NTA JEE Main 4th September 2020 Shift 2

Let $$\lambda \neq 0$$ be in $$R$$. If $$\alpha$$ and $$\beta$$ are the roots of the equation, $$x^2 - x + 2\lambda = 0$$ and $$\alpha$$ and $$\gamma$$ are the roots of the equation, $$3x^2 - 10x + 27\lambda = 0$$, then $$\frac{\beta\gamma}{\lambda}$$ is equal to:

If $$a$$ and $$b$$ are real numbers such that $$(2 + \alpha)^4 = a + b\alpha$$, where $$\alpha = \frac{-1 + i\sqrt{3}}{2}$$, then $$a + b$$ is equal to:

Let $$a_1, a_2, \ldots, a_n$$ be a given A.P. whose common difference is an integer and $$S_n = a_1 + a_2 + \ldots + a_n$$. If $$a_1 = 1, a_n = 300$$ and $$15 \leq n \leq 50$$, then the ordered pair $$(S_{n-4}, a_{n-4})$$ is equal to:

If for some positive integer $$n$$, the coefficients of three consecutive terms in the binomial expansion of $$(1 + x)^{n+5}$$ are in the ratio 5 : 10 : 14, then the largest coefficient in the expansion is:

If the perpendicular bisector of the line segment joining the points $$P(1, 4)$$ and $$Q(k, 3)$$ has $$y$$-intercept equal to $$-4$$, then a value of $$k$$ is:

The circle passing through the intersection of the circles, $$x^2 + y^2 - 6x = 0$$ and $$x^2 + y^2 - 4y = 0$$ having its centre on the line, $$2x - 3y + 12 = 0$$, also passes through the point:

Let $$x = 4$$ be a directrix to an ellipse whose centre is at the origin and its eccentricity is $$\frac{1}{2}$$. If $$P(1, \beta), \beta \gt 0$$ is a point on this ellipse, then the equation of the normal to it at P is:

Contrapositive of the statement:
'If a function $$f$$ is differentiable at $$a$$, then it is also continuous at $$a$$', is

The angle of elevation of a cloud $$C$$ from a point $$P$$, 200 m above a still lake is 30°. If the angle of depression of the image of $$C$$ in the lake from the point $$P$$ is 60°, then $$PC$$ (in m) is equal to

Let $$\bigcup_{i=1}^{50} X_i = \bigcup_{i=1}^{n} Y_i = T$$, where each $$X_i$$ contains 10 elements and each $$Y_i$$ contains 5 elements. If each element of the set $$T$$ is an element of exactly 20 of sets $$X_i$$'s and exactly 6 of sets $$Y_i$$'s then $$n$$ is equal to: