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.
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
− √[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
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) : x^{2} + y^{2} = 25] and B = [(x, y) : x^{2} + 9y^{2} = 144], then A ∩ B contains _______ points.
a) 5
b) 4
c) 3
d) 2
If A = [(x, y) : x^{2} + y^{2} = 25] and B = [(x, y) : x^{2} + 9y^{2} = 144], then A ∩ B contains _______ points.
Q4. If f (x) = a cos (bx + c) + d, then what is the range of f (x)?
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
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 [π^{2}] x + cos[−π^{2}] x, then find the function of the angle.
a) -1
b) 1
c) 0
d) None of the above
f (x) = cos [π^{2}] x + cos[−π^{2}] 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 _____________.
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) = $x+x− $, 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
Let f (x) = \frac{x^2-1}/{x^2+1}
= \frac{x^2+1-2}/{x^2+1}
= 1 − (2 / [x^{2} + 1]) [Because [x^{2} + 1] > 1 also (2 / [x^{2} + 1]) ≤ 2]
So 1 − [2 / [x^{2} + 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) = cos^{2 }x + sin^{4}x for x ∈ R, then what is f (R)?
f (x) = cos^{2 }x + sin^{4}x
y = f (x) = cos^{2 }x + sin^{2}x (1 − cos^{2}x)
y = cos^{2 }x + sin^{2}x − sin^{2}x cos^{2}x
y = 1 − sin^{2}x cos^{2}x
y = 1 − [1 / 4] * [sin^{2}2x]
3 / 4 ≤ f (x) ≤ 1, (Because 0 ≤ sin^{2}2x ≤ 1)
f (R) ∈ [3/4, 1]
Q10. The function f : R → R defined by f (x) = e^{x} is ________.
a) into
b) onto
c) one -one
d) None of these
Function f: R → R is defined by f (x) = e^{x}.
Let x_{1}, x_{2} ∈ R and f (x_{1}) = f (x_{2}) or e^{x1 }= e^{x2} or x_{1 }= x_{2}.
Therefore, f is one-one.
Let f (x) = e^{x} = 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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