NTA JEE Main 31st August 2021 Shift 1

A point $$z$$ moves in the complex plane such that $$\arg\left(\frac{z-2}{z+2}\right) = \frac{\pi}{4}$$, then the minimum value of $$|z - 9\sqrt{2} - 2i|^2$$ is equal to _________.

The number of six letter words (with or without meaning), formed using all the letters of the word 'VOWELS', so that all the consonants never come together, is _________.

If $$\left(\frac{x^6}{4^4}\right) k$$ is the term, independent of $$x$$, in the binomial expansion of $$\left(\frac{x}{4} - \frac{12}{x^2}\right)^{12}$$, then $$k$$ is equal to _________.

If the variable line $$3x + 4y = \alpha$$ lies between the two circles $$(x-1)^2 + (y-1)^2 = 1$$ and $$(x-9)^2 + (y-1)^2 = 4$$, without intercepting a chord on either circle, then the sum of all the integral values of $$\alpha$$ is _________.

The mean of 10 numbers
$$7 \times 8, 10 \times 10, 13 \times 12, 16 \times 14, \ldots$$ is _________.

If $$R$$ is the least value of $$a$$ such that the function $$f(x) = x^2 + ax + 1$$ is increasing on $$[1, 2]$$ and $$S$$ is the greatest value of $$a$$ such that the function $$f(x) = x^2 + ax + 1$$ is decreasing on $$[1, 2]$$, then the value of $$|R - S|$$ is _________.

Let $$[t]$$ denote the greatest integer $$\leq t$$. Then the value of $$8 \cdot \int_{-\frac{1}{2}}^{1} \left([2x] + |x|\right) dx$$ is _________.

If $$x\phi(x) = \int_5^x (3t^2 - 2\phi'(t)) dt$$, $$x > -2$$, $$\phi(0) = 4$$, then $$\phi(2)$$ is _________.

The square of the distance of the point of intersection of the line $$\frac{x-1}{2} = \frac{y-2}{3} = \frac{z+1}{6}$$ and the plane $$2x - y + z = 6$$ from the point $$(-1, -1, 2)$$ is _________.

An electric instrument consists of two units. Each unit must function independently for the instrument to operate. The probability that the first unit functions is 0.9 and that of the second unit is 0.8. The instrument is switched on and it fails to operate. If the probability that only the first unit failed and second unit is functioning is $$p$$, then $$98p$$ is equal to _________.