NTA JEE Main 13th April 2023 Shift 1

Let $$w = z\bar{z} + k_1z + k_2iz + \lambda(1+i)$$, $$k_1, k_2 \in \mathbb{R}$$. Let $$Re(w) = 0$$ be the circle $$C$$ of radius 1 in the first quadrant touching the line $$y = 1$$ and the $$y$$-axis. If the curve $$Im(w) = 0$$ intersects $$C$$ at $$A$$ and $$B$$, then $$30(AB)^2$$ is equal to _____.

The number of seven digit positive integers formed using the digits 1, 2, 3 and 4 only and sum of the digits equal to 12 is _____.

The sum to 20 terms of the series $$2 \cdot 2^2 - 3^2 + 2 \cdot 4^2 - 5^2 + 2 \cdot 6^2 - \ldots$$ is equal to _____.

Let $$\alpha$$ be the constant term in the binomial expansion of $$\left(\sqrt{x} - \frac{6}{x^{3/2}}\right)^n$$, $$n \leq 15$$. If the sum of the coefficients of the remaining terms in the expansion is $$649$$ and the coefficient of $$x^{-n}$$ is $$\lambda\alpha$$, then $$\lambda$$ is equal to _____.

Let $$m_1$$ and $$m_2$$ be the slopes of the tangents drawn from the point $$P(4, 1)$$ to the hyperbola $$H : \frac{y^2}{25} - \frac{x^2}{16} = 1$$. If $$Q$$ is the point from which the tangents drawn to $$H$$ have slopes $$|m_1|$$ and $$|m_2|$$ and they make positive intercepts $$\alpha$$ and $$\beta$$ on the $$x-$$axis, then $$\frac{(PQ)^2}{\alpha\beta}$$ is equal to _____.

Let the mean of the data

If $$S = \left\{x \in \mathbb{R} : \sin^{-1}\left(\frac{x+1}{\sqrt{x^2+2x+2}}\right) - \sin^{-1}\left(\frac{x}{\sqrt{x^2+1}}\right) = \frac{\pi}{4}\right\}$$ then $$\sum_{x \in S}\left(\sin\left((x^2+x+5)\frac{\pi}{2}\right) - \cos\left((x^2+x+5)\pi\right)\right)$$ is equal to _____.

Let for $$x \in \mathbb{R}$$, $$S_0(x) = x$$, $$S_k(x) = C_k x + k\int_0^x S_{k-1}(t)dt$$ where $$C_0 = 1$$, $$C_k = 1 - \int_0^1 S_{k-1}(x)dx$$, $$k = 1, 2, 3, \ldots$$ Then $$S_2(3) + 6C_3$$ is equal to _____.

Let $$\vec{a} = 3\hat{i} + \hat{j} - \hat{k}$$ and $$\vec{c} = 2\hat{i} - 3\hat{j} + 3\hat{k}$$. If $$\vec{b}$$ is a vector such that $$\vec{a} = \vec{b} \times \vec{c}$$ and $$|\vec{b}|^2 = 50$$, then $$\left|72 - |\vec{b} + \vec{c}|^2\right|$$ is equal to _____.

Let the image of the point $$\left(\frac{5}{3}, \frac{5}{3}, \frac{8}{3}\right)$$ in the plane $$x - 2y + z - 2 = 0$$ be $$P$$. If the distance of the point $$Q(6, -2, \alpha)$$, $$\alpha > 0$$, from $$P$$ is $$13$$, then $$\alpha$$ is equal to _____.