Let [x] denote the greatest integer less than or equal to x. Then domain of f(x)=sec^-1(2[x]+1) is:

Solution & Explanation

### Related Formula The domain of sec^-1(y) is given by |y| ge 1, which means: y le -1 quad textor quad y ge 1 ### Core Logic For f(x) = sec^-1(2[x]+1) to be defined: 2[x] + 1 le -1 quad textor quad 2[x] + 1 ge 1 ### Step 1: Solve individual inequalities Case 1: 2[x] + 1 le -1 implies 2[x] le -2 implies [x] le -1 This holds true for all x < 0, i.e., x in (-infty, 0). Case 2: 2[x] + 1 ge 1 implies 2[x] ge 0 implies [x] ge 0 This holds true for all x ge 0, i.e., x in [0, infty). ### Step 2: Take Union of the Solutions textDomain = (-infty, 0) cup [0, infty) = (-infty, infty) ### Pattern Recognition Since [x] covers all integer values and 2[x]+1 forms all odd integer values, the expression inside sec^-1 is always a non-zero integer. Non-zero integers always have absolute value ge 1. Hence, it is valid for all real numbers. ### Evaluation Rubric / Model Answer null ### Chapter Mix Class 11 Mathematics: Functions Class 12 Mathematics: Inverse Trigonometric Functions

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Q6 jee_main_2024_31_jan_morning Composition of Functions
If f(x) = frac4x + 36x - 4, x neq frac23 and (fof)(x) = g(x), where g : mathbbR - left\frac23right\ to mathbbR - left\frac23right\, then (gogog)(4) is equal to
  • A. -frac1920
  • B. frac1920
  • C. -4
  • D. 4

Solution

### Core Logic f(x) = frac4x + 36x - 4 Compute g(x) = f(f(x)): g(x) = frac4left(frac4x + 36x - 4right) + 36left(frac4x + 36x - 4right) - 4 = frac16x + 12 + 18x - 1224x + 18 - 24x + 16 = frac34x34 = x ### Step 1: Composition Evaluation Since g(x) = x, g is the identity function. (gogog)(4) = g(g(g(4))) = 4 ### Evaluation Rubric / Model Answer null ### Chapter Mix Class 12 Maths: Relations and Functions
Q29 jee_main_2024_31_jan_morning Equivalence Relations
Let A = \1, 2, 3, 4\ and R = \(1, 2), (2, 3), (1, 4)\ be a relation on A. Let S be the equivalence relation on A such that R subset S and the number of elements in S is n. Then, the minimum value of n is
Numerical Answer. Answer: 16 to 16

Solution

### Core Logic S must be reflexive, symmetric, and transitive, containing (1,2), (2,3), and (1,4). Symmetric property forces (2,1), (3,2), (4,1) in S. Transitive property: (1,2) and (2,3) implies (1,3) in S. Symmetric implies (3,1) in S. (4,1) and (1,2) implies (4,2) in S. Symmetric implies (2,4) in S. (4,1) and (1,3) implies (4,3) in S. Symmetric implies (3,4) in S. ### Step 1: Universal Relation Since 1 is related to 2, 3, 4 and the relation is an equivalence relation (which creates partitions), all elements 1, 2, 3, and 4 must fall into the same single equivalence class. Thus, S must contain all possible ordered pairs in A times A. ### Step 2: Final Count Number of elements in A times A = 4 times 4 = 16. Minimum value of n is 16. ### Pattern Recognition If a relation connects all elements in a set to each other through a chain, its equivalence closure is the universal relation A times A. ### Evaluation Rubric / Model Answer null ### Chapter Mix Class 12 Maths: Relations and Functions

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