NTA JEE Main 9th January 2019 Shift 2

Let $$A = \{x \in R : x$$ is not a positive integer$$\}$$. Define a function $$f: A \to R$$ as $$f(x) = \frac{2x}{x-1}$$, then $$f$$ is:

Let $$f$$ be a differentiable function from $$R$$ to $$R$$ such that $$|f(x) - f(y)| \leq 2|x - y|^{3/2}$$, for all $$x, y \in R$$. If $$f(0) = 1$$ then $$\int_0^1 f^2(x)dx$$ is equal to:

If $$x = 3\tan t$$ and $$y = 3\sec t$$, then the value of $$\frac{d^2y}{dx^2}$$ at $$t = \frac{\pi}{4}$$, is:

If $$f(x) = \int \frac{(5x^8 + 7x^6)}{(x^2 + 1 + 2x^7)^2}dx$$, $$(x \geq 0)$$, and $$f(0) = 0$$, then the value of $$f(1)$$ is:

If $$\int_0^{\pi/3} \frac{\tan\theta}{\sqrt{2k\sec\theta}} d\theta = 1 - \frac{1}{\sqrt{2}}$$, $$(k \gt 0)$$, then the value of $$k$$ is:

The area of the region $$A = \{(x, y) : 0 \leq y \leq x|x| + 1$$ and $$-1 \leq x \leq 1\}$$ in sq. units, is:

Let $$\vec{a} = \hat{i} + \hat{j} + \sqrt{2}\hat{k}$$, $$\vec{b} = b_1\hat{i} + b_2\hat{j} + \sqrt{2}\hat{k}$$ and $$\vec{c} = 5\hat{i} + \hat{j} + \sqrt{2}\hat{k}$$ be three vectors such that the projection vector of $$\vec{b}$$ on $$\vec{a}$$ is $$|\vec{a}|$$. If $$\vec{a} + \vec{b}$$ is perpendicular to $$\vec{c}$$, then $$|\vec{b}|$$ is equal to:

If the lines $$x = ay + b$$, $$z = cy + d$$ and $$x = a'z + b'$$, $$y = c'z + d'$$ are perpendicular, then:

The equation of the plane containing the straight line $$\frac{x}{2} = \frac{y}{3} = \frac{z}{4}$$ and perpendicular to the plane containing the straight lines $$\frac{x}{3} = \frac{y}{4} = \frac{z}{2}$$ and $$\frac{x}{4} = \frac{y}{2} = \frac{z}{3}$$ is:

An urn contains 5 red and 2 green balls. A ball is drawn at random from the urn. If the drawn ball is green, then a red ball is added to the urn and if the drawn ball is red, then a green ball is added to the urn; the original ball is not returned to the urn. Now, a second ball is drawn at random from it. The probability that the second ball is red, is: