NTA JEE Main 25th July 2022 Shift 2

For $$z \in \mathbb{C}$$, if the minimum value of $$(|z - 3\sqrt{2}| + |z - p\sqrt{2}i|)$$ is $$5\sqrt{2}$$, then a value of $$p$$ is

The sum $$\displaystyle\sum_{n=1}^{21} \dfrac{3}{(4n-1)(4n+3)}$$ is equal to

The remainder when $$(11)^{1011} + (1011)^{11}$$ is divided by $$9$$ is

The value of $$2\sin\dfrac{\pi}{22} \sin\dfrac{3\pi}{22} \sin\dfrac{5\pi}{22} \sin\dfrac{7\pi}{22} \sin\dfrac{9\pi}{22}$$ is

Let the point $$P(\alpha, \beta)$$ be at a unit distance from each of the two lines $$L_1: 3x - 4y + 12 = 0$$, and $$L_2: 8x + 6y + 11 = 0$$. If $$P$$ lies below $$L_1$$ and above $$L_2$$, then $$100(\alpha + \beta)$$ is equal to

The tangents at the points $$A(1, 3)$$ and $$B(1, -1)$$ on the parabola $$y^2 - 2x - 2y = 1$$ meet at the point $$P$$. Then the area (in $$\text{unit}^2$$) of the triangle $$PAB$$ is:

If the ellipse $$\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1$$ meets the line $$\dfrac{x}{7} + \dfrac{y}{2\sqrt{6}} = 1$$ on the $$x$$-axis and the line $$\dfrac{x}{7} - \dfrac{y}{2\sqrt{6}} = 1$$ on the $$y$$-axis, then the eccentricity of the ellipse is

Let the foci of the ellipse $$\dfrac{x^2}{16} + \dfrac{y^2}{7} = 1$$ and the hyperbola $$\dfrac{x^2}{144} - \dfrac{y^2}{\alpha} = \dfrac{1}{25}$$ coincide. Then the length of the latus rectum of the hyperbola is:

$$\displaystyle\lim_{x \to \frac{\pi}{4}} \dfrac{8\sqrt{2} - (\cos x + \sin x)^7}{\sqrt{2} - \sqrt{2}\sin 2x}$$ is equal to

Consider the following statements:
$$P$$: Ramu is intelligent.
$$Q$$: Ramu is rich.
$$R$$: Ramu is not honest.
The negation of the statement "Ramu is intelligent and honest if and only if Ramu is not rich" can be expressed as: