NTA JEE Mains 6th April 2024 Shift 1

For $$\alpha, \beta \in \mathbb{R}$$ and a natural number $$n$$, let $$A_r = \begin{vmatrix} r & 1 & \frac{n^2}{2} + \alpha \\ 2r & 2 & n^2 - \beta \\ 3r - 2 & 3 & \frac{n(3n-1)}{2} \end{vmatrix}$$. Then $$\sum_{r=1}^{n} A_r$$ is

The function $$f: \mathbb{R} \rightarrow \mathbb{R}$$, $$f(x) = \frac{x^2 + 2x - 15}{x^2 - 4x + 9}$$, $$x \in \mathbb{R}$$ is

If $$f(x) = \begin{cases} x^3 \sin\left(\frac{1}{x}\right), & x \neq 0 \\ 0, & x = 0 \end{cases}$$ then

The interval in which the function $$f(x) = x^x, x > 0$$, is strictly increasing is

$$\int_0^{\pi/4} \frac{\cos^2 x \sin^2 x}{(\cos^3 x + \sin^3 x)^2} dx$$ is equal to

Let the area of the region enclosed by the curves $$y = 3x$$, $$2y = 27 - 3x$$ and $$y = 3x - x\sqrt{x}$$ be $$A$$. Then $$10A$$ is equal to

Let $$y = y(x)$$ be the solution of the differential equation $$(1 + x^2)\frac{dy}{dx} + y = e^{\tan^{-1}x}$$, $$y(1) = 0$$. Then $$y(0)$$ is

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

The shortest distance between the lines $$\frac{x-3}{2} = \frac{y+15}{-7} = \frac{z-9}{5}$$ and $$\frac{x+1}{2} = \frac{y-1}{1} = \frac{z-9}{-3}$$ is

A company has two plants $$A$$ and $$B$$ to manufacture motorcycles. 60% motorcycles are manufactured at plant $$A$$ and the remaining are manufactured at plant $$B$$. 80% of the motorcycles manufactured at plant $$A$$ are rated of the standard quality, while 90% of the motorcycles manufactured at plant $$B$$ are rated of the standard quality. A motorcycle picked up randomly from the total production is found to be of the standard quality. If $$p$$ is the probability that it was manufactured at plant $$B$$, then $$126p$$ is