Q. The coordinates of the centroid of a triangle with vertices at (0, 0), (6, 0), and (3, 6) are:
A.
(3, 2)
B.
(3, 3)
C.
(2, 3)
D.
(0, 0)
Show solution
Solution
Centroid = ((x1+x2+x3)/3, (y1+y2+y3)/3) = (9/3, 6/3) = (3, 2).
Correct Answer:
B
— (3, 3)
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Q. The coordinates of the centroid of a triangle with vertices at (2, 3), (4, 5), and (6, 1) are:
A.
(4, 3)
B.
(4, 4)
C.
(3, 3)
D.
(5, 3)
Show solution
Solution
Centroid = ((2+4+6)/3, (3+5+1)/3) = (4, 3).
Correct Answer:
A
— (4, 3)
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Q. The coordinates of the centroid of the triangle with vertices (0, 0), (6, 0), and (3, 6) are:
A.
(3, 2)
B.
(2, 3)
C.
(3, 3)
D.
(0, 0)
Show solution
Solution
Centroid = ((0+6+3)/3, (0+0+6)/3) = (3, 2).
Correct Answer:
A
— (3, 2)
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Q. The coordinates of the centroid of the triangle with vertices (2, 3), (4, 5), and (6, 7) are:
A.
(4, 5)
B.
(3, 4)
C.
(5, 6)
D.
(6, 5)
Show solution
Solution
Centroid = ((2+4+6)/3, (3+5+7)/3) = (4, 5).
Correct Answer:
B
— (3, 4)
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Q. The critical points of the function f(x) = x^3 - 6x^2 + 9x + 1 are:
A.
x = 1, 3
B.
x = 0, 2
C.
x = 2, 4
D.
x = 1, 2
Show solution
Solution
Finding f'(x) = 3x^2 - 12x + 9 and solving gives critical points at x = 1 and x = 3.
Correct Answer:
A
— x = 1, 3
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Q. The distance from the point (1, 2) to the line 2x + 3y - 6 = 0 is:
Show solution
Solution
Distance = |2(1) + 3(2) - 6| / √(2² + 3²) = |2 + 6 - 6| / √13 = 2/√13.
Correct Answer:
B
— 2
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Q. The distance from the point (3, 4) to the line 2x + 3y - 6 = 0 is:
Show solution
Solution
Distance = |2(3) + 3(4) - 6| / √(2² + 3²) = |6 + 12 - 6| / √13 = 12/√13.
Correct Answer:
B
— 2
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Q. The eccentricity of an ellipse is defined as e = c/a. If a = 10 and c = 6, what is the eccentricity?
A.
0.6
B.
0.8
C.
0.4
D.
0.5
Show solution
Solution
Eccentricity e = c/a = 6/10 = 0.6.
Correct Answer:
B
— 0.8
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Q. The equation of a line parallel to y = 2x + 3 and passing through (1, 1) is?
A.
y = 2x - 1
B.
y = 2x + 1
C.
y = 2x + 3
D.
y = 2x - 3
Show solution
Solution
Parallel lines have the same slope. Using point-slope form: y - 1 = 2(x - 1) => y = 2x - 1.
Correct Answer:
A
— y = 2x - 1
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Q. The equation of a line passing through (1, 2) and (3, 6) is:
A.
y = 2x
B.
y = 3x - 1
C.
y = x + 1
D.
y = 4x - 2
Show solution
Solution
Slope = (6-2)/(3-1) = 2. Using point-slope form: y - 2 = 2(x - 1) => y = 2x.
Correct Answer:
A
— y = 2x
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Q. The equation of a line passing through the points (1, 2) and (3, 6) is:
A.
y = 2x
B.
y = 3x - 1
C.
y = x + 1
D.
y = 4x - 2
Show solution
Solution
Slope = (6-2)/(3-1) = 2. Using point-slope form: y - 2 = 2(x - 1) => y = 2x.
Correct Answer:
A
— y = 2x
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Q. The equation of a parabola is given by x^2 = 16y. What is the length of the latus rectum?
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Solution
The length of the latus rectum for the parabola x^2 = 4py is given by 4p. Here, 4p = 16, so p = 4. Thus, the length of the latus rectum is 4p = 16.
Correct Answer:
B
— 8
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Q. The equation of an ellipse is given by 4x^2 + 9y^2 = 36. What is the eccentricity of the ellipse?
A.
0.5
B.
0.6
C.
0.7
D.
0.8
Show solution
Solution
Rewriting gives x^2/9 + y^2/4 = 1. Here, a^2 = 9, b^2 = 4, c = √(a^2 - b^2) = √(9 - 4) = √5. Eccentricity e = c/a = √5/3 ≈ 0.6.
Correct Answer:
B
— 0.6
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Q. The equation of an ellipse with foci at (0, ±c) and major axis along the y-axis is given by?
A.
x^2/a^2 + y^2/b^2 = 1
B.
y^2/a^2 + x^2/b^2 = 1
C.
x^2/b^2 + y^2/a^2 = 1
D.
y^2/b^2 + x^2/a^2 = 1
Show solution
Solution
The equation of an ellipse with foci at (0, ±c) and major axis along the y-axis is y^2/a^2 + x^2/b^2 = 1.
Correct Answer:
B
— y^2/a^2 + x^2/b^2 = 1
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Q. The equation of the directrix of the parabola y^2 = 8x is?
A.
x = -2
B.
x = 2
C.
y = -4
D.
y = 4
Show solution
Solution
The directrix of the parabola y^2 = 8x is given by x = -2.
Correct Answer:
A
— x = -2
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Q. The equation of the line passing through (1, 2) and (3, 6) is:
A.
y = 2x
B.
y = 3x - 1
C.
y = x + 1
D.
y = 4x - 2
Show solution
Solution
Slope = (6-2)/(3-1) = 2. Using point-slope form: y - 2 = 2(x - 1) => y = 2x.
Correct Answer:
A
— y = 2x
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Q. The equation of the line passing through the points (1, 2) and (3, 6) is:
A.
y = 2x
B.
y = 3x - 1
C.
y = 4x - 2
D.
y = x + 1
Show solution
Solution
Slope = (6-2)/(3-1) = 2. Using point-slope form: y - 2 = 2(x - 1) => y = 2x.
Correct Answer:
A
— y = 2x
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Q. The equation of the pair of lines through the origin is given by y = mx. If m1 and m2 are the slopes, what is the condition for them to be perpendicular?
A.
m1 + m2 = 0
B.
m1 * m2 = 1
C.
m1 - m2 = 0
D.
m1 * m2 = -1
Show solution
Solution
For two lines to be perpendicular, the product of their slopes must equal -1.
Correct Answer:
D
— m1 * m2 = -1
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Q. The equation of the pair of lines through the origin with slopes m1 and m2 is given by:
A.
y = mx
B.
y^2 = mx
C.
x^2 + y^2 = 0
D.
x^2 - 2mxy + y^2 = 0
Show solution
Solution
The correct form of the equation representing the lines through the origin is x^2 - 2mxy + y^2 = 0.
Correct Answer:
D
— x^2 - 2mxy + y^2 = 0
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Q. The equation of the pair of lines through the origin with slopes m1 and m2 is:
A.
y = m1x + m2x
B.
y = (m1 + m2)x
C.
y = m1x - m2x
D.
y = m1x * m2x
Show solution
Solution
The equation of the lines can be expressed as y = (m1 + m2)x, representing the sum of the slopes.
Correct Answer:
B
— y = (m1 + m2)x
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Q. The equation of the tangent line to the curve y = x^2 at the point (2, 4) is:
A.
y = 2x
B.
y = 4x - 4
C.
y = 4x - 8
D.
y = x + 2
Show solution
Solution
The slope of the tangent at x = 2 is f'(x) = 2x, so f'(2) = 4. The equation of the tangent line is y - 4 = 4(x - 2), which simplifies to y = 4x - 8.
Correct Answer:
C
— y = 4x - 8
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Q. The equation of the tangent to the curve y = x^2 at the point (2, 4) is:
A.
y = 2x - 4
B.
y = 2x
C.
y = x + 2
D.
y = x^2 - 2
Show solution
Solution
The derivative f'(x) = 2x. At x = 2, f'(2) = 4. The equation of the tangent line is y - 4 = 4(x - 2), which simplifies to y = 2x - 4.
Correct Answer:
A
— y = 2x - 4
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Q. The equation x^2 + 2x + 1 = 0 can be factored as:
A.
(x + 1)(x + 1)
B.
(x - 1)(x - 1)
C.
(x + 2)(x + 1)
D.
(x - 2)(x - 1)
Show solution
Solution
This is a perfect square: (x + 1)^2 = 0.
Correct Answer:
A
— (x + 1)(x + 1)
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Q. The equation x^2 + 4x + 4 = 0 has:
A.
Two distinct roots
B.
One repeated root
C.
No real roots
D.
None of these
Show solution
Solution
The discriminant is 0, indicating one repeated root.
Correct Answer:
B
— One repeated root
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Q. The equation x^2 - 2x + 1 = 0 has:
A.
Two distinct roots
B.
One repeated root
C.
No real roots
D.
Infinitely many roots
Show solution
Solution
The discriminant is 0, indicating one repeated root.
Correct Answer:
B
— One repeated root
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Q. The equation x^2 - 6x + k = 0 has roots that are both positive. What is the range of k?
A.
k < 0
B.
k > 0
C.
k > 9
D.
k < 9
Show solution
Solution
For both roots to be positive, k must be greater than the square of half the coefficient of x: k > (6/2)^2 = 9.
Correct Answer:
C
— k > 9
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Q. The family of curves defined by the equation x^2 + y^2 = r^2 represents:
A.
Ellipses
B.
Hyperbolas
C.
Circles
D.
Parabolas
Show solution
Solution
The equation x^2 + y^2 = r^2 represents a circle with radius r.
Correct Answer:
C
— Circles
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Q. The family of curves defined by the equation y = a(x - h)^2 + k represents which type of function?
A.
Linear
B.
Quadratic
C.
Cubic
D.
Rational
Show solution
Solution
The equation y = a(x - h)^2 + k represents a quadratic function in vertex form.
Correct Answer:
B
— Quadratic
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Q. The family of curves defined by the equation y = a(x - h)^2 + k represents:
A.
Parabolas
B.
Circles
C.
Ellipses
D.
Hyperbolas
Show solution
Solution
The equation y = a(x - h)^2 + k represents a family of parabolas with vertex (h, k).
Correct Answer:
A
— Parabolas
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Q. The family of curves defined by the equation y = ax^2 + bx + c is known as:
A.
Linear functions
B.
Quadratic functions
C.
Polynomial functions
D.
Rational functions
Show solution
Solution
The equation y = ax^2 + bx + c represents a quadratic function.
Correct Answer:
B
— Quadratic functions
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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!
