Q. Find the weighted mean of the numbers 10, 20, and 30 with weights 1, 2, and 3 respectively.
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Solution
Weighted mean = (10*1 + 20*2 + 30*3) / (1 + 2 + 3) = (10 + 40 + 90) / 6 = 140 / 6 = 23.33.
Correct Answer:
B
— 25
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Q. Find the x-coordinate of the point where the function f(x) = 2x^3 - 9x^2 + 12x has a local maximum.
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Solution
f'(x) = 6x^2 - 18x + 12. Setting f'(x) = 0 gives x = 1 and x = 2. f''(1) < 0 indicates a local maximum at x = 1.
Correct Answer:
B
— 2
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Q. Find the x-coordinate of the point where the function f(x) = x^2 - 4x + 5 has a minimum.
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Solution
The vertex occurs at x = -b/(2a) = 4/2 = 2, which is the x-coordinate of the minimum point.
Correct Answer:
A
— 2
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Q. Find the x-coordinate of the point where the function f(x) = x^2 - 4x + 5 has a local minimum.
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Solution
The vertex occurs at x = -b/(2a) = 4/2 = 2. This is where the local minimum occurs.
Correct Answer:
B
— 2
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Q. Find the y-intercept of the line represented by the equation 5x - 2y = 10.
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Solution
Set x = 0: -2y = 10 => y = -5. The y-intercept is (0, -5).
Correct Answer:
B
— 2
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Q. For f(x) = { x^2, x < 1; 2x - 1, x ≥ 1 }, is f differentiable at x = 1?
A.
Yes
B.
No
C.
Only left
D.
Only right
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Solution
f'(1) from left = 2, from right = 2; hence f is differentiable at x = 1.
Correct Answer:
B
— No
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Q. For the data set 10, 20, 30, 40, 50, what is the mean deviation?
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Solution
Mean = 30; Mean Deviation = (|10-30| + |20-30| + |30-30| + |40-30| + |50-30|) / 5 = 10.
Correct Answer:
B
— 15
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Q. For the data set {10, 12, 23, 23, 16, 23, 21}, what is the mode?
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Solution
The mode is the number that appears most frequently, which is 23.
Correct Answer:
C
— 23
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Q. For the data set {12, 15, 20, 22, 25}, what is the mode?
A.
12
B.
15
C.
20
D.
No mode
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Solution
There is no mode as all values appear only once.
Correct Answer:
D
— No mode
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Q. For the data set {2, 4, 6, 8, 10}, what is the mean deviation?
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Solution
Mean = 6; Mean deviation = (|2-6| + |4-6| + |6-6| + |8-6| + |10-6|)/5 = (4 + 2 + 0 + 2 + 4)/5 = 12/5 = 2.4.
Correct Answer:
B
— 1.6
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Q. For the data set {4, 8, 6, 5, 3}, what is the mean?
A.
4.5
B.
5.5
C.
6.0
D.
5.0
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Solution
Mean = (4 + 8 + 6 + 5 + 3) / 5 = 26 / 5 = 5.0.
Correct Answer:
D
— 5.0
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Q. For the data set: 1, 2, 3, 4, 5, what is the interquartile range?
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Solution
Q1 = 2, Q3 = 4; Interquartile Range = Q3 - Q1 = 4 - 2 = 2.
Correct Answer:
B
— 2
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Q. For the data set: 5, 7, 8, 9, 10, what is the mean absolute deviation?
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Solution
Mean = 7.5; MAD = (|5-7.5| + |7-7.5| + |8-7.5| + |9-7.5| + |10-7.5|) / 5 = 1.
Correct Answer:
B
— 2
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Q. For the data set: 5, 7, 8, 9, 10, what is the standard deviation?
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Solution
Mean = 7.5; Variance = [(5-7.5)^2 + (7-7.5)^2 + (8-7.5)^2 + (9-7.5)^2 + (10-7.5)^2] / 5 = 2; Standard Deviation = sqrt(2) = 1.41
Correct Answer:
B
— 2
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Q. For the ellipse defined by the equation 9x^2 + 16y^2 = 144, what are the lengths of the semi-major and semi-minor axes?
A.
3, 4
B.
4, 3
C.
6, 8
D.
8, 6
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Solution
The semi-major axis is 4 and the semi-minor axis is 3 after rewriting the equation in standard form.
Correct Answer:
A
— 3, 4
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Q. For the function f(x) = 2x^3 - 9x^2 + 12x, find the inflection point.
A.
(1, 1)
B.
(2, 2)
C.
(3, 3)
D.
(4, 4)
Show solution
Solution
f''(x) = 12x - 18. Setting f''(x) = 0 gives x = 1.5. The inflection point is (1.5, f(1.5)).
Correct Answer:
B
— (2, 2)
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Q. For the function f(x) = 2x^3 - 9x^2 + 12x, find the intervals where the function is increasing.
A.
(-∞, 1)
B.
(1, 3)
C.
(3, ∞)
D.
(0, 3)
Show solution
Solution
f'(x) = 6x^2 - 18x + 12. Setting f'(x) = 0 gives x = 1 and x = 3. Testing intervals shows f is increasing on (1, 3).
Correct Answer:
B
— (1, 3)
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Q. For the function f(x) = 2x^3 - 9x^2 + 12x, find the local maxima.
A.
(1, 5)
B.
(2, 0)
C.
(3, 0)
D.
(0, 0)
Show solution
Solution
f'(x) = 6x^2 - 18x + 12. Setting f'(x) = 0 gives x = 1 and x = 2. f(1) = 5 is a local maximum.
Correct Answer:
A
— (1, 5)
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Q. For the function f(x) = 3x^2 - 12x + 7, find the coordinates of the vertex.
A.
(2, -5)
B.
(2, -1)
C.
(3, -2)
D.
(1, 1)
Show solution
Solution
The vertex is at x = -b/(2a) = 12/(2*3) = 2. f(2) = 3(2^2) - 12(2) + 7 = -1. So, the vertex is (2, -1).
Correct Answer:
B
— (2, -1)
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Q. For the function f(x) = 3x^3 - 12x^2 + 9, find the x-coordinates of the inflection points.
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Solution
f''(x) = 18x - 24. Setting f''(x) = 0 gives x = 4/3. This is the inflection point.
Correct Answer:
B
— 2
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Q. For the function f(x) = 3x^3 - 12x^2 + 9x, the number of local maxima and minima is:
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Solution
Finding f'(x) = 9x^2 - 24x + 9 and solving gives two critical points. The second derivative test confirms one maximum and one minimum.
Correct Answer:
C
— 2
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Q. For the function f(x) = e^x - x^2, the point of inflection occurs at:
A.
x = 0
B.
x = 1
C.
x = 2
D.
x = -1
Show solution
Solution
To find the point of inflection, we compute f''(x) = e^x - 2. Setting f''(x) = 0 gives e^x = 2, leading to x = ln(2). The closest integer is x = 1.
Correct Answer:
B
— x = 1
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Q. For the function f(x) = ln(x), find the point where it is not differentiable.
A.
x = 0
B.
x = 1
C.
x = -1
D.
x = 2
Show solution
Solution
f(x) = ln(x) is not defined for x ≤ 0, hence not differentiable at x = 0.
Correct Answer:
A
— x = 0
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Q. For the function f(x) = sin(x) + cos(x), find the x-coordinate of the maximum point in the interval [0, 2π].
A.
π/4
B.
3π/4
C.
5π/4
D.
7π/4
Show solution
Solution
f'(x) = cos(x) - sin(x). Setting f'(x) = 0 gives tan(x) = 1, so x = π/4 + nπ. In [0, 2π], the maximum occurs at x = 3π/4.
Correct Answer:
B
— 3π/4
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Q. For the function f(x) = x^2 + 2x + 1, what is f'(x)?
A.
2x + 1
B.
2x + 2
C.
2x
D.
x + 1
Show solution
Solution
f'(x) = 2x + 2.
Correct Answer:
B
— 2x + 2
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Q. For the function f(x) = x^2 + 2x + 3, find the point where it is not differentiable.
A.
x = -1
B.
x = 0
C.
x = 1
D.
It is differentiable everywhere
Show solution
Solution
The function is a polynomial and is differentiable everywhere.
Correct Answer:
D
— It is differentiable everywhere
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Q. For the function f(x) = x^2 + kx + 1 to be differentiable at x = -1, what must k be?
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Solution
Setting the derivative f'(-1) = 0 gives k = 1 for differentiability.
Correct Answer:
C
— 1
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Q. For the function f(x) = x^2 - 2x + 1, find the slope of the tangent line at x = 1.
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Solution
f'(x) = 2x - 2. Thus, f'(1) = 2(1) - 2 = 0.
Correct Answer:
A
— 0
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Q. For the function f(x) = x^2 - 4x + 4, find the point where it is not differentiable.
A.
x = 0
B.
x = 2
C.
x = 4
D.
It is differentiable everywhere
Show solution
Solution
As a polynomial, f(x) is differentiable everywhere, including at x = 2.
Correct Answer:
D
— It is differentiable everywhere
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Q. For the function f(x) = x^2 - 4x + 5, find the minimum value.
Show solution
Solution
The vertex occurs at x = 2. f(2) = 2^2 - 4*2 + 5 = 1, which is the minimum value.
Correct Answer:
B
— 2
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Showing 811 to 840 of 2847 (95 Pages)
Mathematics Syllabus (JEE Main) MCQ & Objective Questions
The Mathematics Syllabus for JEE Main is crucial for students aiming to excel in competitive exams. Understanding this syllabus not only helps in grasping key concepts but also enhances your ability to tackle objective questions effectively. Practicing MCQs and important questions from this syllabus is essential for solid exam preparation, ensuring you are well-equipped to score better in your exams.
What You Will Practise Here
Sets, Relations, and Functions
Complex Numbers and Quadratic Equations
Permutations and Combinations
Binomial Theorem
Sequences and Series
Limits and Derivatives
Statistics and Probability
Exam Relevance
The Mathematics Syllabus (JEE Main) is not only relevant for JEE but also appears in CBSE and State Board examinations. Students can expect a variety of question patterns, including direct MCQs, numerical problems, and conceptual questions. Mastery of this syllabus will prepare you for similar topics in NEET and other competitive exams, making it vital for your overall academic success.
Common Mistakes Students Make
Misinterpreting the questions, especially in word problems.
Overlooking the importance of units and dimensions in problems.
Confusing formulas related to sequences and series.
Neglecting to practice derivations, leading to errors in calculus.
Failing to apply the correct methods for solving probability questions.
FAQs
Question: What are the key topics in the Mathematics Syllabus for JEE Main? Answer: Key topics include Sets, Complex Numbers, Permutations, Binomial Theorem, and Calculus.
Question: How can I improve my performance in Mathematics MCQs? Answer: Regular practice of MCQs and understanding the underlying concepts are essential for improvement.
Now is the time to take charge of your exam preparation! Dive into solving practice MCQs and test your understanding of the Mathematics Syllabus (JEE Main). Your success is just a question away!
