NTA JEE Main 6th September 2020 Shift 1

Let m and M be respectively the minimum and maximum values of $$\begin{vmatrix} \cos^2 x & 1 + \sin^2 x & \sin 2x \\ 1 + \cos^2 x & \sin^2 x & \sin 2x \\ \cos^2 x & \sin^2 x & 1 + \sin 2x \end{vmatrix}$$. Then the ordered pair (m, M) is equal to:

The values of $$\lambda$$ and $$\mu$$ for which the system of linear equations $$x + y + z = 2$$, $$x + 2y + 3z = 5$$, $$x + 3y + \lambda z = \mu$$ has infinitely many solutions, are respectively:

If $$f(x+y) = f(x)f(y)$$ and $$\sum_{x=1}^{\infty} f(x) = 2$$, $$x, y \in N$$, where $$N$$ is the set of all natural numbers, then the value of $$\frac{f(4)}{f(2)}$$ is:

The position of a moving car at time $$t$$ is given by $$f(t) = at^2 + bt + c$$, $$t > 0$$, where $$a$$, $$b$$ and $$c$$ are real numbers greater than 1. Then the average speed of the car over the time interval $$[t_1, t_2]$$ is attained at the point:

If $$I_1 = \int_0^1 (1-x^{50})^{100}\,dx$$ and $$I_2 = \int_0^1 (1-x^{50})^{101}\,dx$$ such that $$I_2 = \alpha I_1$$ then $$\alpha$$ equals to:

$$\lim_{x \to 1}\left(\frac{\int_0^{(x-1)^2} t\cos t^2\,dt}{(x-1)\sin(x-1)}\right)$$

The area (in sq. units) of the region $$A = \{(x, y) : |x| + |y| \leq 1,\; 2y^2 \geq |x|\}$$

The general solution of the differential equation $$\sqrt{1 + x^2 + y^2 + x^2y^2} + xy\frac{dy}{dx} = 0$$ (where C is a constant of integration)

The shortest distance between the lines $$\frac{x-1}{0} = \frac{y+1}{-1} = \frac{z}{1}$$ and $$x + y + z + 1 = 0$$, $$2x - y + z + 3 = 0$$ is:

Out of 11 consecutive natural numbers if three numbers are selected at random (without repetition), then the probability that they are in A.P. with positive common difference is: