NTA JEE Main 25th February 2021 Shift 1

The total number of numbers, lying between 100 and 1000 that can be formed with the digits 1, 2, 3, 4, 5, if the repetition of digits is not allowed and numbers are divisible by either 3 or 5, is ______

Let $$A_1, A_2, A_3, \ldots$$ be squares such that for each $$n \geq 1$$, the length of the side of $$A_n$$ equals the length of diagonal of $$A_{n+1}$$. If the length of $$A_1$$ is 12 cm, then the smallest value of $$n$$ for which area of $$A_n$$ is less than one, is ______

The locus of the point of intersection of the lines $$\left(\sqrt{3}\right)kx + ky - 4\sqrt{3} = 0$$ and $$\sqrt{3}x - y - 4\left(\sqrt{3}\right)k = 0$$ is a conic, whose eccentricity is ______

If $$A = \begin{bmatrix} 0 & -\tan\left(\frac{\theta}{2}\right) \\ \tan\left(\frac{\theta}{2}\right) & 0 \end{bmatrix}$$ and $$(I_2 + A)(I_2 - A)^{-1} = \begin{bmatrix} a & -b \\ b & a \end{bmatrix}$$, then $$13(a^2 + b^2)$$ is equal to ______.

Let $$A = \begin{bmatrix} x & y & z \\ y & z & x \\ z & x & y \end{bmatrix}$$, where $$x, y$$ and $$z$$ are real numbers such that $$x + y + z > 0$$ and $$xyz = 2$$. If $$A^2 = I_3$$, then the value of $$x^3 + y^3 + z^3$$ is ______

If the system of equations
$$kx + y + 2z = 1$$
$$3x - y - 2z = 2$$
$$-2x - 2y - 4z = 3$$
has infinitely many solutions, then $$k$$ is equal to ______.

The number of points, at which the function $$f(x) = |2x + 1| - 3|x + 2| + |x^2 + x - 2|$$, $$x \in R$$ is not differentiable, is ______

Let $$f(x)$$ be a polynomial of degree 6 in $$x$$, in which the coefficient of $$x^6$$ is unity and it has extrema at $$x = -1$$ and $$x = 1$$. If $$\lim_{x \to 0} \frac{f(x)}{x^3} = 1$$, then $$5 \cdot f(2)$$ is equal to ______

The graphs of sine and cosine functions, intersect each other at a number of points and between two consecutive points of intersection, the two graphs enclose the same area $$A$$. Then $$A^4$$ is equal to ______

Let $$\vec{a} = \hat{i} + 2\hat{j} - \hat{k}$$, $$\vec{b} = \hat{i} - \hat{j}$$ and $$\vec{c} = \hat{i} - \hat{j} - \hat{k}$$ be three given vectors. If $$\vec{r}$$ is a vector such that $$\vec{r} \times \vec{a} = \vec{c} \times \vec{a}$$ and $$\vec{r} \cdot \vec{b} = 0$$, then $$\vec{r} \cdot \vec{a}$$ is equal to ______