Q. Find the solution set for the inequality 6 - 3x ≤ 0.
A.
x ≥ 2
B.
x < 2
C.
x ≤ 2
D.
x > 2
Show solution
Solution
6 - 3x ≤ 0 => -3x ≤ -6 => x ≥ 2.
Correct Answer:
A
— x ≥ 2
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Q. Find the solution set for the inequality 7 - 3x > 1.
A.
x < 2
B.
x > 2
C.
x < 3
D.
x > 3
Show solution
Solution
7 - 3x > 1 => -3x > -6 => x < 2.
Correct Answer:
B
— x > 2
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Q. Find the solution set for the inequality 8x + 1 ≤ 5.
A.
x ≤ 0.5
B.
x < 0.5
C.
x ≥ 0.5
D.
x > 0.5
Show solution
Solution
8x + 1 ≤ 5 => 8x ≤ 4 => x ≤ 0.5.
Correct Answer:
A
— x ≤ 0.5
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Q. Find the solution set for the inequality x + 2 > 3.
A.
x > 1
B.
x < 1
C.
x > -1
D.
x < -1
Show solution
Solution
x + 2 > 3 => x > 1.
Correct Answer:
A
— x > 1
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Q. Find the solutions of the equation 2sin(x) + √3 = 0.
A.
x = 5π/6
B.
x = 7π/6
C.
x = π/6
D.
x = 11π/6
Show solution
Solution
Solving gives sin(x) = -√3/2, so x = 7π/6 and 11π/6.
Correct Answer:
B
— x = 7π/6
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Q. Find the solutions of the equation 2sin(x) - 1 = 0 in the interval [0, 2π].
A.
π/6, 5π/6
B.
π/4, 3π/4
C.
π/3, 2π/3
D.
π/2, 3π/2
Show solution
Solution
The solutions are x = π/6 and x = 5π/6.
Correct Answer:
A
— π/6, 5π/6
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Q. Find the solutions of the equation 2sin(x) - 1 = 0.
A.
π/6
B.
5π/6
C.
7π/6
D.
11π/6
Show solution
Solution
The solutions are x = π/6 and x = 5π/6.
Correct Answer:
A
— π/6
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Q. Find the sum of the roots of the equation 3x^2 - 12x + 9 = 0.
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Solution
The sum of the roots is given by -b/a = 12/3 = 4.
Correct Answer:
B
— 4
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Q. Find the unit vector in the direction of the vector (3, 4).
A.
(0.6, 0.8)
B.
(0.8, 0.6)
C.
(1, 1)
D.
(0.5, 0.5)
Show solution
Solution
Magnitude = √(3^2 + 4^2) = 5. Unit vector = (3/5, 4/5) = (0.6, 0.8).
Correct Answer:
A
— (0.6, 0.8)
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Q. Find the unit vector in the direction of the vector (3, 4, 0).
A.
(0.6, 0.8, 0)
B.
(0.3, 0.4, 0)
C.
(1, 1, 0)
D.
(0, 0, 1)
Show solution
Solution
Magnitude = √(3^2 + 4^2) = 5. Unit vector = (3/5, 4/5, 0) = (0.6, 0.8, 0).
Correct Answer:
A
— (0.6, 0.8, 0)
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Q. Find the unit vector in the direction of the vector (4, 3).
A.
(4/5, 3/5)
B.
(3/5, 4/5)
C.
(1, 0)
D.
(0, 1)
Show solution
Solution
Unit vector = (4, 3) / √(4^2 + 3^2) = (4, 3) / 5 = (4/5, 3/5).
Correct Answer:
A
— (4/5, 3/5)
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Q. Find the unit vector in the direction of the vector (6, 8).
A.
(0.6, 0.8)
B.
(0.8, 0.6)
C.
(1, 1)
D.
(0.5, 0.5)
Show solution
Solution
Magnitude = √(6^2 + 8^2) = √(36 + 64) = √100 = 10. Unit vector = (6/10, 8/10) = (0.6, 0.8).
Correct Answer:
A
— (0.6, 0.8)
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Q. Find the unit vector in the direction of the vector v = (4, -3).
A.
(4/5, -3/5)
B.
(3/5, 4/5)
C.
(4/3, -3/4)
D.
(3/4, 4/3)
Show solution
Solution
Magnitude |v| = √(4^2 + (-3)^2) = √(16 + 9) = 5. Unit vector = (4/5, -3/5).
Correct Answer:
A
— (4/5, -3/5)
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Q. Find the value of (1 + 2)^4 using the binomial theorem.
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Solution
Using the binomial theorem, (1 + 2)^4 = C(4,0) * 1^4 * 2^0 + C(4,1) * 1^3 * 2^1 + C(4,2) * 1^2 * 2^2 + C(4,3) * 1^1 * 2^3 + C(4,4) * 1^0 * 2^4 = 1 + 8 + 24 + 32 + 16 = 81.
Correct Answer:
A
— 16
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Q. Find the value of (1 + i)^2.
Show solution
Solution
(1 + i)^2 = 1^2 + 2(1)(i) + i^2 = 1 + 2i - 1 = 2i.
Correct Answer:
B
— 2
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Q. Find the value of (1 + i)^4.
Show solution
Solution
(1 + i)^4 = (√2 e^(iπ/4))^4 = 4 e^(iπ) = 4(-1) = -4.
Correct Answer:
C
— 8
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Q. Find the value of (1 + x)^10 at x = 1. (2048)
A.
10
B.
11
C.
1024
D.
2048
Show solution
Solution
Using the binomial theorem, (1 + 1)^10 = 2^10 = 1024.
Correct Answer:
C
— 1024
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Q. Find the value of (1 + x)^10 at x = 2.
A.
1024
B.
2048
C.
512
D.
256
Show solution
Solution
Using the binomial theorem, (1 + 2)^10 = 3^10 = 59049.
Correct Answer:
B
— 2048
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Q. Find the value of a for which the function f(x) = { ax + 1, x < 1; 2, x = 1; x^2 + a, x > 1 is continuous at x = 1.
Show solution
Solution
Setting ax + 1 = 2 and x^2 + a = 2 at x = 1 gives a = 0.
Correct Answer:
A
— 0
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Q. Find the value of a for which the function f(x) = { ax + 1, x < 1; 3, x = 1; 2x + a, x > 1 is continuous at x = 1.
Show solution
Solution
Setting ax + 1 = 3 and 2x + a = 3 at x = 1 gives a = 2.
Correct Answer:
A
— 1
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Q. Find the value of a for which the function f(x) = { ax + 1, x < 2; 3x - 5, x >= 2 } is continuous at x = 2.
Show solution
Solution
Setting the two pieces equal at x = 2 gives us 2a + 1 = 1. Solving for a gives a = 0.
Correct Answer:
C
— 3
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Q. Find the value of a for which the function f(x) = { ax + 1, x < 2; x^2 - 3, x >= 2 } is continuous at x = 2.
Show solution
Solution
Setting the two pieces equal at x = 2: 2a + 1 = 2^2 - 3. Solving gives a = 2.
Correct Answer:
C
— 3
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Q. Find the value of a for which the function f(x) = { ax + 1, x < 2; x^2 - 4, x >= 2 } is differentiable at x = 2.
Show solution
Solution
Set the left-hand limit equal to the right-hand limit and their derivatives at x = 2.
Correct Answer:
B
— 1
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Q. Find the value of a for which the function f(x) = { x^2 + a, x < 1; 3, x = 1; 2x + 1, x > 1 is continuous at x = 1.
Show solution
Solution
Setting the left limit (1 + a) equal to the right limit (3), we find a = 2.
Correct Answer:
A
— -1
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Q. Find the value of b for which the function f(x) = { x^2 + b, x < 1; 2x + 3, x >= 1 is continuous at x = 1.
Show solution
Solution
Setting 1 + b = 2 + 3 gives b = 4.
Correct Answer:
C
— 2
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Q. Find the value of b for which the function f(x) = { x^2 + b, x < 1; 3x - 1, x >= 1 is continuous at x = 1.
Show solution
Solution
Setting 1 + b = 2 gives b = 1.
Correct Answer:
A
— -1
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Q. Find the value of c such that the function f(x) = { x^2 + c, x < 1; 2x + 1, x >= 1 } is differentiable at x = 1.
Show solution
Solution
Setting the left-hand limit equal to the right-hand limit gives c = 1.
Correct Answer:
B
— 1
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Q. Find the value of c such that the function f(x) = { x^2 + c, x < 2; 4, x >= 2 } is continuous at x = 2.
Show solution
Solution
Setting the two pieces equal at x = 2 gives 4 = 4 + c, hence c = 0.
Correct Answer:
B
— 2
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Q. Find the value of c such that the function f(x) = { x^3 - 3x + 2, x < 1; c, x = 1; x^2 + 1, x > 1 is continuous at x = 1.
Show solution
Solution
To ensure continuity at x = 1, we set the left limit (1 - 3 + 2 = 0) equal to the right limit (1 + 1 = 2), leading to c = 2.
Correct Answer:
C
— 2
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Q. Find the value of c such that the function f(x) = { x^3 - 3x + 2, x < c; 4, x = c; 2x - 1, x > c is continuous at x = c.
Show solution
Solution
Setting limit as x approaches c from left equal to 4 and from right gives c = 1.
Correct Answer:
A
— 1
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Showing 691 to 720 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!
