NTA JEE Main 11th April 2023 Shift 2

Let $$A = \{1, 3, 4, 6, 9\}$$ and $$B = \{2, 4, 5, 8, 10\}$$. Let $$R$$ be a relation defined on $$A \times B$$ such that $$R = \{(a_1, b_1), (a_2, b_2): a_1 \leq b_2 \text{ and } b_1 \leq a_2\}$$. Then the number of elements in the set $$R$$ is

If the system of linear equations
$$7x + 11y + \alpha z = 13$$
$$5x + 4y + 7z = \beta$$
$$175x + 194y + 57z = 361$$
has infinitely many solutions, then $$\alpha + \beta + 2$$ is equal to

If $$\begin{vmatrix} x+1 & x & x \\ x & x+\lambda & x \\ x & x & x+\lambda^2 \end{vmatrix} = \frac{9}{8}(103x + 81)$$, then $$\lambda$$, $$\frac{\lambda}{3}$$ are the roots of the equation

The domain of the function $$f(x) = \frac{1}{\sqrt{[x]^2 - 3[x] - 10}}$$ is (where $$[x]$$ denotes the greatest integer less than or equal to $$x$$)

Let $$f$$ and $$g$$ be two functions defined by $$f(x) = \begin{cases} x + 1, & x < 0 \\ |x - 1|, & x \geq 0 \end{cases}$$ and $$g(x) = \begin{cases} x + 1, & x < 0 \\ 1, & x \geq 0 \end{cases}$$. Then $$(g \circ f)(x)$$ is

Let the function $$f: [0, 2] \to \mathbb{R}$$ be defined as $$f(x) = \begin{cases} e^{\min\{x^2, x-[x]\}}, & x \in [0, 1) \\ e^{[x - \log_e x]}, & x \in [1, 2] \end{cases}$$, where $$[t]$$ denotes the greatest integer less than or equal to $$t$$. Then the value of the integral $$\int_0^2 xf(x)dx$$ is

Let $$y = y(x)$$ be the solution of the differential equation $$\frac{dy}{dx} + \frac{5}{x(x^5+1)}y = \frac{(x^5+1)^2}{x^7}$$, $$x > 0$$. If $$y(1) = 2$$, then $$y(2)$$ is equal to

If four distinct points with position vectors $$\vec{a}, \vec{b}, \vec{c}$$ and $$\vec{d}$$ are coplanar, then $$[\vec{a}\vec{b}\vec{c}]$$ is equal to

Let $$P$$ be the plane passing through the points $$(5, 3, 0)$$, $$(13, 3, -2)$$ and $$(1, 6, 2)$$. For $$\alpha \in \mathbb{N}$$, if the distance of the points $$A(3, 4, \alpha)$$ and $$B(2, \alpha, a)$$ from the plane $$P$$ are 2 and 3 respectively, then the positive value of a is

Let $$S = \{z \in \mathbb{C} - \{i, 2i\}: \frac{z^2 + 8iz - 15}{z^2 - 3iz - 2} \in \mathbb{R}\}$$. $$\alpha - \frac{13}{11}i \in S$$, $$\alpha \in \mathbb{R} - \{0\}$$, then $$242\alpha^2$$ is equal to _______