NTA JEE Main 25th February 2021 Shift 1

A man is observing, from the top of a tower, a boat speeding towards the tower from a certain point A, with uniform speed. At that point, angle of depression of the boat with the man's eye is 30° (Ignore man's height). After sailing for 20 seconds, towards the base of the tower (which is at the level of water), the boat has reached a point B, where the angle of depression is 45°. Then the time taken (in seconds) by the boat from B to reach the base of the tower is:

Let $$f, g : N \to N$$ such that $$f(n + 1) = f(n) + f(1)$$ $$\forall n \in N$$ and $$g$$ be any arbitrary function. Which of the following statements is NOT true?

If Rolle's theorem holds for the function $$f(x) = x^3 - ax^2 + bx - 4$$, $$x \in [1, 2]$$ with $$f'\left(\frac{4}{3}\right) = 0$$, then ordered pair $$(a, b)$$ is equal to:

The value of the integral $$\int \frac{\sin\theta \cdot \sin 2\theta (\sin^6\theta + \sin^4\theta + \sin^2\theta)\sqrt{2\sin^4\theta + 3\sin^2\theta + 6}}{1 - \cos 2\theta} d\theta$$ is (where $$c$$ is a constant of integration)

The value of $$\int_{-1}^{1} x^2 e^{[x^3]} dx$$, where $$[t]$$ denotes the greatest integer $$\leq t$$, is:

If a curve passes through the origin and the slope of the tangent to it at any point $$(x, y)$$ is $$\frac{x^2 - 4x + y + 8}{x - 2}$$, then this curve also passes through the point:

Let $$\alpha$$ be the angle between the lines whose direction cosines satisfy the equations $$l + m - n = 0$$ and $$l^2 + m^2 - n^2 = 0$$. Then the value of $$\sin^4\alpha + \cos^4\alpha$$ is:

The equation of the line through the point (0, 1, 2) and perpendicular to the line $$\frac{x-1}{2} = \frac{y+1}{3} = \frac{z-1}{-2}$$ is:

The coefficients $$a, b$$ and $$c$$ of the quadratic equation, $$ax^2 + bx + c = 0$$ are obtained by throwing a dice three times. The probability that this equation has equal roots is:

When a missile is fired from a ship, the probability that it is intercepted is $$\frac{1}{3}$$ and the probability that the missile hits the target, given that it is not intercepted, is $$\frac{3}{4}$$. If three missiles are fired independently from the ship, then the probability that all three hit the target, is: