NTA JEE Main 11th April 2023 Shift 2

If the line $$l_1: 3y - 2x = 3$$ is the angular bisector of the lines $$l_2: x - y + 1 = 0$$ and $$l_3: \alpha x + \beta y + 17 = 0$$, then $$\alpha^2 + \beta^2 - \alpha - \beta$$ is equal to _______

If the radius of the largest circle with centre (2, 0) inscribed in the ellipse $$x^2 + 4y^2 = 36$$ is $$r$$, then $$12r^2$$ is equal to _______

Let the tangent to the parabola $$y^2 = 12x$$ at the point $$(3, \alpha)$$ be perpendicular to the line $$2x + 2y = 3$$. Then the square of distance of the point $$(6, -4)$$ from the normal to the hyperbola $$\alpha^2x^2 - 9y^2 = 9\alpha^2$$ at its point $$(\alpha - 1, \alpha + 2)$$ is equal to _______

Let $$A = \{1, 2, 3, 4, 5\}$$ and $$B = \{1, 2, 3, 4, 5, 6\}$$. Then the number of functions $$f: A \to B$$ satisfying $$f(1) + f(2) = f(4) - 1$$ is equal to _______

If $$f: \mathbb{R} \to \mathbb{R}$$ be a continuous function satisfying $$\int_0^{\frac{\pi}{2}} f(\sin 2x) \sin x \, dx + \alpha \int_0^{\frac{\pi}{4}} f(\cos 2x) \cos x \, dx = 0$$, then the value of $$\alpha$$ is _______

If $$A$$ is the area in the first quadrant enclosed by the curve $$C: 2x^2 - y + 1 = 0$$, the tangent to $$C$$ at the point $$(1, 3)$$ and the line $$x + y = 1$$, then the value of $$60A$$ is _______

Let $$\vec{a} = \hat{i} + 2\hat{j} + 3\hat{k}$$ and $$\vec{b} = \hat{i} + \hat{j} - \hat{k}$$. If $$\vec{c}$$ is a vector such that $$\vec{a} \cdot \vec{c} = 11$$, $$\vec{b} \cdot (\vec{a} \times \vec{c}) = 27$$ and $$\vec{b} \cdot \vec{c} = -\sqrt{3}|\vec{b}|$$, then $$|\vec{a} \times \vec{c}|^2$$ is equal to _______

Let the line passing through the points $$P(2, -1, 2)$$ and $$Q(5, 3, 4)$$ meet the plane $$x - y + z = 4$$ at the point $$R$$. Then the distance of the point $$R$$ from the plane $$x + 2y + 3z + 2 = 0$$ measured parallel to the line $$\frac{x-7}{2} = \frac{y+3}{2} = \frac{z-2}{1}$$ is _______

Let the line $$L: x = \frac{1-y}{-2} = \frac{z-3}{\lambda}$$, $$\lambda \in \mathbb{R}$$ meet the plane $$P: x + 2y + 3z = 4$$ at the point $$(\alpha, \beta, \gamma)$$. If the angle between the line $$L$$ and the plane $$P$$ is $$\cos^{-1}\left(\sqrt{\frac{5}{14}}\right)$$, then $$\alpha + 2\beta + 6\gamma$$ is equal to _______

Let the probability of getting head for a biased coin be $$\frac{1}{4}$$. It is tossed repeatedly until a head appears. Let $$N$$ be the number of tosses required. If the probability that the equation $$64x^2 + 5Nx + 1 = 0$$ has no real root is $$\frac{p}{q}$$, where $$p$$ and $$q$$ are co-prime, then $$q - p$$ is equal to _______