Mathematics MCQ on Sets Relations and Functions for JEE and Engineering Exam 2022

MCQ on Sets Relations and Functions: To be an expert in JEE Mathematics, it is absolutely necessary to practice and be familiar will all the concepts as well as the questions of different types. This is essential to gain mastery over the subject. We have also often heard the common saying, “Practice Makes a Man Perfect”, hence students have to practice, practice and practice till they master the subject.

MCQ on Sets, Relations And Functions

In this post we are providing you MCQ on Sets Relations and Functions, which will be beneficial for you in upcoming JEE and Engineering Exams.

MCQ on Sets Relations and Functions

Q1. If f: R → S defined by f (x) = sinx − √3 cosx + 1 is onto, then what is the interval of S?
a) -1,3
b) -1,1
c) -1,0
d)1,3

View Answer

− √[1 + (√−3)2] ≤ (sinx − √3 cosx) ≤ √[1 + (√−3)2]

−2 ≤ (sinx − √3cosx) ≤ 2

−2 + 1 ≤ (sinx − √3 cosx + 1) ≤ 2 + 1

−1 ≤ (sinx − √3 cosx + 1) ≤ 3 i.e.,

range = [−1, 3]

For f to be onto S = [−1, 3].

 

Q2. If f (x) = cos (log x), then find the value of f (x) * f (4) − [1 / 2] * [f (x / 4) + f (4x)].
a) 1
b) 4
c) 3
d) 0

View Answer

f (x) = cos (log x)

Now let y = f (x) * f (4) − [1 / 2] * [f (x / 4) + f (4x)]

y = cos (log x) * cos (log 4) − [1 / 2] * [cos log (x / 4) + cos (log 4x)]

y = cos (log x) cos (log 4) − [1 / 2] * [cos (log x −log 4) + cos (log x + log 4)]

y = cos (log x) cos (log 4) − [1 / 2] * [2 cos (log x) cos (log 4)]

y = 0

Q3. If A = [(x, y) : x2 + y2 = 25] and B = [(x, y) : x2 + 9y2 = 144], then A ∩ B contains _______ points.
a) 5
b) 4
c) 3
d) 2

View Answer

If A = [(x, y) : x2 + y2 = 25] and B = [(x, y) : x2 + 9y2 = 144], then A ∩ B contains _______ points.

Q4. If f (x) = a cos (bx + c) + d, then what is the range of f (x)?

View Answer

f (x) = a cos (bx + c) + d ..(i)

For minimum cos (bx + c) = −1

From (i), f (x) = −a + d = (d − a)

For maximum cos (bx + c) = 1

From (i), f (x) = a + d = (d + a)

Range of f (x) = [d − a, d + a]

Q5. In a college of 300 students, every student reads 5 newspapers and every newspaper is read by 60 students. The number of newspapers is ________.
a) 25
b) 20
c) 30
d) 28

View Answer

Let the number of newspapers be x.

If every student reads one newspaper, the number of students would be x (60) = 60x

Since every student reads 5 newspapers, the number of students = [x * 60] / [5] = 300

x = 25

Q6. If f (x) = cos [π2] x + cos[−π2] x, then find the function of the angle.
a) -1
b) 1
c) 0 
d) None of the above

View Answer

f (x) = cos [π2] x + cos[−π2] x

f (x) = cos (9x) + cos (−10x)

= cos (9x) + cos (10x)

= 2 cos (19x / 2) cos (x / 2)

f (π / 2) = 2 cos (19π / 4) cos (π / 4);

f (π / 2) = 2 * −1 / √2 * 1/ √2

= −1

Q7.  If f (x) = 3x − 5, then f−1(x) is _____________.

View Answer

Let f (x) = y ⇒ x = f−1 (y).

Hence, f (x) = y = 3x − 5

⇒ x = frac{y + 5}/{3}

⇒f−1 (y) = x = frac{y + 5}/{3}

f−1 (x) = frac{x+ 5}/{3}

Also, f is one-one and onto, so f−1 exists and is given by f−1 (x) = [x + 5] / [3

Q8. If f (x) = \frac{x^2-1}{x^2+1}, for every real number. Then what is the minimum value of f?
a) Equal to 1
b) Less than 1
c) Greater then 1
d) 0

View Answer

Let f (x) = \frac{x^2-1}/{x^2+1}

\frac{x^2+1-2}/{x^2+1}

= 1 − (2 / [x2 + 1]) [Because [x2 + 1] > 1 also (2 / [x2 + 1]) ≤ 2]

So 1 − [2 / [x2 + 1]] ≥ 1 − 2;

−1 ≤ f (x) < 1

Thus, f (x) has the minimum value equal to 1.

Q9. The function f: R → R is defined by f (x) = cosx + sin4x for x ∈ R, then what is f (R)?

View Answer

f (x) = cosx + sin4x

y = f (x) = cosx + sin2x (1 − cos2x)

y = cosx + sin2x − sin2x cos2x

y = 1 − sin2x cos2x

y = 1 − [1 / 4] * [sin22x]

3 / 4 ≤ f (x) ≤ 1, (Because 0 ≤ sin22x ≤ 1)

f (R) ∈ [3/4, 1]

Q10. The function f : R → R defined by f (x) = ex is ________.
a) into
b) onto
c) one -one
d) None of these

View Answer

Function f: R → R is defined by f (x) = ex.

Let x1, x2 ∈ R and f (x1) = f (x2) or ex1 = ex2 or x= x2.

Therefore, f is one-one.

Let f (x) = ex = y.

Taking log on both sides, we get x = logy.

We know that negative real numbers have no pre-image or the function is not onto and zero is not the image of any real number.

Therefore, function f is into.

 


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