NTA JEE Main 26th June 2022 Shift 1

The sum of the cubes of all the roots of the equation $$x^4 - 3x^3 - 2x^2 + 3x + 1 = 0$$ is ______

There are ten boys $$B_1, B_2, \ldots, B_{10}$$ and five girls $$G_1, G_2, \ldots G_5$$ in a class. Then the number of ways of forming a group consisting of three boys and three girls, if both $$B_1$$ and $$B_2$$ together should not be the members of a group, is ______

Let $$A = \sum_{i=1}^{10}\sum_{j=1}^{10} \min\{i, j\}$$ and $$B = \sum_{i=1}^{10}\sum_{j=1}^{10} \max\{i, j\}$$. Then $$A + B$$ is equal to ______

If $$\sin^2(10°)\sin(20°)\sin(40°)\sin(50°)\sin(70°) = \alpha - \frac{1}{16}\sin(10°)$$, then $$16 + \alpha^{-1}$$ is equal to ______

Let the common tangents to the curves $$4(x^2 + y^2) = 9$$ and $$y^2 = 4x$$ intersect at the point $$Q$$. Let an ellipse, centered at the origin $$O$$, has lengths of semi-minor and semi-major axes equal to $$OQ$$ and $$6$$, respectively. If $$e$$ and $$l$$ respectively denote the eccentricity and the length of the latus rectum of this ellipse, then $$\frac{l}{e^2}$$ is equal to ______

Let $$A = \{n \in N : H.C.F.(n, 45) = 1\}$$ and let $$B = \{2k : k \in \{1, 2, \ldots, 100\}\}$$. Then the sum of all the elements of $$A \cap B$$ is ______

Let $$f(x) = \max\{|x+1|, |x+2|, \ldots, |x+5|\}$$. Then $$\int_{-6}^{0} f(x)dx$$ is equal to ______

The value of the integral $$\frac{48}{\pi^4}\int_0^{\pi}\left(\frac{3\pi x^2}{2} - x^3\right)\frac{\sin x}{1 + \cos^2 x}dx$$ is equal to ______

Let the solution curve $$y = y(x)$$ of the differential equation $$(4 + x^2)dy - 2x(x^2 + 3y + 4)dx = 0$$ pass through the origin. Then $$y(2)$$ is equal to ______

Let $$S = (0, 2\pi) - \left\{\frac{\pi}{2}, \frac{3\pi}{4}, \frac{3\pi}{2}, \frac{7\pi}{4}\right\}$$. Let $$y = y(x), x \in S$$, be the solution curve of the differential equation $$\frac{dy}{dx} = \frac{1}{1 + \sin 2x}, y\left(\frac{\pi}{4}\right) = \frac{1}{2}$$. If the sum of abscissas of all the points of intersection of the curve $$y = y(x)$$ with the curve $$y = \sqrt{2}\sin x$$ is $$\frac{k\pi}{12}$$, then $$k$$ is equal to ______