NTA JEE Main 22nd July 2021 Shift 1

If the domain of the function $$f(x) = \frac{\cos^{-1}\sqrt{x^2 - x + 1}}{\sqrt{\sin^{-1}\left(\frac{2x-1}{2}\right)}}$$ is the interval $$(\alpha, \beta]$$, then $$\alpha + \beta$$ is equal to:

Let $$f : R \to R$$ be defined as $$f(x) = \begin{cases} \frac{x^3}{(1-\cos 2x)^2} \log_e\left(\frac{1+2xe^{-2x}}{(1-xe^{-x})^2}\right), & x \neq 0 \\ \alpha, & x = 0 \end{cases}$$
If $$f$$ is continuous at $$x = 0$$, then $$\alpha$$ is equal to:

Let $$f : R \to R$$ be defined as $$f(x) = \begin{cases} -\frac{4}{3}x^3 + 2x^2 + 3x, & x > 0 \\ 3xe^x, & x \le 0 \end{cases}$$
Then $$f$$ is increasing function in the interval

If $$\int_0^{100\alpha} \frac{\sin^2 x}{e^{\left(\frac{x}{\pi} - \left[\frac{x}{\pi}\right]\right)}} dx = \frac{\alpha\pi^3}{1+4\pi^2}$$, $$\alpha \in R$$ where $$[x]$$ is the greatest integer less than or equal to $$x$$, then the value of $$\alpha$$ is:

Let $$y = y(x)$$ be the solution of the differential equation $$\cosec^2 x \, dy + 2dx = (1 + y\cos 2x) \cosec^2 x \, dx$$, with $$y\left(\frac{\pi}{4}\right) = 0$$. Then, the value of $$(y(0) + 1)^2$$ is equal to:

Let a vector $$\vec{a}$$ be coplanar with vectors $$\vec{b} = 2\hat{i} + \hat{j} + \hat{k}$$ and $$\vec{c} = \hat{i} - \hat{j} + \hat{k}$$. If $$\vec{a}$$ is perpendicular to $$\vec{d} = 3\hat{i} + 2\hat{j} + 6\hat{k}$$, and $$|\vec{a}| = \sqrt{10}$$. Then a possible value of $$[\vec{a} \ \vec{b} \ \vec{c}] + [\vec{a} \ \vec{b} \ \vec{d}] + [\vec{a} \ \vec{c} \ \vec{d}]$$ is equal to:

Let three vectors $$\vec{a}, \vec{b}$$ and $$\vec{c}$$ be such that $$\vec{a} \times \vec{b} = \vec{c}$$, $$\vec{b} \times \vec{c} = \vec{a}$$ and $$|\vec{a}| = 2$$. Then which one of the following is not true?

Let $$L$$ be the line of intersection of planes $$\vec{r} \cdot (\hat{i} - \hat{j} + 2\hat{k}) = 2$$ and $$\vec{r} \cdot (2\hat{i} + \hat{j} - \hat{k}) = 2$$. If $$P(\alpha, \beta, \gamma)$$ is the foot of perpendicular on $$L$$ from the point $$(1, 2, 0)$$, then the value of $$35(\alpha + \beta + \gamma)$$ is equal to:

If the shortest distance between the straight lines $$3(x-1) = 6(y-2) = 2(z-1)$$ and $$4(x-2) = 2(y-\lambda) = (z-3)$$, $$\lambda \in R$$ is $$\frac{1}{\sqrt{38}}$$, then the integral value of $$\lambda$$ is equal to:

Four dice are thrown simultaneously and the numbers shown on these dice are recorded in $$2 \times 2$$ matrices. The probability that such formed matrices have all different entries and are non-singular, is: