NTA JEE Main 31st January 2023 Shift 1

Let 5 digit numbers be constructed using the digits 0, 2, 3, 4, 7, 9 with repetition allowed, and are arranged in ascending order with serial numbers. Then the serial number of the number 42923 is ______.

Let $$a_1, a_2, \ldots, a_n$$ be in A.P. If $$a_5 = 2a_7$$ and $$a_{11} = 18$$, then $$12\left(\dfrac{1}{\sqrt{a_{10}} + \sqrt{a_{11}}} + \dfrac{1}{\sqrt{a_{11}} + \sqrt{a_{12}}} + \ldots + \dfrac{1}{\sqrt{a_{17}} + \sqrt{a_{18}}}\right)$$ is equal to ______.

Number of 4-digit numbers that are less than or equal to 2800 and either divisible by 3 or by 11, is equal to ______.

Let $$\alpha > 0$$, be the smallest number such that the expansion of $$x^{\frac{2}{3}} + \dfrac{2}{x^3}^{30}$$ has a term $$\beta x^{-\alpha}$$, $$\beta \in N$$. Then $$\alpha$$ is equal to ______.

The remainder on dividing $$5^{99}$$ by 11 is ______.

If the variance of the frequency distribution

is 3, then $$\alpha$$ is equal to ______.

Let for $$x \in R$$, $$f(x) = \dfrac{x+x}{2}$$ and $$g(x) = \begin{cases} x, & x < 0 \\ x^2, & x \ge 0 \end{cases}$$. Then area bounded by the curve $$y = f \circ g(x)$$ and the lines $$y = 0, 2y - x = 15$$ is equal to ______.

Let $$\vec{a}$$ and $$\vec{b}$$ be two vectors such that $$\vec{a} = \sqrt{14}$$, $$\vec{b} = \sqrt{6}$$ and $$\vec{a} \times \vec{b} = \sqrt{48}$$. Then $$(\vec{a} \cdot \vec{b})^2$$ is equal to ______.

Let the line $$L$$: $$\dfrac{x-1}{2} = \dfrac{y+1}{-1} = \dfrac{z-3}{1}$$ intersect the plane $$2x + y + 3z = 16$$ at the point $$P$$. Let the point $$Q$$ be the foot of perpendicular from the point $$R(1, -1, -3)$$ on the line $$L$$. If $$\alpha$$ is the area of triangle $$PQR$$, then $$\alpha^2$$ is equal to ______.

Let $$\theta$$ be the angle between the planes $$P_1 = \vec{r} \cdot (\hat{i} + \hat{j} + 2\hat{k}) = 9$$ and $$P_2 = \vec{r} \cdot (2\hat{i} - \hat{j} + \hat{k}) = 15$$. Let L be the line that meets $$P_2$$ at the point (4, -2, 5) and makes angle $$\theta$$ with the normal of $$P_2$$. If $$\alpha$$ is the angle between L and $$P_2$$ then $$\tan^2\theta \cot^2\alpha$$ is equal to ______.