NTA JEE Main 25th July 2022 Shift 2

If the mean deviation about median for the numbers $$3, 5, 7, 2k, 12, 16, 21, 24$$ arranged in the ascending order, is $$6$$ then the median is

The number of real values of $$\lambda$$, such that the system of linear equations
$$2x - 3y + 5z = 9$$
$$x + 3y - z = -18$$
$$3x - y + (\lambda^2 - |\lambda|)z = 16$$
has no solutions, is

The number of bijective functions $$f(\{1, 3, 5, 7, \ldots, 99\}) \to \{2, 4, 6, 8, \ldots, 100\}$$ if $$f(3) > f(5) > f(7) \ldots > f(99)$$ is

$$\displaystyle\lim_{n \to \infty} \dfrac{1}{2n}\left(\dfrac{1}{\sqrt{1 - \frac{1}{2n}}} + \dfrac{1}{\sqrt{1 - \frac{2}{2n}}} + \dfrac{1}{\sqrt{1 - \frac{3}{2n}}} + \ldots + \dfrac{1}{\sqrt{1 - \frac{2n-1}{2n}}}\right)$$ is equal to

Let $$[t]$$ denote the greatest integer less than or equal to $$t$$. Then the value of the integral $$\displaystyle\int_{-3}^{101} ([\sin(\pi x)] + e^{[\cos(2\pi x)]}) dx$$ is equal to

Let a smooth curve $$y = f(x)$$ be such that the slope of the tangent at any point $$(x, y)$$ on it is directly proportional to $$\left(\dfrac{-y}{x}\right)$$. If the curve passes through the points $$(1, 2)$$ and $$(8, 1)$$, then $$\left|y\left(\dfrac{1}{8}\right)\right|$$ is equal to

Let $$\vec{a} = \hat{i} - \hat{j} + 2\hat{k}$$ and let $$\vec{b}$$ be a vector such that $$\vec{a} \times \vec{b} = 2\hat{i} - \hat{k}$$ and $$\vec{a} \cdot \vec{b} = 3$$. Then the projection of $$\vec{b}$$ on the vector $$\vec{a} - \vec{b}$$ is:

A plane $$E$$ is perpendicular to the two planes $$2x - 2y + z = 0$$ and $$x - y + 2z = 4$$, and passes through the point $$P(1, -1, 1)$$. If the distance of the plane $$E$$ from the point $$Q(a, a, 2)$$ is $$3\sqrt{2}$$, then $$(PQ)^2$$ is equal to

The shortest distance between the lines $$\dfrac{x+7}{-6} = \dfrac{y-6}{7} = z$$ and $$\dfrac{7-x}{2} = y - 2 = z - 6$$ is

If $$A$$ and $$B$$ are two events such that $$P(A) = \dfrac{1}{3}$$, $$P(B) = \dfrac{1}{5}$$ and $$P(A \cup B) = \dfrac{1}{2}$$, then $$P\left(\dfrac{A}{B'}\right) + P\left(\dfrac{B}{A'}\right)$$ is equal to