NTA JEE Mains 8th April 2024 Shift 2

The number of distinct real roots of the equation $$|x + 1||x + 3| - 4|x + 2| + 5 = 0$$, is _____

An arithmetic progression is written in the following way:

The sum of all the terms of the $$10^{th}$$ row is _____

Let a ray of light passing through the point $$(3, 10)$$ reflects on the line $$2x + y = 6$$ and the reflected ray passes through the point $$(7, 2)$$. If the equation of the incident ray is $$ax + by + 1 = 0$$, then $$a^2 + b^2 + 3ab$$ is equal to _____

Let S be the focus of the hyperbola $$\frac{x^2}{3} - \frac{y^2}{5} = 1$$, on the positive x-axis. Let C be the circle with its centre at $$A(\sqrt{6}, \sqrt{5})$$ and passing through the point S. If O is the origin and SAB is a diameter of C, then the square of the area of the triangle OSB is equal to _____

If $$\alpha = \lim_{x \to 0^+} \left(\frac{e^{\sqrt{\tan x}} - e^{\sqrt{x}}}{\sqrt{\tan x} - \sqrt{x}}\right)$$ and $$\beta = \lim_{x \to 0} (1 + \sin x)^{\frac{1}{2}\cot x}$$ are the roots of the quadratic equation $$ax^2 + bx - \sqrt{e} = 0$$, then $$12\log_e(a + b)$$ is equal to _____

Let $$a, b, c \in \mathbb{N}$$ and $$a < b < c$$. Let the mean, the mean deviation about the mean and the variance of the 5 observations $$9, 25, a, b, c$$ be $$18, 4$$ and $$\frac{136}{5}$$, respectively. Then $$2a + b - c$$ is equal to _____

Let A be the region enclosed by the parabola $$y^2 = 2x$$ and the line $$x = 24$$. Then the maximum area of the rectangle inscribed in the region A is _____

If $$\int \frac{1}{\sqrt[5]{(x-1)^4(x+3)^6}} dx = A\left(\frac{\alpha x - 1}{\beta x + 3}\right)^B + C$$, where C is the constant of integration, then the value of $$\alpha + \beta + 20AB$$ is _____

Let $$\alpha|x| = |y|e^{xy - \beta}$$, $$\alpha, \beta \in \mathbb{N}$$ be the solution of the differential equation $$x\,dy - y\,dx + xy(x\,dy + y\,dx) = 0$$, $$y(1) = 2$$. Then $$\alpha + \beta$$ is equal to _____

Let $$P(\alpha, \beta, \gamma)$$ be the image of the point $$Q(1, 6, 4)$$ in the line $$\frac{x}{1} = \frac{y - 1}{2} = \frac{z - 2}{3}$$. Then $$2\alpha + \beta + \gamma$$ is equal to _____