NTA JEE Main 25th January 2023 Shift 1

Let $$y(x) = (1+x)(1+x^2)(1+x^4)(1+x^8)(1+x^{16})$$. Then $$y' - y''$$ at $$x = -1$$ is equal to

Let $$x = 2$$ be a local minima of the function $$f(x) = 2x^4 - 18x^2 + 8x + 12$$, $$x \in (-4, 4)$$. If $$M$$ is local maximum value of the function $$f$$ in $$(-4, 4)$$, then $$M =$$

The minimum value of the function $$f(x) = \int_0^2 e^{|x-t|} dt$$ is

Let $$y = y(x)$$ be the solution curve of the differential equation $$\frac{dy}{dx} = \frac{y}{x}(1 - xy^2(1 + \log_e x))$$, $$x \gt 0$$, $$y(1) = 3$$. Then $$\frac{y^2(x)}{9}$$ is equal to:

The distance of the point $$P(4, 6, -2)$$ from the line passing through the point $$(-3, 2, 3)$$ and parallel to a line with direction ratios $$3, 3, -1$$ is equal to:

Let M be the maximum value of the product of two positive integers when their sum is 66. Let the sample space $$S = \{x \in \mathbb{Z} : x(66-x) \geq \frac{5}{9}M\}$$ and the event $$A = \{x \in S : x$$ is a multiple of 3$$\}$$. Then P(A) is equal to

Let $$S = \{\alpha : \log_2(9^{2\alpha-4} + 13) - \log_2(\frac{5}{2} \cdot 3^{2\alpha-4} + 1) = 2\}$$. Then the maximum value of $$\beta$$ for which the equation $$x^2 - 2(\sum_{\alpha \in s} \alpha)^2 x + \sum_{\alpha \in s}(\alpha+1)^2\beta = 0$$ has real roots, is _____.

Let $$x$$ and $$y$$ be distinct integers where $$1 \leq x \leq 25$$ and $$1 \leq y \leq 25$$. Then, the number of ways of choosing $$x$$ and $$y$$, such that $$x + y$$ is divisible by 5, is _____.

Let $$S = \{1, 2, 3, 5, 7, 10, 11\}$$. The number of non-empty subsets of $$S$$ that have the sum of all elements a multiple of 3, is _____.

Let $$A_1, A_2, A_3$$ be the three A.P. with the same common difference $$d$$ and having their first terms as $$A, A+1, A+2$$, respectively. Let $$a, b, c$$ be the 7$$^{th}$$, 9$$^{th}$$, 17$$^{th}$$ terms of $$A_1$$, $$A_2$$, $$A_3$$, respectively such that $$\begin{vmatrix} a & 7 & 1 \\ 2b & 17 & 1 \\ c & 17 & 1 \end{vmatrix} + 70 = 0$$. If $$a = 29$$, then the sum of first 20 terms of an AP whose first term is $$c - a - b$$ and common difference is $$\frac{d}{12}$$, is equal to _____.