NTA JEE Main 27th July 2021 Shift 2

The number of real roots of the equation $$e^{4x} - e^{3x} - 4e^{2x} - e^x + 1 = 0$$ is equal to _________

If the real part of the complex number $$z = \frac{3 + 2i\cos\theta}{1 - 3i\cos\theta}$$, $$\theta \in \left(0, \frac{\pi}{2}\right)$$ is zero, then the value of $$\sin^2 3\theta + \cos^2 \theta$$ is equal to _________

Let $$n$$ be a non-negative integer. Then the number of divisors of the form $$4n + 1$$ of the number $$(10)^{10} \cdot (11)^{11} \cdot (13)^{13}$$ is equal to _________.

Let $$E$$ be an ellipse whose axes are parallel to the co-ordinates axes, having its centre at $$(3, -4)$$, one focus at $$(4, -4)$$ and one vertex at $$(5, -4)$$. If $$mx - y = 4$$, $$m \gt 0$$ is a tangent to the ellipse $$E$$, then the value of $$5m^2$$ is equal to _________.

Let $$A = \{n \in N \mid n^2 \leq n + 10000\}$$, $$B = \{3k + 1 \mid k \in N\}$$ and $$C = \{2k \mid k \in N\}$$, then the sum of all the elements of the set $$A \cap (B - C)$$ is equal to _________.

If $$A = \begin{bmatrix} 1 & 1 & 1 \\ 0 & 1 & 1 \\ 0 & 0 & 1 \end{bmatrix}$$ and $$M = A + A^2 + A^3 + \ldots + A^{20}$$, then the sum of all the elements of the matrix $$M$$ is equal to _________.

If $$\int_0^\pi (\sin^3 x) e^{-\sin^2 x} dx = \alpha - \frac{\beta}{e} \int_0^1 \sqrt{t} \, e^t dt$$, then $$\alpha + \beta$$ is equal to _________

Let $$y = y(x)$$ be the solution of the differential equation $$dy = e^{\alpha x + y} dx$$; $$\alpha \in N$$. If $$y(\log_e 2) = \log_e 2$$ and $$y(0) = \log_e\left(\frac{1}{2}\right)$$, then the value of $$\alpha$$ is equal to _________.

Let $$\vec{a} = \hat{i} - \alpha\hat{j} + \beta\hat{k}$$, $$\vec{b} = 3\hat{i} + \beta\hat{j} - \alpha\hat{k}$$ and $$\vec{c} = -\alpha\hat{i} - 2\hat{j} + \hat{k}$$, where $$\alpha$$ and $$\beta$$ are integers. If $$\vec{a} \cdot \vec{b} = -1$$ and $$\vec{b} \cdot \vec{c} = 10$$, then $$(\vec{a} \times \vec{b}) \cdot \vec{c}$$ is equal to _________.

The distance of the point $$P(3, 4, 4)$$ from the point of intersection of the line joining the points $$Q(3, -4, -5)$$ and $$R(2, -3, 1)$$ and the plane $$2x + y + z = 7$$, is equal to _________.