NTA JEE Main 25th June 2022 Shift 1

For a natural number $$n$$, let $$\alpha_n = 19^n - 12^n$$. Then, the value of $$\frac{31\alpha_9 - \alpha_{10}}{57\alpha_8}$$ is ______.

The number of 3-digit odd numbers, whose sum of digits is a multiple of $$7$$, is ______.

The greatest integer less than or equal to the sum of first $$100$$ terms of the sequence $$\frac{1}{3}, \frac{5}{9}, \frac{19}{27}, \frac{65}{81}, \ldots$$ is equal to ______.

Let $$C_r$$ denote the binomial coefficient of $$x^r$$ in the expansion of $$(1 + x)^{10}$$. If for $$\alpha, \beta \in R$$,
$$C_{1} + 3 \cdot 2C_{2} + 5 \cdot 3C_{3} + \ldots $$ upto 10 terms $$= \frac{\alpha \times 2^{11}}{2^{\beta} - 1}(C_{0} + \frac{C_{1}}{2} + \frac{C_{2}}{3} + \ldots $$ upto 10 terms ) then the value of $$\alpha + \beta $$ is equal to ______.

The number of values of $$x$$ in the interval $$\left[\frac{\pi}{4}, \frac{7\pi}{4}\right]$$ for which $$14\csc^2 x - 2\sin^2 x = 21 - 4\cos^2 x$$ holds, is ______.

Let the abscissae of the two points $$P$$ and $$Q$$ be the roots of $$2x^2 - rx + p = 0$$ and the ordinates of $$P$$ and $$Q$$ be the roots of $$x^2 - sx - q = 0$$. If the equation of the circle described on $$PQ$$ as diameter is $$2x^2 + y^2 - 11x - 14y - 22 = 0$$, then $$2r + s - 2q + p$$ is equal to ______.

Let $$A$$ be a $$3 \times 3$$ matrix having entries from the set $$\{-1, 0, 1\}$$. The number of all such matrices $$A$$ having sum of all the entries equal to $$5$$, is ______.

Let $$f : R \to R$$ be a function defined by $$f(x)=21-f\left(\frac{25}{x}\right)+f\left(\frac{1}{x^{50}}\right)$$. If the function $$g(x) = f(f(f(x))) + f(f(x))$$, then the greatest integer less than or equal to $$g(1)$$ is ______.

Let $$\theta$$ be the angle between the vectors $$\vec{a}$$ and $$\vec{b}$$, where $$|\vec{a}| = 4, |\vec{b}| = 3$$ and $$\theta \in \left[\frac{\pi}{4}, \frac{\pi}{3}\right]$$. Then $$|\vec{a} - \vec{b} \times \vec{a} + \vec{b}|^2 + 4|\vec{a} \cdot \vec{b}|^2$$ is equal to ______.

Let the lines $$L_1 : \vec{r} = \lambda\hat{i} + 2\hat{j} + 3\hat{k}, \lambda \in R$$ and $$L_2 : \vec{r} = \hat{i} + 3\hat{j} + \hat{k} + \mu(\hat{i} + \hat{j} + 5\hat{k}); \mu \in R$$, intersect at the point $$S$$. If a plane $$ax + by - z + d = 0$$ passes through $$S$$ and is parallel to the lines $$L_1$$ and $$L_2$$, then the value of $$a + b + d$$ is equal to ______.