NTA JEE Main 24th June 2022 Shift 1

The set of all values of $$k$$ for which $$(\tan^{-1}x)^3 + (\cot^{-1}x)^3 = k\pi^3, x \in R$$, is the interval

The domain of $$f(x) = \frac{\cos^{-1}\left(\frac{x^2 - 5x + 6}{x^2 - 9}\right)}{\log(x^2 - 3x + 2)}$$ is

For the function $$f(x) = 4\log_e(x-1) - 2x^2 + 4x + 5, x > 1$$, which one of the following is NOT correct?

If the tangent at the point $$(x_1, y_1)$$ on the curve $$y = x^3 + 3x^2 + 5$$ passes through the origin, then $$(x_1, y_1)$$ does NOT lie on the curve

The sum of absolute maximum and absolute minimum values of the function $$f(x) = |2x^2 + 3x - 2| + \sin x \cos x$$ in the interval $$[0, 1]$$ is

The surface area of a balloon of spherical shape being inflated, increases at a constant rate. If initially, the radius of balloon is $$3$$ units and after $$5$$ seconds, it becomes $$7$$ units, then its radius after $$9$$ seconds is

If $$x = x(y)$$ is the solution of the differential equation $$y\frac{dx}{dy} = 2x + y^3(y+1)e^y, x(1) = 0$$; then $$x(e)$$ is equal to

Let $$\hat{a}, \hat{b}$$ be unit vectors. If $$\vec{c}$$ be a vector such that the angle between $$\hat{a}$$ and $$\vec{c}$$ is $$\frac{\pi}{12}$$, and $$\hat{b} = \vec{c} + 2(\vec{c} \times \hat{a})$$, then $$|6\vec{c}|^2$$ is equal to:

Bag $$A$$ contains $$2$$ white, $$1$$ black and $$3$$ red balls and bag $$B$$ contains $$3$$ black, $$2$$ red and $$n$$ white balls. One bag is chosen at random and $$2$$ balls drawn from it at random are found to be $$1$$ red and $$1$$ black. If the probability that both balls come from Bag $$A$$ is $$\frac{6}{11}$$, then $$n$$ is equal to

If a random variable $$X$$ follows the Binomial distribution $$B(33, p)$$ such that $$3P(X = 0) = P(X = 1)$$, then the value of $$\frac{P(X = 15)}{P(X = 18)} - \frac{P(X = 16)}{P(X = 17)}$$ is equal to