NTA JEE Main 24th June 2022 Shift 1

If the sum of the squares of the reciprocals of the roots $$\alpha$$ and $$\beta$$ of the equation $$3x^2 + \lambda x - 1 = 0$$ is $$15$$, then $$6(\alpha^3 + \beta^3)^2$$ is equal to

Let $$A = \{z \in \mathbb{C} : 1 \leqslant |z - (1+i)| \leqslant 2\}$$ and $$B = \{z \in A : |z - (1-i)| = 1\}$$. Then, $$B$$

If $$\{a_i\}_{i=1}^{n}$$, where $$n$$ is an even integer, is an arithmetic progression with common difference $$1$$, and $$\sum_{i=1}^{n} a_i = 192$$, $$\sum_{i=1}^{n/2} a_{2i} = 120$$, then $$n$$ is equal to

The remainder when $$3^{2022}$$ is divided by $$5$$ is

Let $$S = \left\{\theta \in [-\pi, \pi] - \left\{\pm \frac{\pi}{2}\right\} : \sin\theta \tan\theta + \tan\theta = \sin 2\theta\right\}$$. If $$T = \sum_{\theta \in S} \cos 2\theta$$, then $$T + n(S)$$ is equal to

Let $$x^2 + y^2 + Ax + By + C = 0$$ be a circle passing through $$(0, 6)$$ and touching the parabola $$y = x^2$$ at $$(2, 4)$$. Then $$A + C$$ is equal to ______

Let $$\lambda x - 2y = \mu$$ be a tangent to the hyperbola $$a^2x^2 - y^2 = b^2$$. Then $$\left(\frac{\lambda}{a}\right)^2 - \left(\frac{\mu}{b}\right)^2$$ is equal to

The number of choices for $$\Delta \in \{\wedge, \vee, \Rightarrow, \Leftrightarrow\}$$, such that $$(p \Delta q) \Rightarrow ((p \Delta \sim q) \vee ((\sim p) \Delta q))$$ is a tautology, is

Let $$S = \{\sqrt{n} : 1 \leqslant n \leqslant 50$$ and $$n$$ is odd$$\}$$. Let $$a \in S$$ and $$A = \begin{bmatrix} 1 & 0 & a \\ -1 & 1 & 0 \\ -a & 0 & 1 \end{bmatrix}$$. If $$\sum_{a \in S} \det(\text{adj } A) = 100\lambda$$, then $$\lambda$$ is equal to

The number of values of $$\alpha$$ for which the system of equations
$$x + y + z = \alpha$$
$$\alpha x + 2\alpha y + 3z = -1$$
$$x + 3\alpha y + 5z = 4$$
is inconsistent, is