NTA JEE Mains 06th April 2024 Shift 2

If $$z_1, z_2$$ are two distinct complex numbers such that $$\left|\frac{z_1 - 2z_2}{\frac{1}{2} - z_1\bar{z}_2}\right| = 2$$, then

Let $$0 \leq r \leq n$$. If $$^{n+1}C_{r+1} : ^{n}C_{r} : ^{n-1}C_{r-1} = 55 : 35 : 21$$, then $$2n + 5r$$ is equal to:

If all the words with or without meaning made using all the letters of the word "NAGPUR" are arranged as in a dictionary, then the word at $$315^{th}$$ position in this arrangement is :

Let $$ABC$$ be an equilateral triangle. A new triangle is formed by joining the middle points of all sides of the triangle $$ABC$$ and the same process is repeated infinitely many times. If $$P$$ is the sum of perimeters and $$Q$$ is be the sum of areas of all the triangles formed in this process, then :

A software company sets up $$m$$ number of computer systems to finish an assignment in 17 days. If 4 computer systems crashed on the start of the second day, 4 more computer systems crashed on the start of the third day and so on, then it took 8 more days to finish the assignment. The value of $$m$$ is equal to:

If $$P(6, 1)$$ be the orthocentre of the triangle whose vertices are $$A(5, -2)$$, $$B(8, 3)$$ and $$C(h, k)$$, then the point $$C$$ lies on the circle:

If the locus of the point, whose distances from the point $$(2, 1)$$ and $$(1, 3)$$ are in the ratio $$5 : 4$$, is $$ax^2 + by^2 + cxy + dx + ey + 170 = 0$$, then the value of $$a^2 + 2b + 3c + 4d + e$$ is equal to :

$$\lim_{n \to \infty} \frac{(1^2 - 1)(n-1) + (2^2 - 2)(n-2) + \cdots + ((n-1)^2 - (n-1)) \cdot 1}{(1^3 + 2^3 + \cdots + n^3) - (1^2 + 2^2 + \cdots + n^2)}$$ is equal to :

Let $$A = \{1, 2, 3, 4, 5\}$$. Let $$R$$ be a relation on $$A$$ defined by $$xRy$$ if and only if $$4x \leq 5y$$. Let $$m$$ be the number of elements in $$R$$ and $$n$$ be the minimum number of elements from $$A \times A$$ that are required to be added to $$R$$ to make it a symmetric relation. Then $$m + n$$ is equal to :

If $$A$$ is a square matrix of order 3 such that $$\det(A) = 3$$ and $$\det(\text{adj}(-4 \text{adj}(-3 \text{adj}(3 \text{adj}((2A)^{-1}))))) = 2^m 3^n$$, then $$m + 2n$$ is equal to :