Q. Find the coefficient of x^4 in the expansion of (3x - 2)^6.
A.
540
B.
720
C.
810
D.
960
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Solution
Using the binomial theorem, the coefficient of x^4 in (3x - 2)^6 is given by 6C4 * (3)^4 * (-2)^2 = 15 * 81 * 4 = 4860.
Correct Answer:
C
— 810
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Q. Find the coefficient of x^5 in the expansion of (x + 1)^8.
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Solution
The coefficient of x^5 is C(8,5) = 56.
Correct Answer:
B
— 70
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Q. Find the coefficient of x^5 in the expansion of (x + 3)^8.
A.
56
B.
168
C.
336
D.
672
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Solution
The coefficient of x^5 is C(8,5) * (3)^3 = 56 * 27 = 1512.
Correct Answer:
B
— 168
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Q. Find the coefficient of x^5 in the expansion of (x - 3)^7.
A.
-1890
B.
-2187
C.
-2401
D.
-2430
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Solution
The coefficient of x^5 is C(7,5) * (-3)^2 = 21 * 9 = -1890.
Correct Answer:
A
— -1890
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Q. Find the condition for the lines represented by the equation 2x^2 + 3xy + y^2 = 0 to be parallel.
A.
D = 0
B.
D > 0
C.
D < 0
D.
D = 1
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Solution
For the lines to be parallel, the discriminant D must be equal to 0.
Correct Answer:
A
— D = 0
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Q. Find the condition for the lines represented by the equation ax^2 + 2hxy + by^2 = 0 to be perpendicular.
A.
ab + h^2 = 0
B.
ab - h^2 = 0
C.
a + b = 0
D.
a - b = 0
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Solution
The condition for the lines to be perpendicular is given by the relation ab + h^2 = 0.
Correct Answer:
A
— ab + h^2 = 0
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Q. Find the condition for the lines represented by the equation ax^2 + 2hxy + by^2 = 0 to be parallel.
A.
h^2 = ab
B.
h^2 > ab
C.
h^2 < ab
D.
h^2 = 0
Show solution
Solution
The condition for the lines to be parallel is given by h^2 = ab.
Correct Answer:
A
— h^2 = ab
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Q. Find the conjugate of the complex number z = 5 - 6i.
A.
5 + 6i
B.
5 - 6i
C.
-5 + 6i
D.
-5 - 6i
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Solution
The conjugate of z = 5 - 6i is z̅ = 5 + 6i.
Correct Answer:
A
— 5 + 6i
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Q. Find the coordinates of the centroid of the triangle with vertices at (0, 0), (6, 0), and (3, 6).
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. Find the coordinates of the centroid of the triangle with vertices at (1, 2), (3, 4), and (5, 6).
A.
(3, 4)
B.
(2, 3)
C.
(4, 5)
D.
(5, 6)
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Solution
Centroid = ((1+3+5)/3, (2+4+6)/3) = (3, 4).
Correct Answer:
B
— (2, 3)
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Q. Find the coordinates of the focus of the parabola y^2 = -12x.
A.
(-3, 0)
B.
(-2, 0)
C.
(3, 0)
D.
(2, 0)
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Solution
The equation y^2 = -12x can be rewritten as (y - 0)^2 = 4p(x - 0) with p = -3, so the focus is at (-3, 0).
Correct Answer:
A
— (-3, 0)
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Q. Find the coordinates of the foot of the perpendicular from the point (1, 2) to the line 2x - 3y + 6 = 0.
A.
(0, 2)
B.
(1, 1)
C.
(2, 0)
D.
(3, -1)
Show solution
Solution
Using the formula for foot of perpendicular, we find the coordinates to be (1, 1).
Correct Answer:
B
— (1, 1)
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Q. Find the coordinates of the foot of the perpendicular from the point (3, 4) to the line 2x + 3y - 6 = 0.
A.
(2, 0)
B.
(1, 1)
C.
(0, 2)
D.
(3, 2)
Show solution
Solution
Using the formula for foot of perpendicular, we find the coordinates to be (3, 2).
Correct Answer:
D
— (3, 2)
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Q. Find the coordinates of the point on the curve y = x^3 - 3x + 2 where the slope of the tangent is 0.
A.
(1, 0)
B.
(0, 2)
C.
(2, 0)
D.
(3, 2)
Show solution
Solution
f'(x) = 3x^2 - 3. Setting f'(x) = 0 gives x^2 = 1, so x = 1 or x = -1. f(1) = 0, f(-1) = 4. The point is (1, 0).
Correct Answer:
A
— (1, 0)
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Q. Find the coordinates of the point on the curve y = x^3 - 3x + 2 where the tangent is horizontal.
A.
(0, 2)
B.
(1, 0)
C.
(2, 0)
D.
(3, 2)
Show solution
Solution
f'(x) = 3x^2 - 3. Setting f'(x) = 0 gives x = 1. The point is (1, 0).
Correct Answer:
B
— (1, 0)
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Q. Find the coordinates of the point where the function f(x) = 3x^2 - 12x + 9 has a local maximum.
A.
(2, 3)
B.
(3, 0)
C.
(1, 1)
D.
(0, 9)
Show solution
Solution
f'(x) = 6x - 12. Setting f'(x) = 0 gives x = 2. f(2) = 3(2^2) - 12(2) + 9 = 3.
Correct Answer:
A
— (2, 3)
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Q. Find the critical points of f(x) = x^3 - 3x^2 + 4.
A.
(0, 4)
B.
(1, 2)
C.
(2, 0)
D.
(3, 1)
Show solution
Solution
Setting f'(x) = 3x^2 - 6x = 0 gives x(x - 2) = 0, so critical points are x = 0 and x = 2. Evaluating f(1) = 2.
Correct Answer:
B
— (1, 2)
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Q. Find the critical points of the function f(x) = 3x^4 - 8x^3 + 6.
A.
(0, 6)
B.
(2, -2)
C.
(1, 1)
D.
(3, 0)
Show solution
Solution
f'(x) = 12x^3 - 24x^2. Setting f'(x) = 0 gives x^2(12x - 24) = 0, so x = 0 or x = 2. f(2) = 3(2^4) - 8(2^3) + 6 = -2.
Correct Answer:
B
— (2, -2)
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Q. Find the critical points of the function f(x) = x^3 - 6x^2 + 9x.
A.
(0, 0)
B.
(3, 0)
C.
(2, 0)
D.
(1, 0)
Show solution
Solution
f'(x) = 3x^2 - 12x + 9. Setting f'(x) = 0 gives x = 1 and x = 3. Critical points are (1, f(1)) and (3, f(3)).
Correct Answer:
B
— (3, 0)
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Q. Find the cross product of vectors A = (1, 2, 3) and B = (4, 5, 6).
A.
(-3, 6, -3)
B.
(0, 0, 0)
C.
(3, -6, 3)
D.
(1, -2, 1)
Show solution
Solution
Cross product A × B = |i j k| |1 2 3| |4 5 6| = (-3, 6, -3).
Correct Answer:
A
— (-3, 6, -3)
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Q. Find the derivative of f(x) = 1/x.
A.
-1/x^2
B.
1/x^2
C.
-2/x^2
D.
1/x
Show solution
Solution
Using the power rule, f'(x) = -1/x^2.
Correct Answer:
A
— -1/x^2
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Q. Find the derivative of f(x) = 3x^2 + 5x - 7.
A.
6x + 5
B.
3x + 5
C.
6x - 5
D.
3x^2 + 5
Show solution
Solution
Using the power rule, f'(x) = d/dx(3x^2) + d/dx(5x) - d/dx(7) = 6x + 5.
Correct Answer:
A
— 6x + 5
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Q. Find the derivative of f(x) = 5x^4 - 3x + 2.
A.
20x^3 - 3
B.
15x^3 - 3
C.
20x^4 - 3
D.
5x^3 - 3
Show solution
Solution
Using the power rule, f'(x) = 20x^3 - 3.
Correct Answer:
A
— 20x^3 - 3
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Q. Find the derivative of f(x) = e^(2x) at x = 0.
Show solution
Solution
f'(x) = 2e^(2x). At x = 0, f'(0) = 2e^0 = 2.
Correct Answer:
B
— 2
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Q. Find the derivative of f(x) = e^(2x).
A.
2e^(2x)
B.
e^(2x)
C.
2xe^(2x)
D.
e^(x)
Show solution
Solution
Using the chain rule, f'(x) = 2e^(2x).
Correct Answer:
A
— 2e^(2x)
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Q. Find the derivative of f(x) = e^(x^2).
A.
2xe^(x^2)
B.
e^(x^2)
C.
x e^(x^2)
D.
2e^(x^2)
Show solution
Solution
Using the chain rule, f'(x) = e^(x^2) * 2x = 2x e^(x^2).
Correct Answer:
A
— 2xe^(x^2)
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Q. Find the derivative of f(x) = e^x * ln(x) at x = 1.
Show solution
Solution
Using the product rule, f'(x) = e^x * ln(x) + e^x/x. At x = 1, this simplifies to 0.
Correct Answer:
A
— 1
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Q. Find the derivative of f(x) = e^x * sin(x) at x = 0.
Show solution
Solution
Using the product rule, f'(0) = e^0 * sin(0) + e^0 * cos(0) = 0 + 1 = 1.
Correct Answer:
A
— 1
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Q. Find the derivative of f(x) = ln(x^2 + 1) at x = 1.
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Solution
f'(x) = (2x)/(x^2 + 1). At x = 1, f'(1) = (2*1)/(1^2 + 1) = 2/2 = 1.
Correct Answer:
B
— 1
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Q. Find the derivative of f(x) = ln(x^2 + 1).
A.
2x/(x^2 + 1)
B.
1/(x^2 + 1)
C.
2/(x^2 + 1)
D.
x/(x^2 + 1)
Show solution
Solution
Using the chain rule, f'(x) = d/dx(ln(x^2 + 1)) = (2x)/(x^2 + 1).
Correct Answer:
A
— 2x/(x^2 + 1)
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Showing 481 to 510 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!
