NTA JEE Mains 04th April 2024 Shift 2

The area (in sq. units) of the region $$S = \{z \in \mathbb{C} : |z - 1| \leq 2; (z + \bar{z}) + i(z - \bar{z}) \leq 2, \text{Im}(z) \geq 0\}$$ is

The value of $$\frac{1 \times 2^2 + 2 \times 3^2 + \ldots + 100 \times (101)^2}{1^2 \times 2 + 2^2 \times 3 + \ldots + 100^2 \times 101}$$ is

Let three real numbers $$a, b, c$$ be in arithmetic progression and $$a + 1, b, c + 3$$ be in geometric progression. If $$a > 10$$ and the arithmetic mean of $$a, b$$ and $$c$$ is $$8$$, then the cube of the geometric mean of $$a, b$$ and $$c$$ is

If the coefficients of $$x^4$$, $$x^5$$ and $$x^6$$ in the expansion of $$(1 + x)^n$$ are in the arithmetic progression, then the maximum value of $$n$$ is:

Let $$C$$ be a circle with radius $$\sqrt{10}$$ units and centre at the origin. Let the line $$x + y = 2$$ intersects the circle $$C$$ at the points $$P$$ and $$Q$$. Let $$MN$$ be a chord of $$C$$ of length $$2$$ unit and slope $$-1$$. Then, a distance (in units) between the chord $$PQ$$ and the chord $$MN$$ is

Let $$PQ$$ be a chord of the parabola $$y^2 = 12x$$ and the midpoint of $$PQ$$ be at $$(4, 1)$$. Then, which of the following point lies on the line passing through the points $$P$$ and $$Q$$?

Consider a hyperbola $$H$$ having centre at the origin and foci on the $$x$$-axis. Let $$C_1$$ be the circle touching the hyperbola $$H$$ and having the centre at the origin. Let $$C_2$$ be the circle touching the hyperbola $$H$$ at its vertex and having the centre at one of its foci. If areas (in sq units) of $$C_1$$ and $$C_2$$ are $$36\pi$$ and $$4\pi$$, respectively, then the length (in units) of latus rectum of $$H$$ is

Let $$f(x) = \int_0^x (t + \sin(1 - e^t))dt$$, $$x \in \mathbb{R}$$. Then, $$\lim_{x \to 0} \frac{f(x)}{x^3}$$ is equal to

If the mean of the following probability distribution of a random variable $$X$$:

is $$\frac{46}{9}$$, then the variance of the distribution is

Let a relation $$R$$ on $$\mathbb{N} \times \mathbb{N}$$ be defined as: $$(x_1, y_1) R (x_2, y_2)$$ if and only if $$x_1 \leq x_2$$ or $$y_1 \leq y_2$$. Consider the two statements: (I) $$R$$ is reflexive but not symmetric. (II) $$R$$ is transitive. Then which one of the following is true?