NTA JEE Main 25th July 2021 Shift 1

If $$\alpha, \beta$$ are roots of the equation $$x^2 + 5\sqrt{2}x + 10 = 0$$, $$\alpha > \beta$$ and $$P_n = \alpha^n - \beta^n$$ for each positive integer $$n$$, then the value of $$\frac{P_{17}P_{20} + 5\sqrt{2}P_{17}P_{19}}{P_{18}P_{19} + 5\sqrt{2}P_{18}^2}$$ is equal to ___.

There are 5 students in class 10, 6 students in class 11 and 8 students in class 12. If the number of ways, in which 10 students can be selected from them so as to include at least 2 students from each class and at most 5 students from the total 11 students of classes 10 and 11 is 100k, then k is equal to ___.

If the value of $$\left(1 + \frac{2}{3} + \frac{6}{3^2} + \frac{10}{3^3} + \ldots \text{ upto } \infty\right)^{ \log_{(0.25)}\left(\frac{1}{3} + \frac{1}{3^2} + \frac{1}{3^3} + \ldots \text{ upto } \infty\right)}$$ is $$l$$, then $$l^2$$ is equal to ___.

The ratio of the coefficient of the middle term in the expansion of $$(1+x)^{20}$$ and the sum of the coefficients of two middle terms in expansion of $$(1+x)^{19}$$ is ___.

The term independent of $$x$$ in the expansion of $$\left(\frac{x+1}{x^{2/3} - x^{1/3} + 1} - \frac{x-1}{x - x^{1/2}}\right)^{10}$$, where $$x \neq 0, 1$$ is equal to ___.

Consider the following frequency distribution:

If the sum of all frequencies is 584 and median is 45, then $$|\alpha - \beta|$$ is equal to ___.

Let $$M = A = \begin{pmatrix} a & b \\ c & d \end{pmatrix} : a, b, c, d \in \{\pm 3, \pm 2, \pm 1, 0\}$$. Define $$f: M \to Z$$, as $$f(A) = \det A$$, for all $$A \in M$$ where $$Z$$ is set of all integers. Then the number of $$A \in M$$ such that $$f(A) = 15$$ is equal to ___.

Let $$S = \{n \in N, \begin{pmatrix} 0 & i \\ 1 & 0 \end{pmatrix}^n \begin{pmatrix} a & b \\ c & d \end{pmatrix}= \begin{pmatrix} a & b \\ c & d \end{pmatrix}$$ $$\forall a, b, c, d \in R$$, where $$i = \sqrt{-1}\}$$. Then the number of 2-digit numbers in the set $$S$$ is ___.

Let $$y = y(x)$$ be solution of the following differential equation
$$e^y \frac{dy}{dx} - 2e^y \sin x + \sin x \cos^2 x = 0$$, $$y\left(\frac{\pi}{2}\right) = 0$$.
If $$y(0) = \log_e \alpha + \beta e^{-2}$$, then $$4(\alpha + \beta)$$ is equal to ___.

Let $$\vec{p} = 2\hat{i} + 3\hat{j} + \hat{k}$$ and $$\vec{q} = \hat{i} + 2\hat{j} + \hat{k}$$ be two vectors. If a vector $$\vec{r} = \alpha\hat{i} + \beta\hat{j} + \gamma\hat{k}$$ is perpendicular to each of the vectors $$(\vec{p} + \vec{q})$$ and $$(\vec{p} - \vec{q})$$, and $$|\vec{r}| = \sqrt{3}$$, then $$|\alpha| + |\beta| + |\gamma|$$ is equal to ___.