NTA JEE Main 4th September 2020 Shift 1

Let [t] denote the greatest integer $$\leq$$ t. Then the equation in x, $$[x]^2 + 2[x+2] - 7 = 0$$ has:

Let $$\alpha$$ and $$\beta$$ be the roots of $$x^2 - 3x + p = 0$$ and $$\gamma$$ and $$\delta$$ be the roots of $$x^2 - 6x + q = 0$$. If $$\alpha, \beta, \gamma, \delta$$ form a geometric progression. Then ratio $$(2q + p) : (2q - p)$$ is

Let $$u = \frac{2z+i}{z-ki}$$, $$z = x + iy$$ and $$k \gt 0$$. If the curve represented by Re(u) + Im(u) = 1 intersects the y-axis at points P and Q where PQ = 5 then the value of k is

If $$1 + (1 - 2^2 \cdot 1) + (1 - 4^2 \cdot 3) + (1 - 6^2 \cdot 5) + \ldots + (1 - 20^2 \cdot 19) = \alpha - 220\beta$$, then an ordered pair $$(\alpha, \beta)$$ is equal to:

The value of $$\sum_{r=0}^{20} {}^{50-r}C_6$$ is equal to:

A triangle ABC lying in the first quadrant has two vertices as $$A(1, 2)$$ and $$B(3, 1)$$. If $$\angle BAC = 90°$$, and ar($$\triangle ABC$$) = $$5\sqrt{5}$$ sq. units, then the abscissa of the vertex C is:

Let $$\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$$ $$(a > b)$$ be a given ellipse, length of whose latus rectum is 10. If its eccentricity is the maximum value of the function, $$\phi(t) = \frac{5}{12} + t - t^2$$, then $$a^2 + b^2$$ is equal to:

Let $$P(3, 3)$$ be a point on the hyperbola, $$\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$$. If the normal to it at P intersects the $$x$$-axis at (9, 0) and $$e$$ is its eccentricity, then the ordered pair $$(a^2, e^2)$$ is equal to:

Given the following two statements:
$$(S_1)$$ : $$(q \vee p) \to (p \leftrightarrow \sim q)$$ is a tautology
$$(S_2)$$ : $$\sim q \wedge (\sim p \leftrightarrow q)$$ is a fallacy. Then:

The mean and variance of 8 observations are 10 and 13.5, respectively. If 6 of these observations are 5, 7, 10, 12, 14, 15, then the absolute difference of the remaining two observations is: