NTA JEE Main 20th July 2021 Shift 1

Let $$[x]$$ denote the greatest integer $$\le x$$, where $$x \in R$$. If the domain of the real valued function $$f(x) = \sqrt{\frac{|x|-2}{|x|-3}}$$ is $$(-\infty, a) \cup [b, c) \cup [4, \infty)$$, $$a < b < c$$, then the value of $$a + b + c$$ is:

Let a function $$f : R \to R$$ be defined as,
$$$f(x) = \begin{cases} \sin x - e^x & \text{if } x \le 0 \\ a + [-x] & \text{if } 0 < x < 1 \\ 2x - b & \text{if } x \ge 1 \end{cases}$$$
Where $$[x]$$ is the greatest integer less than or equal to $$x$$. If $$f$$ is continuous on $$R$$, then $$(a + b)$$ is equal to:

Let $$A = [a_{ij}]$$ be a $$3 \times 3$$ matrix, where $$a_{ij} = \begin{cases} 1, & \text{if } i = j \\ -x, & \text{if } |i-j| = 1 \\ 2x+1, & \text{otherwise} \end{cases}$$
Let a function $$f : R \to R$$ be defined as $$f(x) = \det(A)$$. Then the sum of maximum and minimum values of $$f$$ on $$R$$ is equal to:

Let $$a$$ be a real number such that the function $$f(x) = ax^2 + 6x - 15$$, $$x \in R$$ is increasing in $$\left(-\infty, \frac{3}{4}\right)$$ and decreasing in $$\left(\frac{3}{4}, \infty\right)$$. Then the function $$g(x) = ax^2 - 6x + 15$$, $$x \in R$$ has a

Let $$a$$ be a positive real number such that $$\int_0^a e^{x-[x]} dx = 10e - 9$$ where $$[x]$$ is the greatest integer less than or equal to $$x$$. Then, $$a$$ is equal to:

The value of the integral $$\int_{-1}^{1} \log_e\left(\sqrt{1-x} + \sqrt{1+x}\right)dx$$ is equal to:

Let $$y = y(x)$$ be the solution of the differential equation $$x \tan\left(\frac{y}{x}\right) dy = \left(y \tan\left(\frac{y}{x}\right) - x\right) dx$$, $$-1 \le x \le 1$$, $$y\left(\frac{1}{2}\right) = \frac{\pi}{6}$$. Then the area of the region bounded by the curves $$x = 0$$, $$x = \frac{1}{\sqrt{2}}$$ and $$y = y(x)$$ in the upper half plane is:

Let $$y = y(x)$$ be the solution of the differential equation $$e^x\sqrt{1-y^2}dx + \left(\frac{y}{x}\right)dy = 0$$, $$y(1) = -1$$. Then the value of $$(y(3))^2$$ is equal to:

Let $$\vec{a} = 2\hat{i} + \hat{j} - 2\hat{k}$$ and $$\vec{b} = \hat{i} + \hat{j}$$. If $$\vec{c}$$ is a vector such that $$\vec{a} \cdot \vec{c} = |\vec{c}|$$, $$|\vec{c} - \vec{a}| = 2\sqrt{2}$$ and the angle between $$(\vec{a} \times \vec{b})$$ and $$\vec{c}$$ is $$\frac{\pi}{6}$$, then the value of $$|(\vec{a} \times \vec{b}) \times \vec{c}|$$ is:

Words with or without meaning are to be formed using all the letters of the word EXAMINATION. The probability that the letter M appears at the fourth position in any such word is: