 2\hat{i} - \hat{j} + 2\hat{k}$$ and $$\vec{b} = \hat{i} + 2\hat{j} - \hat{k}$$. Let a vector $$\vec{v}$$ be in the plane containing $$\vec{a}$$ and $$\vec{b}$$. If $$\vec{v}$$ is perpendicular to the vector $$3\hat{i} + 2\hat{j} - \hat{k}$$ and its projection on $$\vec{a}$$ is 19 units, then $$|2\vec{v}|^2$$ is equal to _________.

Let $$X$$ be a random variable with distribution.

If the mean of $$X$$ is 2.3 and variance of $$X$$ is $$\sigma^2$$, then $$100\sigma^2$$ is equal to _________.