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Q1. A={1, 2, 3, 4, …, 14}. Let R be a relation from A to A defined by R = {(a, b):3a – b = 0, a, b∈ A} then domain of R is

• {1, 2, 3, 4}              
• {3, 6, 9, 12}
• {3, 6, 9}                                
• {1, 2, 3}              

Solution

Here  (a, b) ∈ R, 3a = b, Therefore R = {(1, 3), (2, 6), (3, 9), (4, 12)}     Domain of relation R is the set of I elements of the ordered pairs Domain R= {1, 2, 3, 4}  

Q2. If n(A)=2, n(B)=2 . What is the total number of relations from A to B?

• 4
• 32
• 12
• 16

Solution

The number of relations from a set with m elements to a set with n elements is given by 2^m.n Hence, No. of relations from A to B is 2^2x2 = 16

Q3. For a constant function f(x)=k where k is a real number ; {k} is

• the domain
• a superset of the range
• the range of f
• the co-domain of f

Solution

Range of a function is the set of all images of the elements of the domain of the function. In this case each x from the domain is mapped with the number k. So {k} becomes the range of the function.

Q4. A relation R defined on the set A = {1,2,3,5} as {(x, y): x+y >10: x,y ∈ A }is

• Universal relation 
• Identity relation
• Empty relation  
• Zero relation

Solution

Here R = {(x, y): x + y >10: x, y ∈ A }  Sum of none of the two elements of A is more than 10 and hence R is an empty relation.

Q5. If f and g are two functions over real numbers defined as f(x) = 3x + 1, g(x) = x² + 2, then find f-g

• 3x + x²
• 3x – x²-1
• x²+1-3x
• x²+1

Solution

f - g = (3x + 1) – (x² + 2) = 3x – x²-1

Q6. Let R be a relation defined on the set of N natural numbers as R = {(x, y): y is a factor of x, x, y∈  N} then,

• (9, 5) ∈ R                                  
• (3, 9) ∈ R
• (4, 2) ∈ R     
• (2, 4) ∈ R

Solution

R = {(x, y): y is a factor of x, x, y  ∈ N} Now if y is a factor of x then y divides x exactly.  Since 2 divides 4 so (4, 2) ∈ R       

Q7. If f(x) = x² - x + 1, g(x) = 7x - 3, be two real functions then (f - g)(8) is

• 57
• 53
• 4
• 64

Solution

For functions f : X  R and g : X  R, we  have (f - g)(x) = f(x) - g(x), x  X   Hence, (f - g)(8) = f(8) - g(8) = 4

Q8. The domain and range of the function f: R →R defined by: f = {(x+1, x+5): x ∈ {0, 1, 2, 3, 4, 5}}

• Domain = {1, 2, 3, 4, 5, 6}; Range = {5, 6, 7, 8, 9, 10}
• Domain = {1, 2, 3, 4, 5}; Range = {5, 6, 7, 8, 9}
• Domain = {2, 3, 4, 5, 6}; Range = {6, 7, 8, 9, 10}
• Domain = R ; Range = R

Solution

Domain = {x+1, x ∈ {0, 1, 2, 3, 4, 5}} ={1, 2, 3, 4, 5, 6}; Range= {x+5, x ∈ {0, 1, 2, 3, 4, 5}} = {5, 6, 7, 8, 9, 10}

Q9. Find the domain of the function f(x) = x |x|

• R⁺
• R
• R – {0}
• R – {-1}

Solution

Domain of the function f is all real numbers. Since f(x) =x|x| is defined for all real values of x .  

Q10. If (x + 2, y - 2) = (3, 1) then the values of x and y are: 

• 1, 3
• 3, 5
• 5, 3
• 1, -3

Solution

Two ordered pairs are equal, if and only if the corresponding first elements are equal and the second elements are also equal. Hence, x = 1 and y = 3.

Q11. If A = {1,2,3}, B = {a,b,c}, which of the following is a function?

•  R = {1,,a), (2, a), (1, c)}
• R = {(1, c), (2, b), (2, c)}
• R = {(1,b), (1, c), (3, a)}
• R = {(1, b), (2, c), (3, a)}

Solution

 R = {(1, b), (2, c), (3, a)} is a function  Since all the elements of the set A has an image in B  and no element of A is connected to two elements of B.  

Q12. Let A = {1, 2, 3, …, 30} and R be the relation “is one-fifth of”  on A. Then the range of R is

• {5, 10, 15, 20, 25, 30}
• {1, 2, 3, 4, 5, 6}
• {1, 2, 3, 4, 5}
• {1, 2, 3, 4, 5, 6, 10, 15, 20, 25, 30}

Solution

The range of the relation R is the set of all second elements of the ordered pair in a relation R. Here R = {(1, 5), (2, 10), (3, 15), (4, 20), (5, 25), (6, 30)} Range is {5, 10, 15, 20, 25, 30}

Q13. The graph of the modulus function is symmetric to the line

• x=0
• y =0
• y=x
• y=-x

Solution

It can be observed from the graph that the modulus function is symmetrical about y-axis or x = 0

Q14. For the relation R= {(1,2),(2,3),(4,5),(3,6)} The domain is

• {1,2,3,4}
• {2,3,5,6}
• {1,3,5,6}
• {1,2,3,4,5,6}

Solution

The domain of a relation is the set of all first entries in the ordered pairs of the relation.

Q15. If f(x) = x² and g(x) = x are two functions from R to R then (fg)(2) is

•     4
•     2
• 1
• 8

Solution

fg(2 )=  f(2).g(2) = 2².2 = 8

Q16. If A = {1, 2, 3, 4, 5}, let relation R be defined on A as R={(x, y): y = x+3, x, y∈ A } Then R is

• {(1, 4), (2, 5)}                            
• {(1, 4), (2, 5), (3, 6)}
• {(1, 4), (2, 5), (3, 6), (4, 7), (5, 8)}   
• {(1, 2), (2, 3), (3, 4)}

Solution

A = {1, 2, 3, 4, 5}. A relation R is defined as R={(x, y): y = x+3, x, y∈ A } 1 R 4 and 2 R 5 and 3R 6 but 6∉ A.    So R = {(1, 4), (2, 5)}

Q17. If ordered pair (a + 2b, 9) = (7, 3a + 2b), then the values of a and b are 

• 7, 9
• 1, 3
• 3, 1
• 9, 7

Solution

Two ordered pairs are equal, if and only if the corresponding first elements are equal and the second elements are also equal. Hence, a + 2b = 7 and 3a + 2b = 9. Solving the linear equations, we get a = 1 and b = 3

Q18. If n(A) = 3 and n(B)  =4 then find the number of elements in the power set of A × B

• 24
• 4096
• 8
• 64

Solution

As n(  A × B) = 12 and so n( P( A × B)) = 2¹² = 4096

Q19. If A = {1, 2, 3 }, B = {4, 5, 6 }, which of the following is not a relation from A to B

• R₁ = {(1, 4), (1, 5),(1, 6)}
• R₂ = {(1, 5), (2, 4), (3, 6)}
• R₃ = {(1, 4), (1, 5), (3, 6), (2, 6), (3, 4)}
• R₄ = {(4, 2), (2, 6), (5, 1), (2, 4)}

Solution

R₄ is a set of ordered pairs where each first element does not come from A.

Q20. Given A = {1, 2} and B = {5, 6, 7} then  A x B =

•  {(5, 1), (5, 2), (6, 1), (6, 2), (7, 1), (7, 2)}
•  Φ
•  {(1, 5), (1, 6), (1, 7), (2, 5), (2, 6), (2, 7)}
•  { 1, 6), (1, 7), (2, 5), (2, 6), (2, 7)}

Solution

   A x B = {(1, 5), (1, 6), (1, 7), (2, 5), (2, 6), (2, 7)}

Q21. Which of the following relations is not a function?

• R= {(1,2), (3,4),(2,1),(5,1)}
• R= {(2,1), (4,4)(3,1),(5,1)}
• R= {(1,2), (3,4)(2,1),(5,2)}
• R={(1,2), (1,4)(3,1),(5,1)}

Solution

For a relation to be a function, every element of the domain must be connected to a unique element of the co-domain. In this case, (d) has the element 1 being connected to two different elements 2 and 4.Thus, it’ can’t be a function.

Q22. f(x) = x is called

• an identity function
• a constant function
• step function
• inverse function

Solution

In this function every element x is mapped with itself; hence it’s the identity function.

Q23. Let R be a relation defined as R = {(x, y): y = 2x, x is natural number < 5} then Range of R is given as ,

• {2, 4, 6, 8, 10}
• {1, 2, 3, 4}
• {2, 4, 6, 8}
• N

Solution

R = {(x, y): y = 2x, x is natural number < 5}  Relation R in roster form is { (1,2),(2,4),(3,6),(4,8)} Range of relation R is the set of II elements of the ordered pairs { (1,2),(2,4),(3,6),(4,8)}  So range is  {2, 4, 6, 8}

Q24. If n(A) = p and n(B) = q, then how many relations are there from A to B.

• pq 
• 2^pq
• (pq)²                           
• 3^pq

Solution

The number of relations from a set A with m elements to a set B with n elements is given by 2^mn  So given n(A) = p and n(B) = q Then 2^pq relations are defined from A to B.    

Q25. If A × B = {(x, a), (x, b), (y,a), (y, b)} then A =

• {x,y}      
• {a,b}
• {x,a}
• {x,x,y,y}

Solution

Given A × B = {(x, a), (x, b), (y, a), (y, b)} Set A is represented by the first elements of ordered pairs so  A = {x, y}

Q26. If f(x) = x² – x + 1; g(x) = 7x – 3, be two real functions then (f + g)(3) is

• 25
• 3
• 7
• 18

Solution

For functions f : X  R and g : X  R, we have (f + g)(x) = f(x) + g(x), x  X Hence, (f + g)(3) = f(3) + g(3) = 25

Q27. Which of the relations is not a function?

• one-one
• many one
• one many
• domain = codomain

Solution

For a relation to be a function one element of the domain must not be connected with more than one elements of the co-domain. Hence the answer.

Q28. If f(x) = x² and g(x) = cosx, which of the following is true?

• f + g is even function
• f + g is an odd function
• f - g is an odd function
• f + g is not defined

Solution

Thus, its an even function.

Q29. R = {(x, y): │x –y │is divisible by 4, x,y are natural numbers < 15} then, which of the elements is related to 1.

• 6,9             
• 5,9,13
• 8,13            
• 5,9,15

Solution

  The elements of R are of the type (1, 5), (2, 6) etc. the elements of R with first entry of the ordered pair as 1 are  (1, 5), (1, 9), (1, 13). So 5,9,13 are related to 1.  

Q30. Which is not true for the graph of the real function y = x²:

• The graph of the given function is a parabola.
• The parabola will open upward.
• The least value of x²  is one and will be so when x = 1.
• For this function domain = Real numbers

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