NTA JEE Main 24th February 2021 Shift 1

If $$f: R \to R$$ is a function defined by $$f(x) = x - 1\cos\frac{2x-1}{2}\pi$$, where $$[\cdot]$$ denotes the greatest integer function, then $$f$$ is:

The function $$f(x) = \frac{4x^3 - 3x^2}{6} - 2\sin x + (2x - 1)\cos x$$:

If the tangent to the curve $$y = x^3$$ at the point $$P(t, t^3)$$ meets the curve again at $$Q$$, then the ordinate of the point which divides $$PQ$$ internally in the ratio 1 : 2 is:

If $$\int \frac{\cos x - \sin x}{\sqrt{8 - \sin 2x}} dx = a\sin^{-1}\frac{\sin x + \cos x}{b} + c$$, where $$c$$ is a constant of integration, then the ordered pair $$(a, b)$$ is equal to:

$$\lim_{x \to 0} \frac{\int_0^{x^2} \sin\sqrt{ t} \, dt}{x^3}$$ is equal to:

The area (in sq. units) of the part of the circle $$x^2 + y^2 = 36$$, which is outside the parabola $$y^2 = 9x$$, is equal to

The population $$P = P(t)$$ at time $$t$$ of a certain species follows the differential equation $$\frac{dP}{dt} = 0.5P - 450$$. If $$P(0) = 850$$, then the time at which population becomes zero is:

The distance of the point (1, 1, 9) from the point of intersection of the line $$\frac{x - 3}{1} = \frac{y - 4}{2} = \frac{z - 5}{2}$$ and the plane $$x + y + z = 17$$ is:

The equation of the plane passing through the point (1, 2, -3) and perpendicular to the planes $$3x + y - 2z = 5$$ and $$2x - 5y - z = 7$$, is

An ordinary dice is rolled for a certain number of times. If the probability of getting an odd number 2 times is equal to the probability of getting an even number 3 times, then the probability of getting an odd number for odd number of times is: