NTA JEE Main 31st January 2023 Shift 2

Among the relations
$$S = \{(a,b): a, b \in R - \{0\}, 2 + \dfrac{a}{b} > 0\}$$ and $$T = \{(a,b): a, b \in R, a^2 - b^2 \in Z\}$$,

If a point $$P(\alpha, \beta, \gamma)$$ satisfying $$(\alpha \; \beta \; \gamma) \begin{pmatrix} 2 & 10 & 8 \\ 9 & 3 & 8 \\ 8 & 4 & 8 \end{pmatrix} = (0 \; 0 \; 0)$$ lies on the plane $$2x + 4y + 3z = 5$$, then $$6\alpha + 9\beta + 7\gamma$$ is equal to

Let $$(a, b) \subset (0, 2\pi)$$ be the largest interval for which $$\sin^{-1}(\sin\theta) - \cos^{-1}(\sin\theta) > 0$$, $$\theta \in (0, 2\pi)$$, holds. If $$\alpha x^2 + \beta x + \sin^{-1}(x^2 - 6x + 10) + \cos^{-1}(x^2 - 6x + 10) = 0$$ and $$\alpha - \beta = b - a$$, then $$\alpha$$ is equal to:

Let $$f: R - \{2, 6\} \to R$$ be real valued function defined as $$f(x) = \dfrac{x^2+2x+1}{x^2-8x+12}$$. Then range of $$f$$ is

The absolute minimum value, of the function $$f(x) = |x^2 - x + 1| + [x^2 - x + 1]$$, where $$[t]$$ denotes the greatest integer function, in the interval $$[-1, 2]$$, is

Let $$y = y(x)$$ be the solution of the differential equation $$(3y^2 - 5x^2)y\,dx + 2x(x^2 - y^2)\,dy = 0$$ such that $$y(1) = 1$$. Then $$|(y(2))^3 - 12y(2)|$$ is equal to:

Let $$\vec{a} = \hat{i} + 2\hat{j} + 3\hat{k}$$, $$\vec{b} = \hat{i} - \hat{j} + 2\hat{k}$$ and $$\vec{c} = 5\hat{i} - 3\hat{j} + 3\hat{k}$$, be three vectors. If $$\vec{r}$$ is a vector such that, $$\vec{r} \times \vec{b} = \vec{c} \times \vec{b}$$ and $$\vec{r} \cdot \vec{a} = 0$$, then $$25|\vec{r}|^2$$ is equal to

Let the plane $$P: 8x + \alpha_1 y + \alpha_2 z + 12 = 0$$ be parallel to the line $$L: \dfrac{x+2}{2} = \dfrac{y-3}{3} = \dfrac{z+4}{5}$$. If the intercept of $$P$$ on the y-axis is 1, then the distance between $$P$$ and $$L$$ is

Let $$P$$ be the plane, passing through the point $$(1, -1, -5)$$ and perpendicular to the line joining the points $$(4, 1, -3)$$ and $$(2, 4, 3)$$. Then the distance of $$P$$ from the point $$(3, -2, 2)$$ is

The foot of perpendicular from the origin $$O$$ to a plane $$P$$ which meets the co-ordinate axes at the point $$A, B, C$$ is $$(2, a, 4)$$, $$a \in N$$. If the volume of the tetrahedron $$OABC$$ is 144 unit$$^3$$, then which of the following points is NOT on $$P$$?