NTA JEE Mains 1st Feb 2023 Shift 1

Let $$S = \{x : x \in \mathbb{R} \text{ and } (\sqrt{3} + \sqrt{2})^{x^2-4} + (\sqrt{3} - \sqrt{2})^{x^2-4} = 10\}$$. Then $$n(S)$$ is equal to

If the center and radius of the circle $$\l\left|?\frac{z-2}{z-3} \right| = 2$$ are respectively $$(?lpha, ?eta)$$ and $$\gamma$$, then $$3(?lpha + ?eta + \gamma)$$ is equal to

The sum to 10 terms of the series $$\f\frac{1}{1+1^2+1^4} + \f\frac{2}{1+2^2+2^4} + \f\frac{3}{1+3^2+3^4} + \ldots$$ is :-

The value of $$\frac{1}{1!50!} + \frac{1}{3!48!} + \frac{1}{5!46!} + \ldots + \frac{1}{49!2!} + \frac{1}{51!1!}$$ is

The combined equation of the two lines $$ax + by + c = 0$$ and $$a'x + b'y + c' = 0$$ can be written as $$(ax + by + c)(a'x + b'y + c') = 0$$. The equation of the angle bisectors of the lines represented by the equation $$2x^2 + xy - 3y^2 = 0$$ is

If the orthocentre of the triangle, whose vertices are $$(1, 2), (2, 3)$$ and $$(3, 1)$$ is $$(\alpha, \beta)$$, then the quadratic equation whose roots are $$\alpha + 4\beta$$ and $$4\alpha + \beta$$, is

The negation of the expression $$q \vee ((\sim q) \wedge p)$$ is equivalent to

The mean and variance of 5 observations are 5 and 8 respectively. If 3 observations are 1, 3, 5, then the sum of cubes of the remaining two observations is

For a triangle $$ABC$$, the value of $$\cos 2A + \cos 2B + \cos 2C$$ is least. If its inradius is 3 and incentre is $$M$$, then which of the following is NOT correct?

Let $$R$$ be a relation on $$\mathbb{R}$$, given by $$R = \{(a, b) : 3a - 3b + \sqrt{7} \text{ is an irrational number}\}$$. Then $$R$$ is