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Test: Relations & Functions- 1 - JEE MCQ


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25 Questions MCQ Test Mathematics (Maths) for JEE Main & Advanced - Test: Relations & Functions- 1

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Test: Relations & Functions- 1 - Question 1

The range of the function f(x) = 7-x Px-3 is 

Detailed Solution for Test: Relations & Functions- 1 - Question 1

Here, 0 ≤ x- 3 ≤ 7 - x  
⇒0 ≤ x - 3 and x - 3 ≤ 7 - x
By solvation, we will get 3 ≤ x ≤ 5
So x = 3,4,5 find the values of 7-x Px - 3 by substituting the values of x
at x = 3 4P0 = 1
at x = 4 3P1 = 3 
at x = 5 2P2 = 2

Test: Relations & Functions- 1 - Question 2

Let R be the relation over the set of straight lines of a plane such that l1 R l2 ⇔ l1 ⊥ l2. Then, R is

Detailed Solution for Test: Relations & Functions- 1 - Question 2

To be reflexive, a line must be perpendicular to itself, but which is not true. So, R is not reflexive
For symmetric, if  l1 R l2 ⇒ l1 ⊥ l2.
⇒  l2 ⊥ l1 ⇒ l1 R l2 hence symmetric
For transitive,  if l1 R l2 and l2 R l3
⇒ l1 R l2  and l2 R l3  does not imply that l1 ⊥ l3 hence not transitive.

Test: Relations & Functions- 1 - Question 3

The binary relation S = Φ (empty set) on set A = {1, 2, 3} is

Detailed Solution for Test: Relations & Functions- 1 - Question 3

Reflexive : A relation is reflexive if every element of set is paired with itself. Here none of the element of A is paired with themselves, so S is not reflexive.
Symmetric : This property says that if there is a pair (a, b) in S, then there must be a pair (b, a) in S. Since there is no pair here in S, this is trivially true, so S is symmetric.
Transitive : This says that if there are pairs (a, b) and (b, c) in S, then there must be pair (a,c) in S. Again, this condition is trivially true, so S is transitive.

Test: Relations & Functions- 1 - Question 4

The void relation (a subset of A x A) on a non empty set A is:

Detailed Solution for Test: Relations & Functions- 1 - Question 4

The relation { } ⊂ A x A on a is surely not reflexive. However, neither symmetry nor transitivity is contradicted. So { } is a transitive and symmetry relation on A.

Test: Relations & Functions- 1 - Question 5

The domain of the function f = {(1, 3), (3, 5), (2, 6)} is

Detailed Solution for Test: Relations & Functions- 1 - Question 5

The domain in ordered pair (x,y) is represented by x-coordinate. Therefore, the domain of the given function is given by : {1, 3, 2}.

Test: Relations & Functions- 1 - Question 6

The domain of the function 

Detailed Solution for Test: Relations & Functions- 1 - Question 6

x - 1 ≥ 0 and 6 – x ≥ 0 ⇒ 1 ≤ x ≤ 6.

Test: Relations & Functions- 1 - Question 7

Let R be the relation on N defined as x R y if x + 2 y = 8. The domain of R is

Detailed Solution for Test: Relations & Functions- 1 - Question 7

As x R y if x + 2y = 8, therefore, domain of the relation R is given by x = 8 – 2y ∈ N.
When y = 1, 
⇒ x = 6 ,when y = 2, 
⇒ x = 4, when y = 3, 
⇒ x = 2.
therefore domain is {2, 4, 6}.

Test: Relations & Functions- 1 - Question 8

The range of  is 

Detailed Solution for Test: Relations & Functions- 1 - Question 8

We have , 



Therefore, range of f(x) is {-1}.

Test: Relations & Functions- 1 - Question 9

The function f(x) = sin x2 is

Detailed Solution for Test: Relations & Functions- 1 - Question 9

For even function: f(-x) = f(x) , 
therefore, f(− x)
 = sin (− x)2 = sin x2 = f(x).

Test: Relations & Functions- 1 - Question 10

If A = {1, 2, 3}, then the relation R = {(1, 2), (2, 3), (1, 3) in A is

Detailed Solution for Test: Relations & Functions- 1 - Question 10

A relation R on a non empty set A is said to be transitive if fxRy and y Rz ⇒ xRz, for all x ∈ R. Here, (1, 2) and (2, 3) belongs to R implies that (1, 3) belongs to R.

Test: Relations & Functions- 1 - Question 11

A relation R in a set A is called transitive, if

Detailed Solution for Test: Relations & Functions- 1 - Question 11

A relation R on a non empty set A is said to be transitive if fx Ry and yRz ⇒ x Rz, for all x ∈ R.

Test: Relations & Functions- 1 - Question 12

The range of the function f(x) =|x−1| is

Detailed Solution for Test: Relations & Functions- 1 - Question 12

We have, f(x) = |x−1|, which always gives non-negative values of f(x) for all x ∈ R.Therefore range of the given function is all non-negative real numbers i.e. [0,∞).

Test: Relations & Functions- 1 - Question 13

The range of the function f(x) = 7 − x Px − 3 is

Detailed Solution for Test: Relations & Functions- 1 - Question 13

7 − x ≥ 1,x − 3 ≥ 0
and 7 − x ≥ x − 3
⇒ x ≤ 6, x ≥ 3, x ≤ 5
Thus 3 ≤ x ≤ 5
∴ Range ={4P0, 3P1, 2P2}
={1,3,2}

Test: Relations & Functions- 1 - Question 14

The range of the function   is

Detailed Solution for Test: Relations & Functions- 1 - Question 14

 

f(x) = (x+2) / |x+2|, where x ≠ -2

Step 1: Consider the Sign of (x+2)

  • If x + 2 > 0 (i.e., x > -2), then |x+2| = x+2, so:

f(x) = (x+2) / (x+2) = 1

  • If x + 2 < 0 (i.e., x < -2), then |x+2| = -(x+2), so:

f(x) = (x+2) / -(x+2) = -1

Since x ≠ -2, the function is defined for all values except x = -2, and the function only takes values 1 and -1.

Step 2: Determine the Range

The function only outputs two values: {1, -1}.

Option (d) {1, -1}

Test: Relations & Functions- 1 - Question 15

If f: (0, π) → R is given by (x)=∑nk=1 [1 + sin kx], [x] denotes the greatest integer function, then the range of f(x) is

Detailed Solution for Test: Relations & Functions- 1 - Question 15


for some k then x = π/2k, hence sinx, sin2x,…, sin(k−1)x will lie between 0 and 1 so [sinjx] = 0 1 ≤ j ≤ k − 1;sinkx=1 so f(x) can be n+1 or n.

Test: Relations & Functions- 1 - Question 16

Let A = {a, b, c} and R = {(a, a), (b, b), (c, c), (b, c)} be a relation on A. Here, R is

Detailed Solution for Test: Relations & Functions- 1 - Question 16

Correct Answer :- b

Explanation:- A = {a, b, c} and R = {(a, a), (b, b), (c, c), (b, c)}

Any relation R is reflexive if fx Rx for all x ∈ R. Here ,(a, a), (b, b), (c, c) ∈ R. Therefore , R is reflexive.

For the transitive, in the relation R there should be (a,c)

Hence it is not transitive.

Test: Relations & Functions- 1 - Question 17

Let f(x) = x4 − 2x2 + 5 be defined on [−2, 2] 
Statement-1: The range of f(x) is [2,13]
Statement-2: The greatest value of f is attained at x = 2

Detailed Solution for Test: Relations & Functions- 1 - Question 17

f′(x) = 4x3 − 4x = 4x(x − 1)(x + 1).
The critical points of f are 0,−1,1 But f(0) = 5,f(1) = 4,f(−1) =  4,f(2) = 13.
So the range of f is [4,13] and grea, test value of f is at x=2

Test: Relations & Functions- 1 - Question 18

A relation R from C to R is defined by x Ry iff |x| = y. Which of the following is correct?

Detailed Solution for Test: Relations & Functions- 1 - Question 18

Test: Relations & Functions- 1 - Question 19

The range of the function defined by 

Detailed Solution for Test: Relations & Functions- 1 - Question 19


Clearly the range is (−∞,−5)∪[0,1]∪(5,∞)

Test: Relations & Functions- 1 - Question 20

A relation R in a set A is said to be an equivalence relation if

Detailed Solution for Test: Relations & Functions- 1 - Question 20

A relation R on a non empty set A is said to be reflexive iff xRx for all x ∈ R . .
A relation R on a non empty set A is said to be symmetric if fx Ry ⇔ y Rx, for all x , y ∈ R .
A relation R on a non empty set A is said to be transitive if fx Ry and y Rz ⇒ x Rz, for all x ∈ R.
An equivalence relation satisfies all these three properties.

Test: Relations & Functions- 1 - Question 21

If x ∈ R, then the range of is

Detailed Solution for Test: Relations & Functions- 1 - Question 21


Hence, option (c) is correct.

Test: Relations & Functions- 1 - Question 22

Let f: R → R be a mapping such that f(x) = . Then f is

Detailed Solution for Test: Relations & Functions- 1 - Question 22

Correct answer is D.

Test: Relations & Functions- 1 - Question 23

If f:R → R and is defined by for each x ∈ R, then the range of f is

Detailed Solution for Test: Relations & Functions- 1 - Question 23

 Given,

∵ −1 ≤ cos3x ≤ 1
⇒ 1≤ − cos3x ≤ −1
⇒  2+1 ≤ 2 − cos3x ≤ 2 − 1
⇒  3 ≤ 2 − cos3x ≤ 1

∴ Range of f is [1/3,1]

Test: Relations & Functions- 1 - Question 24

Which of the following is a polynomial function?

Detailed Solution for Test: Relations & Functions- 1 - Question 24

A polynomial function has all exponents as integral whole numbers. 

Test: Relations & Functions- 1 - Question 25

If (m, n) represents the domain of the function defined as 

Detailed Solution for Test: Relations & Functions- 1 - Question 25


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