NTA JEE Main 27th August 2021 Shift 1

If $$A = \{x \in R : |x-2| > 1\}$$, $$B = \{x \in R : \sqrt{x^2 - 3} > 1\}$$, $$C = \{x \in R : |x-4| \geq 2\}$$ and $$Z$$ is the set of all integers, then the number of subsets of the set $$(A \cap B \cap C)^c \cap Z$$ is _________.

A number is called a palindrome if it reads the same backward as well as forward. For example 285582 is a six digit palindrome. The number of six digit palindromes, which are divisible by 55, is _________.

Let the equation $$x^2 + y^2 + px + (1-p)y + 5 = 0$$ represent circles of varying radius $$r \in (0, 5]$$. Then the number of elements in the set $$S = \{q : q = p^2$$ and $$q$$ is an integer$$\}$$ is _________.

If the minimum area of the triangle formed by a tangent to the ellipse $$\frac{x^2}{b^2} + \frac{y^2}{4a^2} = 1$$ and the co-ordinate axis is $$kab$$, then $$k$$ is equal to _________.

Let $$n$$ be an odd natural number such that the variance of 1, 2, 3, 4, ..., n is 14. Then $$n$$ is equal to _________.

If the system of linear equations
$$2x + y - z = 3$$
$$x - y - z = \alpha$$
$$3x + 3y + \beta z = 3$$
has infinitely many solutions, then $$|\alpha + \beta - \alpha\beta|$$ is equal to _________.

If $$y^{1/4} + y^{-1/4} = 2x$$, and $$(x^2 - 1)\frac{d^2y}{dx^2} + \alpha x\frac{dy}{dx} + \beta y = 0$$, then $$|\alpha - \beta|$$ is equal to _________.

The number of distinct real roots of the equation $$3x^4 + 4x^3 - 12x^2 + 4 = 0$$ is _________.

If $$\int \frac{dx}{(x^2+x+1)^2} = a\tan^{-1}\left(\frac{2x+1}{\sqrt{3}}\right) + b\left(\frac{2x+1}{x^2+x+1}\right) + C$$,$$x > 0$$ where $$C$$ is the constant of integration, then the value of $$9\left(\sqrt{3}a + b\right)$$ is equal to _________.

Let $$\vec{a} = \hat{i} + 5\hat{j} + \alpha\hat{k}$$, $$\vec{b} = \hat{i} + 3\hat{j} + \beta\hat{k}$$ and $$\vec{c} = -\hat{i} + 2\hat{j} - 3\hat{k}$$ be three vectors such that, $$|\vec{b} \times \vec{c}| = 5\sqrt{3}$$ and $$\vec{a}$$ is perpendicular to $$\vec{b}$$. Then the greatest amongst the values of $$|\vec{a}|^2$$ is _________.