NTA JEE Main 11th April 2023 Shift 1

If $$a$$ and $$b$$ are the roots of the equation $$x^2 - 7x - 1 = 0$$, then the value of $$\frac{a^{21} + b^{21} + a^{17} + b^{17}}{a^{19} + b^{19}}$$ is equal to _______

In an examination, 5 students have been allotted their seats as per their roll numbers. The number of ways, in which none of the students sits on the allotted seat, is _______

Let $$S = 109 + \frac{108}{5} + \frac{107}{5^2} + \ldots + \frac{2}{5^{107}} + \frac{1}{5^{108}}$$. Then the value of $$16S - (25)^{-54}$$ is equal to _______

The number of integral terms in the expansion of $$\left(3^{\frac{1}{2}} + 5^{\frac{1}{4}}\right)^{680}$$ is equal to _______

The mean of the coefficients of $$x, x^2, \ldots, x^7$$ in the binomial expression of $$(2 + x)^9$$ is _______

Let $$H_n: \frac{x^2}{1+n} - \frac{y^2}{3+n} = 1$$, $$n \in \mathbb{N}$$. Let $$k$$ be the smallest even value of $$n$$ such that the eccentricity of $$H_k$$ is a rational number. If $$l$$ is the length of the latus rectum of $$H_k$$, then $$21l$$ is equal to _______

The number of ordered triplets of the truth values of $$p, q$$ and $$r$$ such that the truth value of the statement $$p \vee q \wedge p \vee r \Rightarrow q \vee r$$ is True, is equal to _______

Let $$A = \begin{pmatrix} 0 & 1 & 2 \\ a & 0 & 3 \\ 1 & c & 0 \end{pmatrix}$$, where $$a, c \in \mathbb{R}$$. If $$A^3 = A$$ and the positive value of $$a$$ belongs to the interval $$(n-1, n]$$, where $$n \in \mathbb{N}$$, then $$n$$ is equal to _______.

For $$m, n > 0$$, let $$\alpha(m, n) = \int_0^2 t^m(1 + 3t)^n dt$$. If $$11\alpha(10, 6) + 18\alpha(11, 5) = p \cdot 14^6$$, then $$p$$ is equal to _______

Let a line $$L$$ pass through the origin and be perpendicular to the lines
$$L_1: \vec{r} = (\hat{i} - 11\hat{j} - 7\hat{k}) + \lambda(\hat{i} + 2\hat{j} + 3\hat{k})$$, $$\lambda \in \mathbb{R}$$ and
$$L_2: \vec{r} = (-\hat{i} + \hat{k}) + \mu(2\hat{i} + 2\hat{j} + \hat{k})$$, $$\mu \in \mathbb{R}$$. If $$P$$ is the point of intersection of $$L$$ and $$L_1$$, and $$Q(\alpha, \beta, \gamma)$$ is the foot of perpendicular from $$P$$ on $$L_2$$, then $$9\alpha + \beta + \gamma$$ is equal to _______