NTA JEE Main 27th July 2021 Shift 2

Let $$N$$ be the set of natural numbers and a relation $$R$$ on $$N$$ be defined by $$R = \{(x, y) \in N \times N : x^3 - 3x^2y - xy^2 + 3y^3 = 0\}$$. Then the relation $$R$$ is:

Let $$A$$ and $$B$$ be two $$3 \times 3$$ real matrices such that $$(A^2 - B^2)$$ is invertible matrix. If $$A^5 = B^5$$ and $$A^3B^2 = A^2B^3$$, then the value of the determinant of the matrix $$A^3 + B^3$$ is equal to:

Let $$f : R \rightarrow R$$ be defined as $$f(x + y) + f(x - y) = 2f(x)f(y)$$, $$f\left(\frac{1}{2}\right) = -1$$. Then the value of $$\sum_{k=1}^{20} \frac{1}{\sin(k)\sin(k + f(k))}$$ is equal to:

Let $$f : [0, \infty) \rightarrow [0, 3]$$ be a function defined by $$f(x) = \begin{cases} \max\{\sin t : 0 \leq t \leq \pi\}, & x \in [0, \pi] \\ 2 + \cos x, & x > \pi \end{cases}$$. Then which of the following is true?

Let $$f : (a, b) \rightarrow R$$ be twice differentiable function such that $$f(x) = \int_a^x g(t) \, dt$$ for a differentiable function $$g(x)$$. If $$f(x) = 0$$ has exactly five distinct roots in $$(a, b)$$, then $$g(x)g'(x) = 0$$ has at least:

The area of the region bounded by $$y - x = 2$$ and $$x^2 = y$$ is equal to:

Let $$y = y(x)$$ be the solution of the differential equation $$(x - x^3)dy = (y + yx^2 - 3x^4)dx$$, $$x \gt 2$$. If $$y(3) = 3$$, then $$y(4)$$ is equal to:

Let $$\vec{a}, \vec{b}$$ and $$\vec{c}$$ be three vectors such that $$\vec{a} = \vec{b} \times (\vec{b} \times \vec{c})$$. If magnitudes of the vectors $$\vec{a}, \vec{b}$$ and $$\vec{c}$$ are $$\sqrt{2}, 1$$ and 2 respectively and the angle between $$\vec{b}$$ and $$\vec{c}$$ is $$\theta$$ $$(0 < \theta < \frac{\pi}{2})$$, then the value of $$1 + \tan \theta$$ is equal to:

For real numbers $$\alpha$$ and $$\beta \neq 0$$, if the point of intersection of the straight lines $$\frac{x - \alpha}{1} = \frac{y - 1}{2} = \frac{z - 1}{3}$$ and $$\frac{x - 4}{\beta} = \frac{y - 6}{3} = \frac{z - 7}{3}$$ lies on the plane $$x + 2y - z = 8$$, then $$\alpha - \beta$$ is equal to:

A student appeared in an examination consisting of 8 true-false type questions. The student guesses the answers with equal probability. The smallest value of $$n$$, so that the probability of guessing at least $$n$$ correct answers is less than $$\frac{1}{2}$$, is: