NTA JEE Main 10th January 2019 Shift 1

Let, $$f: R \to R$$ be a function such that $$f(x) = x^3 + x^2f'(1) + xf''(2) + f'''(3)$$, $$\forall x \in R$$. Then $$f(2)$$ equals:

The shortest distance between the point $$\left(\frac{3}{2}, 0\right)$$ and the curve $$y = \sqrt{x}$$, $$(x > 0)$$, is:

Let, $$n \geq 2$$ be a natural number and $$0 \lt \theta \lt \frac{\pi}{2}$$. Then $$\int \frac{(\sin^n\theta - \sin\theta)^{1/n} \cos\theta}{\sin^{n+1}\theta} d\theta$$, is equal to:

Let $$I = \int_a^b (x^4 - 2x^2)dx$$. If $$I$$ is minimum then the ordered pair $$(a, b)$$ is:

If the area enclosed between the curves $$y = kx^2$$ and $$x = ky^2$$, $$(k \gt 0)$$, is 1 sq. unit. Then $$k$$ is:

If $$\frac{dy}{dx} + \frac{3}{\cos^2 x}y = \frac{1}{\cos^2 x}$$, $$x \in \left(-\frac{\pi}{3}, \frac{\pi}{3}\right)$$, and $$y\left(\frac{\pi}{4}\right) = \frac{4}{3}$$, then $$y\left(-\frac{\pi}{4}\right)$$ equals:

Let $$\vec{a} = 2\hat{i} + \lambda_1\hat{j} + 3\hat{k}$$, $$\vec{b} = 4\hat{i} + (3-\lambda_2)\hat{j} + 6\hat{k}$$ and $$\vec{c} = 3\hat{i} + 6\hat{j} + (\lambda_3 - 1)\hat{k}$$ be three vectors such that $$\vec{b} = 2\vec{a}$$ and $$\vec{a}$$ is perpendicular to $$\vec{c}$$. Then a possible value of $$(\lambda_1, \lambda_2, \lambda_3)$$ is:

Let $$A$$ be a point on the line $$\vec{r} = (1-3\mu)\hat{i} + (\mu-1)\hat{j} + (2+5\mu)\hat{k}$$ and $$B(3, 2, 6)$$ be a point in the space. Then the value of $$\mu$$ for which the vector $$\vec{AB}$$ is parallel to the plane $$x - 4y + 3z = 1$$ is:

The plane passing through the point $$(4, -1, 2)$$ and parallel to the lines $$\frac{x+2}{3} = \frac{y-2}{-1} = \frac{z+1}{2}$$ and $$\frac{x-2}{1} = \frac{y-3}{2} = \frac{z-4}{3}$$ also passes through the point:

An unbiased coin is tossed. If the outcome is a head then a pair of unbiased dice is rolled and the sum of the numbers obtained on them is noted. If the toss of the coin results in tail then a card from a well-shuffled pack of nine cards numbered 1, 2, 3, ..., 9 is randomly picked and the number on the card is noted. The probability that the noted number is either 7 or 8 is: