NTA JEE Main 10th January 2019 Shift 2

The tangent to the curve, $$y = xe^{x^2}$$ passing through the point $$(1, e)$$ also passes through the point:

If $$\int x^5 e^{-4x^3}dx = \frac{1}{48}e^{-4x^3}f(x) + C$$, where $$C$$ is a constant of integration, then $$f(x)$$ is equal to:

The value of $$\int_{-\pi/2}^{\pi/2} \frac{dx}{[x] + [\sin x] + 4}$$, where $$[t]$$ denotes the greatest integer less than or equal to $$t$$, is:

If $$\int_0^x f(t)dt = x^2 + \int_x^1 t^2 f(t)dt$$, then $$f'\left(\frac{1}{2}\right)$$ is:

A curve amongst the family of curves represented by the differential equation, $$(x^2 - y^2)dx + 2xy \; dy = 0$$ which passes through $$(1, 1)$$, is:

Let $$f(x)$$ be a differentiable function such that $$f'(x) = 7 - \frac{3}{4}\frac{f(x)}{x}$$, $$(x > 0)$$ and $$f(1) \neq 4$$. Then $$\lim_{x \to 0^+} xf\left(\frac{1}{x}\right)$$:

Let $$\vec{\alpha} = (\lambda - 2)\vec{a} + \vec{b}$$ and $$\vec{\beta} = (4\lambda - 2)\vec{a} + 3\vec{b}$$, be two given vectors where vectors $$\vec{a}$$ and $$\vec{b}$$ are non-collinear. The value of $$\lambda$$ for which vectors $$\vec{\alpha}$$ and $$\vec{\beta}$$ are collinear, is:

The plane which bisects the line segment joining the points $$(-3, -3, 4)$$ and $$(3, 7, 6)$$ at right angles, passes through which one of the following points?

On which of the following lines lies the point of intersection of the line, $$\frac{x-4}{2} = \frac{y-5}{2} = \frac{z-3}{1}$$ and the plane, $$x + y + z = 2$$?

If the probability of hitting a target by a shooter, in any shot is $$\frac{1}{3}$$, then the minimum number of independent shots at the target required by him so that the probability of hitting the target at least once is greater than $$\frac{5}{6}$$, is: