JEE Main 11th January 2019 Shift 2

Let x, y be positive real numbers and m, n positive integers. The maximum value of the expression $$\frac{x^m y^n}{(1+x^{2m})(1+y^{2n})}$$ is:

If $$\int \frac{x+1}{\sqrt{2x-1}} dx = f(x)\sqrt{2x-1} + C$$, where C is a constant of integration, then $$f(x)$$ is equal to:

The integral $$\int_{\pi/6}^{\pi/4} \frac{dx}{\sin 2x(\tan^5 x + \cot^5 x)}$$ equals:

The area (in sq. units) in the first quadrant bounded by the parabola, $$y = x^2 + 1$$, the tangent to it at the point (2, 5) and the coordinate axes is:

The solution of the differential equation, $$\frac{dy}{dx} = (x - y)^2$$, when $$y(1) = 1$$, is:

Let $$\sqrt{3}\hat{i} + \hat{j}$$, $$\hat{i} + \sqrt{3}\hat{j}$$ and $$\beta\hat{i} + (1 - \beta)\hat{j}$$ respectively be the position vectors of the points A, B and C with respect to the origin O. If the distance of C from the bisector of the acute angle between OA and OB is $$\frac{\sqrt{3}}{\sqrt{2}}$$, then the sum of all possible values of $$\beta$$ is:

Two lines $$\frac{x-3}{1} = \frac{y+1}{3} = \frac{z-6}{-1}$$ and $$\frac{x+5}{7} = \frac{y-2}{-6} = \frac{z-3}{4}$$ intersect at the point R. The reflection of R in the xy-plane has coordinates:

If the point $$(2, \alpha, \beta)$$ lies on the plane which passes through the points (3,4,2) and (7,0,6) and is perpendicular to the plane $$2x - 5y = 15$$, then $$2\alpha - 3\beta$$ is equal to:

Let $$S = \{1, 2, \ldots, 20\}$$. A subset B of S is said to be "nice" if the sum of the elements of B is 203. Then the probability that a randomly chosen subset of S is "nice" is:

A bag contains 30 white balls and 10 red balls. 16 balls are drawn one by one randomly from the bag with replacement. If X be the number of white balls drawn, then $$\left(\frac{\text{mean of X}}{\text{standard deviation of X}}\right)$$ is equal to: