NTA JEE Main 22nd July 2021 Shift 1

If the digits are not allowed to repeat in any number formed by using the digits 0, 2, 4, 6, 8, then the number of all numbers greater than 10,000 is equal to ___.

The sum of all the elements in the set $$\{n \in \{1, 2, \ldots, 100\} | \text{H.C.F. of } n \text{ and } 2040 \text{ is } 1\}$$ is equal to ___.

If the constant term, in binomial expansion of $$\left(2x^r + \frac{1}{x^2}\right)^{10}$$ is 180, then $$r$$ is equal to ___.

The number of elements in the set $$\{n \in \{1, 2, 3, \ldots, 100\} | (11)^n > (10)^n + (9)^n\}$$ is ___.

Consider the following frequency distribution:

If mean = $$\frac{309}{22}$$ and median = 14, then the value $$(a-b)^2$$ is equal to ___.

Let $$A = \begin{bmatrix} 0 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 1 \end{bmatrix}$$. Then the number of $$3 \times 3$$ matrices $$B$$ with entries from the set $$\{1, 2, 3, 4, 5\}$$ and satisfying $$AB = BA$$ is ___.

Let $$A = \{0, 1, 2, 3, 4, 5, 6, 7\}$$. Then the number of bijective functions $$f : A \to A$$ such that $$f(1) + f(2) = 3 - f(3)$$ is equal to ___.

Let $$f : R \to R$$ be a function defined as $$f(x) = \begin{cases} 3\left(1 - \frac{|x|}{2}\right) & \text{if } |x| \le 2 \\ 0 & \text{if } |x| > 2 \end{cases}$$
Let $$g : R \to R$$ be given by $$g(x) = f(x+2) - f(x-2)$$. If $$n$$ and $$m$$ denote the number of points in $$R$$ where $$g$$ is not continuous and not differentiable, respectively, then $$n + m$$ is equal to ___.

The area (in sq. units) of the region bounded by the curves $$x^2 + 2y - 1 = 0$$, $$y^2 + 4x - 4 = 0$$ and $$y^2 - 4x - 4 = 0$$ in the upper half plane is ___.

Let $$y = y(x)$$ be the solution of the differential equation $$\left((x+2)e^{\left(\frac{y+1}{x+2}\right)} + (y+1)\right)dx = (x+2)dy$$, $$y(1) = 1$$. If the domain of $$y = y(x)$$ is an open interval $$(\alpha, \beta)$$, then $$|\alpha + \beta|$$ is equal to ___.