NTA JEE Main 10th April 2023 Shift 1

Let $$a$$, $$b$$, $$c$$ be the three distinct positive real numbers such that $$2a^{\log_e a} = bc^{\log_e b}$$ and $$b^{\log_e 2} = a^{\log_e c}$$. Then $$6a + 5bc$$ is equal to _______.

The number of permutations, of the digits 1, 2, 3, ..., 7 without repetition, which neither contain the string 153 nor the string 2467, is _______.

Some couples participated in a mixed doubles badminton tournament. If the number of matches played, so that no couple played in a match, is 840, then the total numbers of persons, who participated in the tournament, is _______.

The sum of all those terms, of the arithmetic progression 3, 8, 13, ..., 373, which are not divisible by 3, is equal to _______.

The coefficient of $$x^7$$ in $$(1 - x + 2x^3)^{10}$$ is _______.

Let a common tangent to the curves $$y^2 = 4x$$ and $$x - 4^2 + y^2 = 16$$ touch the curves at the points $$P$$ and $$Q$$. Then $$PQ^2$$ is equal to _______.

If the mean of the frequency distribution

The number of elements in the set $$\{n \in \mathbb{Z}: |n^2 - 10n + 19| < 6\}$$ is _______.

Let $$f: [-2, 2] \to \mathbb{R}$$ be defined by $$f(x) = \begin{cases} x[x], & -2 < x < 0 \\ (x - 1)[x], & 0 \leq x \leq 2 \end{cases}$$ where $$[x]$$ denotes the greatest integer function. If $$m$$ and $$n$$ respectively are the number of points in $$(-2, 2)$$ at which $$y = |f(x)|$$ is not continuous and not differentiable, then $$m + n$$ is equal to _______.

Let $$y = px$$ be the parabola passing through the points $$(-1, 0)$$, $$(0, 0)$$, $$(1, 0)$$ and $$(1, 0)$$. If the area of the region $$\{(x, y): (x+1)^2 + (y-1)^2 \leq 1, y \leq px\}$$ is $$A$$, then $$12\pi - 4A$$ is equal to _______.