NTA JEE Main 31st January 2023 Shift 1

The number of real roots of the equation $$\sqrt{x^2 - 4x + 3} + \sqrt{x^2 - 9} = \sqrt{4x^2 - 14x + 6}$$, is:

For all $$z \in C$$ on the curve $$C_1$$: $$|z| = 4$$, let the locus of the point $$z + \dfrac{1}{z}$$ be the curve $$C_2$$. Then

If the sum and product of four positive consecutive terms of a G.P., are 126 and 1296, respectively, then the sum of common ratios of all such GPs is

Let a circle $$C_1$$ be obtained on rolling the circle $$x^2 + y^2 - 4x - 6y + 11 = 0$$ upwards 4 units on the tangent T to it at the point (3, 2). Let $$C_2$$ be the image of $$C_1$$ in T. Let $$A$$ and $$B$$ be the centers of circles $$C_1$$ and $$C_2$$ respectively, and $$M$$ and $$N$$ be respectively the feet of perpendiculars drawn from $$A$$ and $$B$$ on the x-axis. Then the area of the trapezium AMNB is:

If the maximum distance of normal to the ellipse $$\dfrac{x^2}{4} + \dfrac{y^2}{b^2} = 1, b < 2$$, from the origin is 1, then the eccentricity of the ellipse is:

Consider:
S1: $$p \Rightarrow q \lor p \land \sim q$$ is a tautology.
S2: $$\sim p \Rightarrow \sim q \land \sim p \lor q$$ is a contradiction.
Then

Let $$R$$ be a relation on $$N \times N$$ defined by $$a, b R c, d$$ if and only if $$abd - c = bca - d$$. Then $$R$$ is

Let $$A = \begin{matrix} 1 & 0 & 0 \\ 0 & 4 & -1 \\ 0 & 12 & -3 \end{matrix}$$. Then the sum of the diagonal elements of the matrix $$A + I^{11}$$ is equal to:

For the system of linear equations
$$x + y + z = 6$$
$$\alpha x + \beta y + 7z = 3$$
$$x + 2y + 3z = 14$$
which of the following is NOT true?

If $$\sin^{-1}\dfrac{\alpha}{17} + \cos^{-1}\dfrac{4}{5} - \tan^{-1}\dfrac{77}{36} = 0$$, $$0 < \alpha < 13$$, then $$\sin^{-1}\sin\alpha + \cos^{-1}\cos\alpha$$ is equal to