NTA JEE Main 30th January 2023 Shift 2

If the value of real number $$\alpha \gt 0$$ for which $$x^2 - 5\alpha x + 1 = 0$$ and $$x^2 - \alpha x - 5 = 0$$ have a common real roots is $$\frac{3}{\sqrt{2\beta}}$$ then $$\beta$$ is equal to ______.

The number of seven digits odd numbers, that can be formed using all the seven digits 1, 2, 2, 2, 3, 3, 5 is ______.

The $$8^{th}$$ common term of the series
$$S_1 = 3 + 7 + 11 + 15 + 19 + \ldots$$
$$S_2 = 1 + 6 + 11 + 16 + 21 + \ldots$$ is

$$50^{th}$$ root of a number $$x$$ is $$12$$ and $$50^{th}$$ root of another number $$y$$ is $$18$$. Then the remainder obtained on dividing $$(x + y)$$ by $$25$$ is ______.

Let $$P(a_1, b_1)$$ and $$Q(a_2, b_2)$$ be two distinct points on a circle with center $$C(\sqrt{2}, \sqrt{3})$$. Let $$O$$ be the origin and $$OC$$ be perpendicular to both $$CP$$ and $$CQ$$. If the area of the triangle $$OCP$$ is $$\frac{\sqrt{35}}{2}$$, then $$a_1^2 + a_2^2 + b_1^2 + b_2^2$$ is equal to ______.

Let $$A = \{1, 2, 3, 5, 8, 9\}$$. Then the number of possible functions $$f : A \to A$$ such that $$f(m \cdot n) = f(m) \cdot f(n)$$ for every $$m, n \in A$$ with $$m \cdot n \in A$$ is equal to

If $$\int \sqrt{\sec 2x - 1} dx = \alpha \log_e \left|\cos 2x + \beta + \sqrt{\cos 2x\left(1 + \cos\frac{1}{\beta}x\right)}\right|$$ + constant, then $$\beta - \alpha$$ is equal to

Let $$A$$ be the area of the region $$\{(x,y) : y \geq x^2, y \geq (1-x)^2, y \leq 2x(1-x)\}$$. Then $$540A$$ is equal to

Let a line $$L$$ pass through the point $$P(2, 3, 1)$$ and be parallel to the line $$x + 3y - 2z - 2 = 0 = x - y + 2z$$. If the distance of $$L$$ from the point $$(5, 3, 8)$$ is $$\alpha$$, then $$3\alpha^2$$ is equal to ______.

A bag contains six balls of different colours. Two balls are drawn in succession with replacement. The probability that both the balls are of the same colour is $$p$$. Next four balls are drawn in succession with replacement and the probability that exactly three balls are of the same colours is $$q$$. If $$p : q = m : n$$, where $$m$$ and $$n$$ are co-prime, then $$m + n$$ is equal to