NTA JEE Main 8th April2023 Shift 1

The largest natural number $$n$$ such that $$3n$$ divides 66! is ______.

Let $$[t]$$ denote the greatest integer $$\le t$$. If the constant term in the expansion of $$\left(3x^2 - \dfrac{1}{2x^5}\right)^7$$ is $$\alpha$$ then $$[\alpha]$$ is equal to ______.

Consider a circle $$C_1: x^2 + y^2 - 4x - 2y = \alpha - 5$$. Let its mirror image in the line $$y = 2x + 1$$ be another circle $$C_2: 5x^2 + 5y^2 - 10fx - 10gy + 36 = 0$$. Let $$r$$ be the radius of $$C_2$$. Then $$\alpha + r$$ is equal to ______.

Let the mean and variance of 8 numbers x, y, 10, 12, 6, 12, 4, 8 be 9 and 9.25 respectively. If $$x > y$$, then $$3x - 2y$$ is equal to ______.

Let $$A = \{0, 3, 4, 6, 7, 8, 9, 10\}$$ and $$R$$ be the relation defined on $$A$$ such that $$R\{(x,y) \in A \times A: x-y$$ is odd positive integer or $$x-y = 2\}$$. The minimum number of elements that must be added to the relation $$R$$, so that it is a symmetric relation, is equal to ______.

If $$a_\alpha$$ is the greatest term in the sequence $$a_n = \dfrac{n^3}{n^4 + 147}$$, $$n = 1, 2, 3, \ldots$$, then $$\alpha$$ is equal to ______.

Let $$[t]$$ denote the greatest integer $$\le t$$. Then $$\dfrac{2}{\pi} \int_{\pi/6}^{5\pi/6} (8[\csc x] - 5[\cot x]) dx$$ is equal to ______.

If the solution curve of the differential equation $$(y-2\log_e x)dx + (x\log_e x^2)dy = 0$$, $$x \gt 1$$ passes through the points $$(e, \dfrac{4}{3})$$ and $$(e^4, \alpha)$$, then $$\alpha$$ is equal to ______.

Let $$\vec{a} = 6\hat{i} + 9\hat{j} + 12\hat{k}$$, $$\vec{b} = \alpha\hat{i} + 11\hat{j} - 2\hat{k}$$ and $$\vec{c}$$ be vectors such that $$\vec{a} \times \vec{c} = \vec{a} \times \vec{b}$$. If $$\vec{a} \cdot \vec{c} = -12$$, and $$\vec{c} \cdot (\hat{i} - 2\hat{j} + \hat{k}) = 5$$ then $$\vec{c} \cdot (\hat{i} + \hat{j} + \hat{k})$$ is equal to ______.

Let $$\lambda_1, \lambda_2$$ be the values of $$\lambda$$ for which the points $$\left(\dfrac{5}{2}, 1, \lambda\right)$$ and $$(-2, 0, 1)$$ are at equal distance from the plane $$2x + 3y - 6z + 7$$. If $$\lambda_1 > \lambda_2$$ then the distance of the point $$(\lambda_1 - \lambda_2, \lambda_2, \lambda_1)$$ from the line $$\dfrac{x-5}{1} = \dfrac{y-1}{2} = \dfrac{z+7}{2}$$ is ______.