NTA JEE Main 26th August 2021 Shift 1

Out of all the patients in a hospital 89% are found to be suffering from heart ailment and 98% are suffering from lungs infection. If $$K$$% of them are suffering from both ailments, then $$K$$ can not belong to the set:

If $$A = \begin{bmatrix} \frac{1}{\sqrt{5}} & \frac{2}{\sqrt{5}} \\ \frac{-2}{\sqrt{5}} & \frac{1}{\sqrt{5}} \end{bmatrix}$$, $$B = \begin{bmatrix} 1 & 0 \\ i & 1 \end{bmatrix}$$, $$i = \sqrt{-1}$$, and $$Q = A^T B A$$, then the inverse of the matrix $$AQ^{2021}A^T$$ is equal to:

Let $$\theta \in \left(0, \frac{\pi}{2}\right)$$. If the system of linear equations
$$(1 + \cos^2 \theta)x + \sin^2 \theta y + 4\sin 3\theta z = 0$$
$$\cos^2 \theta x + (1 + \sin^2 \theta)y + 4\sin 3\theta z = 0$$
$$\cos^2 \theta x + \sin^2 \theta y + (1 + 4\sin 3\theta)z = 0$$
has a non-trivial solution, then the value of $$\theta$$ is:

Let $$f(x) = \cos\left(2\tan^{-1}\sin\left(\cot^{-1}\sqrt{\frac{1-x}{x}}\right)\right)$$, $$0 \lt x \lt 1$$. Then:

The value of $$\lim_{n \to \infty} \frac{1}{n} \sum_{r=0}^{2n-1} \frac{n^2}{n^2 + 4r^2}$$ is:

The value of $$\int_{\frac{-1}{\sqrt{2}}}^{\frac{1}{\sqrt{2}}} \left(\left(\frac{x+1}{x-1}\right)^2 + \left(\frac{x-1}{x+1}\right)^2 - 2\right)^{\frac{1}{2}} dx$$ is:

Let $$y = y(x)$$ be a solution curve of the differential equation $$(y+1)\tan^2 x \, dx + \tan x \, dy + y \, dx = 0$$, $$x \in \left(0, \frac{\pi}{2}\right)$$. If $$\lim_{x \to 0^+} xy(x) = 1$$, then the value of $$y\left(\frac{\pi}{4}\right)$$ is:

Let $$\vec{a} = \hat{i} + \hat{j} + \hat{k}$$ and $$\vec{b} = \hat{j} - \hat{k}$$. If $$\vec{c}$$ is a vector such that $$\vec{a} \times \vec{c} = \vec{b}$$ and $$\vec{a} \cdot \vec{c} = 3$$, then $$\vec{a} \cdot (\vec{b} \times \vec{c})$$ is equal to:

A plane $$P$$ contains the line $$x + 2y + 3z + 1 = 0 = x - y - z - 6$$, and is perpendicular to the plane $$-2x + y + z + 8 = 0$$. Then which of the following points lies on $$P$$?

Let A and B be independent events such that P(A) = p, P(B) = 2p. The largest value of p, for which P(exactly one of A, B occurs) = $$\frac{5}{9}$$, is: