PYQs Homework Notes Mocks Board Daily Homework JEE Mains Maths Question type Is a function on the natural numbers one-one or onto This type of question, asked 4 times: 2021 Paper 3, 28 Jun 2022, 27 Jul 2022, 27 Jan 2024. All 4 questions below, oldest first.
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Question 1 2021 Paper 3
70 Let f , g : N → N f,g:N \rightarrow N f , g : N → N such that f ( n + 1 ) = f ( n ) + f ( 1 ) ∀ n ∈ N f(n + 1) = f(n) + f(1)\ \forall n \in N f ( n + 1 ) = f ( n ) + f ( 1 ) ∀ n ∈ N and g g g be any arbitrary function. Which of the following statements is NOT true? A. B. If fog is one-one, then g g g is one-one C. If g g g is onto, then fog is one-one D. If f f f is onto, then f ( n ) = n ∀ n ∈ N f(n) = n\forall n \in N f ( n ) = n ∀ n ∈ N Tap an option to check your answer.
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Question 2 28 Jun 2022
62 Let a function f : N → N f:\mathbb{N}\rightarrow\mathbb{N} f : N → N be defined by f ( n ) = [ 2 n , n = 2 , 4 , 6 , 8 , … . . n − 1 , n = 3 , 7 , 11 , 15 , … . . n + 1 2 , n = 1 , 5 , 9 , 13 , … . . f(n) =\left\lbrack \begin{matrix} 2n, & n= 2,4,6,8,\ldots.. \\ n- 1, & n= 3,7,11,15,\ldots.. \\ \frac{n+ 1}{2}, & n= 1,5,9,13,\ldots.. \end{matrix} \right. f ( n ) = 2 n , n − 1 , 2 n + 1 , n = 2 , 4 , 6 , 8 , … .. n = 3 , 7 , 11 , 15 , … .. n = 1 , 5 , 9 , 13 , … .. then, f f f is A. one-one but not onto
B. onto but not one-one
C. neither one-one nor onto
D. one-one and onto
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Question 3 27 Jul 2022
62 Let f , g : N − { 1 } → N \mathbf{f,g:N - \{ 1\} \rightarrow N} f , g : N − { 1 } → N be functions defined by f ( a ) = α f(a) = \alpha f ( a ) = α , where α \alpha α is the maximum of the powers of those primes p p p such that p α p^{\alpha} p α divides a a a , and g ( a ) = a + 1 g(a) = a + 1 g ( a ) = a + 1 , for all a ∈ N − { 1 } \mathbf{a \in N - \{ 1\}} a ∈ N − { 1 } . Then, the function f + g f + g f + g is A. one-one but not onto
B. onto but not one-one
C. both one-one and onto
D. neither one-one nor onto
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Question 4 27 Jan 2024
80 The function f : N − { 1 } → N \mathbf{f:N - \{ 1\} \rightarrow N} f : N − { 1 } → N ; defined by f ( n ) = f(n) = f ( n ) = the highest prime factor of n n n , is : A. both one-one and onto
B. one-one only
C. onto only
D. neither one-one nor onto
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