JEE Main 12th January 2019 Shift 1

If a variable line $$3x + 4y - \lambda = 0$$ is such that the two circles $$x^2 + y^2 - 2x - 2y + 1 = 0$$ and $$x^2 + y^2 - 18x - 2y + 78 = 0$$ are on its opposite sides, then the set of all values of $$\lambda$$ is the interval:

Let $$P(4, -4)$$ and $$Q(9, 6)$$ be two points on the parabola, $$y^2 = 4x$$ and let X be any point on the arc POQ of this parabola, where O is the vertex of this parabola, such that the area of $$\Delta PXQ$$ is maximum. Then this maximum area (in sq. units) is:

If the vertices of a hyperbola be at $$(-2, 0)$$ and $$(2, 0)$$ and one of its foci be at $$(-3, 0)$$, then which one of the following points does not lie on this hyperbola?

$$\lim_{x \to \frac{\pi}{4}} \frac{\cot^3 x - \tan x}{\cos\left(x + \frac{\pi}{4}\right)}$$ is

The Boolean expression $$((p \wedge q) \vee (p \vee \sim q)) \wedge (\sim p \wedge \sim q)$$ is equivalent to

If the sum of the deviations of 50 observations from 30 is 50, then the mean of these observations is:

Let $$P = \begin{bmatrix} 1 & 0 & 0 \\ 3 & 1 & 0 \\ 9 & 3 & 1 \end{bmatrix}$$ and $$Q = [q_{ij}]$$ be two $$3 \times 3$$ matrices such that $$Q - P^5 = I_3$$. Then $$\frac{q_{21} + q_{31}}{q_{32}}$$ is equal to:

An ordered pair $$(\alpha, \beta)$$ for which the system of linear equations $$(1 + \alpha)x + \beta y + z = 2$$, $$\alpha x + (1 + \beta)y + z = 3$$, $$\alpha x + \beta y + 2z = 2$$ has a unique solution, is:

Considering only the principal values of inverse functions, the set $$A = \{x \ge 0 : \tan^{-1}(2x) + \tan^{-1}(3x) = \frac{\pi}{4}\}$$

Let S be the set of all points in $$(-\pi, \pi)$$ at which the function, $$f(x) = \min\{\sin x, \cos x\}$$ is not differentiable. Then S is a subset of which of the following?