NTA JEE Mains 05th April 2024 Shift 2

Let $$S_1 = \{z \in C : |z| \leq 5\}$$, $$S_2 = \left\{z \in C : \text{Im}\left(\frac{z + 1 - \sqrt{3}i}{1 - \sqrt{3}i}\right) \geq 0\right\}$$ and $$S_3 = \{z \in C : \text{Re}(z) \geq 0\}$$. Then the area of the region $$S_1 \cap S_2 \cap S_3$$ is :

60 words can be made using all the letters of the word BHBJO, with or without meaning. If these words are written as in a dictionary, then the $$50^{th}$$ word is :

For $$x \geq 0$$, the least value of $$K$$, for which $$4^{1+x} + 4^{1-x}, \frac{K}{2}, 16^x + 16^{-x}$$ are three consecutive terms of an A.P., is equal to :

If the constant term in the expansion of $$\left(\frac{\sqrt[5]{3}}{x} + \frac{2x}{\sqrt[3]{5}}\right)^{12}$$, $$x \neq 0$$, is $$\alpha \times 2^8 \times \sqrt[5]{3}$$, then $$25\alpha$$ is equal to :

Let $$A(-1, 1)$$ and $$B(2, 3)$$ be two points and $$P$$ be a variable point above the line $$AB$$ such that the area of $$\triangle PAB$$ is 10. If the locus of $$P$$ is $$ax + by = 15$$, then $$5a + 2b$$ is :

Let $$ABCD$$ and $$AEFG$$ be squares of side 4 and 2 units, respectively. The point $$E$$ is on the line segment $$AB$$ and the point $$F$$ is on the diagonal $$AC$$. Then the radius $$r$$ of the circle passing through the point $$F$$ and touching the line segments $$BC$$ and $$CD$$ satisfies:

Let the circle $$C_1 : x^2 + y^2 - 2(x + y) + 1 = 0$$ and $$C_2$$ be a circle having centre at $$(-1, 0)$$ and radius 2. If the line of the common chord of $$C_1$$ and $$C_2$$ intersects the $$y$$-axis at the point $$P$$, then the square of the distance of $$P$$ from the centre of $$C_1$$ is :

Let the set $$S = \{2, 4, 8, 16, \ldots, 512\}$$ be partitioned into 3 sets $$A, B, C$$ with equal number of elements such that $$A \cup B \cup C = S$$ and $$A \cap B = B \cap C = A \cap C = \phi$$. The maximum number of such possible partitions of $$S$$ is equal to:

Let $$\alpha\beta \neq 0$$ and $$A = \begin{bmatrix} \beta & \alpha & 3 \\ \alpha & \alpha & \beta \\ -\beta & \alpha & 2\alpha \end{bmatrix}$$. If $$B = \begin{bmatrix} 3\alpha & -9 & 3\alpha \\ -\alpha & 7 & -2\alpha \\ -2\alpha & 5 & -2\beta \end{bmatrix}$$ is the matrix of cofactors of the elements of $$A$$, then $$\det(AB)$$ is equal to :

The values of $$m, n$$, for which the system of equations $$x + y + z = 4$$, $$2x + 5y + 5z = 17$$, $$x + 2y + mz = n$$ has infinitely many solutions, satisfy the equation: