NTA JEE Main 25th February 2021 Shift 2

The total number of two digit numbers $$'n'$$, such that $$3^n + 7^n$$ is a multiple of 10, is ______.

If the remainder when $$x$$ is divided by 4 is 3, then the remainder when $$(2020 + x)^{2022}$$ is divided by 8 is ______.

A line is a common tangent to the circle $$(x - 3)^2 + y^2 = 9$$ and the parabola $$y^2 = 4x$$. If the two points of contact $$(a, b)$$ and $$(c, d)$$ are distinct and lie in the first quadrant, then $$2(a + c)$$ is equal to ______.

If $$\lim_{x \to 0} \frac{ax - (e^{4x} - 1)}{ax(e^{4x} - 1)}$$ exists and is equal to $$b$$, then the value of $$a - 2b$$ is ______.

A function $$f$$ is defined on $$[-3, 3]$$ as
$$f(x) = \begin{cases} \min\{|x|, 2 - x^2\}, & -2 \leq x \leq 2 \\ [|x|], & 2 < |x| \leq 3 \end{cases}$$
where $$[x]$$ denotes the greatest integer $$\leq x$$. The number of points, where $$f$$ is not differentiable in $$(-3, 3)$$ is ______.

If the curves $$x = y^4$$ and $$xy = k$$ cut at right angles, then $$(4k)^6$$ is equal to ______.

The value of $$\int_{-2}^{2} |3x^2 - 3x - 6| dx$$ is ______

If the curve, $$y = y(x)$$ represented by the solution of the differential equation $$(2xy^2 - y)dx + x \, dy = 0$$, passes through the intersection of the lines, $$2x - 3y = 1$$ and $$3x + 2y = 8$$, then $$|y(1)|$$ is equal to ______.

Let $$\vec{a} = \hat{i} + \alpha\hat{j} + 3\hat{k}$$ and $$\vec{b} = 3\hat{i} - \alpha\hat{j} + \hat{k}$$. If the area of the parallelogram whose adjacent sides are represented by the vectors $$\vec{a}$$ and $$\vec{b}$$ is $$8\sqrt{3}$$ square units, then $$\vec{a} \cdot \vec{b}$$ is equal to ______.

A line $$l$$ passing through origin is perpendicular to the lines
$$l_1: \vec{r} = (3 + t)\hat{i} + (-1 + 2t)\hat{j} + (4 + 2t)\hat{k}$$
$$l_2: \vec{r} = (3 + 2s)\hat{i} + (3 + 2s)\hat{j} + (2 + s)\hat{k}$$
If the co-ordinates of the point in the first octant on $$l_2$$ at a distance of $$\sqrt{17}$$ from the point of intersection of $$l$$ and $$l_1$$ are $$(a, b, c)$$, then $$18(a + b + c)$$ is equal to ______.