NTA JEE Main 8th January 2020 Shift 1

If the equation $$x^2 + bx + 45 = 0$$, $$b \in R$$ has conjugate complex roots and they satisfy $$|z + 1| = 2\sqrt{10}$$, then

Let $$f : R \rightarrow R$$ be such that for all $$x \in R$$ $$(2^{1+x} + 2^{1-x})$$, $$f(x)$$ and $$(3^x + 3^{-x})$$ are in A.P., then the minimum value of $$f(x)$$ is

If a, b and c are the greatest values of $$^{19}C_p$$, $$^{20}C_q$$ and $$^{21}C_r$$ respectively, then:

Let two points be $$A(1, -1)$$ and $$B(0, 2)$$. If a point P(x', y') be such that the area of $$\triangle PAB = 5$$ sq. units and it lies on the line $$3x + y - 4\lambda = 0$$, then a value of $$\lambda$$ is

The locus of a point which divides the line segment joining the point $$(0, -1)$$ and a point on the parabola $$x^2 = 4y$$ internally in the ratio 1 : 2 is:

For $$a > 0$$, let the curves $$C_1 : y^2 = ax$$ and $$C_2 : x^2 = ay$$ intersect at origin O and a point P. Let the line $$x = b$$ ($$0 < b < a$$) intersect the chord OP and the x-axis at points Q and R, respectively. If the line $$x = b$$ bisects the area bounded by the curves, $$C_1$$ and $$C_2$$, and the area of $$\triangle OQR = \frac{1}{2}$$, then 'a' satisfies the equation:

Let the line $$y = mx$$ and the ellipse $$2x^2 + y^2 = 1$$ intersect at a point P in the first quadrant. If the normal to this ellipse at P meets the co-ordinate axes at $$\left(-\frac{1}{3\sqrt{2}}, 0\right)$$ and $$(0, \beta)$$, then $$\beta$$ is equal to

$$\lim_{x \to 0} \left(\frac{3x^2+2}{7x^2+2}\right)^{\frac{1}{x^2}}$$ is equal to

Which one of the following is a tautology?

The mean and the standard deviation (s.d.) of 10 observations are 20 and 2 respectively. Each of these 10 observations is multiplied by $$p$$ and then reduced by $$q$$, where $$p \neq 0$$ and $$q \neq 0$$. If the new mean and new s.d. become half of their original values, then $$q$$ is equal to