NTA JEE Main 7th January 2020 Shift 1

Let $$\alpha$$ and $$\beta$$ be two real roots of the equation $$(k + 1)\tan^2 x - \sqrt{2} \cdot \lambda \tan x = (1 - k)$$, where $$k(\neq -1)$$ and $$\lambda$$ are real numbers. If $$\tan^2(\alpha + \beta) = 50$$, then a value of $$\lambda$$ is

If $$\text{Re}\left(\frac{z-1}{2z+i}\right) = 1$$, where $$z = x + iy$$, then the point $$(x, y)$$ lies on a

Total number of 6-digit numbers in which only and all the five digits 1, 3, 5, 7 and 9 appears, is

Five numbers are in A.P., whose sum is 25 and product is 2520. If one of these five numbers is $$-\frac{1}{2}$$, then the greatest number amongst them is

The greatest positive integer $$k$$, for which $$49^k + 1$$ is a factor of the sum $$49^{125} + 49^{124} + \ldots + 49^2 + 49 + 1$$, is

If $$y = mx + 4$$ is a tangent to both the parabolas, $$y^2 = 4x$$ and $$x^2 = 2by$$, then $$b$$ is equal to

If the distance between the foci of an ellipse is 6 and the distance between its directrices is 12, then the length of its latus rectum is

For two statements $$p$$ and $$q$$, the logical statement $$(p \rightarrow q) \wedge (q \rightarrow \sim p)$$ is equivalent to

Let $$\alpha$$ be a root of the equation $$x^2 + x + 1 = 0$$ and the matrix $$A = \frac{1}{\sqrt{3}}\begin{bmatrix} 1 & 1 & 1 \\ 1 & \alpha & \alpha^2 \\ 1 & \alpha^2 & \alpha^4 \end{bmatrix}$$, then the matrix $$A^{31}$$ is equal to

If the system of linear equations
$$2x + 2ay + az = 0$$
$$2x + 3by + bz = 0$$
$$2x + 4cy + cz = 0$$,
where $$a, b, c \in R$$ are non-zero and distinct; has a non-zero solution, then