NTA JEE Main 28th June 2022 Shift 1

The total number of 5-digit numbers, formed by using the digits $$1, 2, 3, 5, 6, 7$$ without repetition, which are multiple of $$6$$, is

Let $$A_1, A_2, A_3, \ldots$$ be an increasing geometric progression of positive real numbers. If $$A_1 A_3 A_5 A_7 = \frac{1}{1296}$$ and $$A_2 + A_4 = \frac{7}{36}$$, then the value of $$A_6 + A_8 + A_{10}$$ is equal to

If $$\sum_{k=1}^{31} \left(^{31}C_{k}\right) \left(^{31}C_{k-1} \right) - \sum_{k=1}^{30} \left(^{30}C_{k}\right) \left(^{30}C_{k-1} \right)= \frac{\alpha(60!)}{(30!)(31!)}$$, where $$\alpha \in R$$, then the value of $$16\alpha$$ is equal to

If the tangents drawn at the point $$O(0,0)$$ and $$P(1+\sqrt{5}, 2)$$ on the circle $$x^2 + y^2 - 2x - 4y = 0$$ intersect at the point $$Q$$, then the area of the triangle $$OPQ$$ is equal to

Let the eccentricity of the hyperbola $$H : \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$$ be $$\sqrt{\frac{5}{2}}$$ and length of its latus rectum be $$6\sqrt{2}$$. If $$y = 2x + c$$ is a tangent to the hyperbola $$H$$, then the value of $$c^2$$ is equal to

Let $$p, q, r$$ be three logical statements. Consider the compound statements
$$S_1 : ((\sim p) \vee q) \vee ((\sim p) \vee r)$$ and $$S_2 : p \to (q \vee r)$$
Then, which of the following is NOT true?

Let $$AB$$ and $$PQ$$ be two vertical poles, $$160$$ m apart from each other. Let $$C$$ be the middle point of $$B$$ and $$Q$$, which are feet of these two poles. Let $$\frac{\pi}{8}$$ and $$\theta$$ be the angles of elevation from $$C$$ to $$P$$ and $$A$$, respectively. If the height of pole $$PQ$$ is twice the height of pole $$AB$$, then $$\tan^2\theta$$ is equal to

Let $$A$$ be a matrix of order $$3 \times 3$$ and $$\det(A) = 2$$. Then $$\det(\det(A) \text{ adj}(5 \text{ adj}(A^3)))$$ is equal to

If the system of linear equations
$$2x + 3y - z = -2$$
$$x + y + z = 4$$
$$x - y + |\lambda|z = 4\lambda - 4$$ where $$\lambda \in \mathbb{R}$$,
has no solution, then

Let a function $$f : \mathbb{N} \to \mathbb{N}$$ be defined by
$$f(n) = \begin{cases} 2n, & n = 2, 4, 6, 8, \ldots \\ n-1, & n = 3, 7, 11, 15, \ldots \\ \frac{n+1}{2}, & n = 1, 5, 9, 13, \ldots \end{cases}$$
then, $$f$$ is