NTA JEE Main 24th February 2021 Shift 2

Let $$a, b, c$$ be in arithmetic progression. Let the centroid of the triangle with vertices $$(a, c)$$, $$(2, b)$$ and $$(a, b)$$ be $$\left(\frac{10}{3}, \frac{7}{3}\right)$$. If $$\alpha, \beta$$ are the roots of the equation $$ax^2 + bx + 1 = 0$$, then the value of $$\alpha^2 + \beta^2 - \alpha\beta$$ is:

If $$n \geq 2$$ is a positive integer, then the sum of the series $${}^{n+1}C_2 + 2({}^2C_2 + {}^3C_2 + {}^4C_2 + \ldots + {}^nC_2)$$ is

If $$P$$ is a point on the parabola $$y = x^2 + 4$$ which is closest to the straight line $$y = 4x - 1$$, then the co-ordinates of $$P$$ are:

The negation of the statement $$\sim p \wedge (p \vee q)$$ is:

For the statements $$p$$ and $$q$$, consider the following compound statements:
$$(a)$$ $$(\sim q \wedge (p \to q)) \to \sim p$$
$$(b)$$ $$((p \vee q) \wedge \sim p) \to q$$
Then which of the following statements is correct?

The angle of elevation of a jet plane from a point A on the ground is 60°. After a flight of 20 seconds at the speed of 432 km/hour, the angle of elevation changes to 30°. If the jet plane is flying at a constant height, then its height is:

For the system of linear equations:
$$x - 2y = 1$$, $$x - y + kz = -2$$, $$ky + 4z = 6$$, $$k \in R$$
Consider the following statements:
(A) The system has unique solution if $$k \neq 2, k \neq -2$$.
(B) The system has unique solution if $$k = -2$$.
(C) The system has unique solution if $$k = 2$$.
(D) The system has no-solution if $$k = 2$$.
(E) The system has infinite number of solutions if $$k \neq -2$$.
Which of the following statements are correct?

Let $$A$$ and $$B$$ be $$3 \times 3$$ real matrices such that $$A$$ is a symmetric matrix and $$B$$ is a skew-symmetric matrix. Then the system of linear equations $$(A^2B^2 - B^2A^2)X = O$$, where $$X$$ is a $$3 \times 1$$ column matrix of unknown variables and $$O$$ is a $$3 \times 1$$ null matrix, has:

A possible value of $$\tan\left(\frac{1}{4}\sin^{-1}\frac{\sqrt{63}}{8}\right)$$ is:

For which of the following curves, the line $$x + \sqrt{3}y = 2\sqrt{3}$$ is the tangent at the point $$\left(\frac{3\sqrt{3}}{2}, \frac{1}{2}\right)$$?