NTA JEE Main 11th April 2023 Shift 2

The number of points, where the curve $$f(x) = e^{8x} - e^{6x} - 3e^{4x} - e^{2x} + 1$$, $$x \in \mathbb{R}$$ cuts $$x$$-axis, is equal to

For $$a \in \mathbb{C}$$, let $$A = \{z \in \mathbb{C}: \text{Re}(a + \bar{z}) > \text{Im}(\bar{a} + z)\}$$ and $$B = \{z \in \mathbb{C}: \text{Re}(a + \bar{z}) < \text{Im}(\bar{a} + z)\}$$. Then among the two statements:
$$(S1)$$: If $$\text{Re}(a), \text{Im}(a) > 0$$, then the set $$A$$ contains all the real numbers
$$(S2)$$: If $$\text{Re}(a), \text{Im}(a) < 0$$, then the set $$B$$ contains all the real numbers,

If the letters of the word MATHS are permuted and all possible words so formed are arranged as in a dictionary with serial numbers, then the serial number of the word THAMS is

Let $$a, b, c$$ and $$d$$ be positive real numbers such that $$a + b + c + d = 11$$. If the maximum value of $$a^5b^3c^2d$$ is $$3750\beta$$, then the value of $$\beta$$ is

For $$k \in \mathbb{N}$$, if the sum of the series $$1 + \frac{4}{k} + \frac{8}{k^2} + \frac{13}{k^3} + \frac{19}{k^4} + \ldots$$ is 10, then the value of $$k$$ is

If the 1011$$^{th}$$ term from the end in the binomial expansion of $$\left(\frac{4x}{5} - \frac{5}{2x}\right)^{2022}$$ is 1024 times 1011$$^{th}$$ term from the beginning, then $$32|x|$$ is equal to

The sum of the coefficients of three consecutive terms in the binomial expansion of $$(1 + x)^{n+2}$$, which are in the ratio 1 : 3 : 5, is equal to

The converse of $$((-p) \wedge q) \Rightarrow r$$ is

Let the mean of 6 observations 1, 2, 4, 5, $$x$$ and $$y$$ be 5 and their variance be 10. Then their mean deviation about the mean is equal to

The angle of elevation of the top $$P$$ of a tower from the feet of one person standing due south of the tower is 45$$^\circ$$ and from the feet of another person standing due west of the tower is 30$$^\circ$$. If the height of the tower is 5 meters, then the distance (in meters) between the two persons is equal to