Q. For the parabola y^2 = 20x, what is the coordinates of the vertex?
A.
(0, 0)
B.
(5, 0)
C.
(0, 5)
D.
(10, 0)
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Solution
The vertex of the parabola y^2 = 4px is at (0, 0). Here, p = 5, but the vertex remains at (0, 0).
Correct Answer:
A
— (0, 0)
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Q. For the quadratic equation 2x^2 - 4x + k = 0 to have real roots, what is the condition on k?
A.
k >= 0
B.
k <= 0
C.
k >= 2
D.
k <= 2
Show solution
Solution
The discriminant must be non-negative: (-4)^2 - 4*2*k >= 0, which simplifies to k <= 2.
Correct Answer:
C
— k >= 2
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Q. For the quadratic equation ax^2 + bx + c = 0, if a = 1, b = -3, and c = 2, what are the roots?
A.
1 and 2
B.
2 and 1
C.
3 and 0
D.
0 and 3
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Solution
The roots can be found using the quadratic formula: x = (3 ± √(9-8))/2 = 1 and 2.
Correct Answer:
A
— 1 and 2
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Q. For the quadratic equation x^2 + 2x + 1 = 0, what is the nature of the roots?
A.
Real and distinct
B.
Real and equal
C.
Complex
D.
None of the above
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Solution
The discriminant is 0, indicating that the roots are real and equal.
Correct Answer:
B
— Real and equal
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Q. For the quadratic equation x^2 + 2x + 1 = 0, what is the vertex of the parabola?
A.
(-1, 0)
B.
(-1, 1)
C.
(0, 1)
D.
(1, 1)
Show solution
Solution
The vertex can be found using the formula (-b/2a, f(-b/2a)). Here, vertex is (-1, 0).
Correct Answer:
A
— (-1, 0)
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Q. For the quadratic equation x^2 + 2x + k = 0 to have no real roots, k must be:
A.
< 0
B.
≥ 0
C.
≤ 0
D.
> 0
Show solution
Solution
The discriminant must be negative: 2^2 - 4*1*k < 0 => 4 < 4k => k > 1.
Correct Answer:
A
— < 0
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Q. For the quadratic equation x^2 + 4x + 4 = 0, what is the nature of the roots?
A.
Real and distinct
B.
Real and equal
C.
Complex
D.
None of the above
Show solution
Solution
The discriminant is 0 (b^2 - 4ac = 16 - 16 = 0), indicating real and equal roots.
Correct Answer:
B
— Real and equal
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Q. For the quadratic equation x^2 + 4x + k = 0 to have no real roots, k must be:
Show solution
Solution
The discriminant must be negative: 4^2 - 4*1*k < 0 => 16 < 4k => k > 4.
Correct Answer:
A
— 0
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Q. For the quadratic equation x^2 + 4x + k = 0 to have real roots, what is the condition on k?
A.
k >= 4
B.
k <= 4
C.
k > 0
D.
k < 0
Show solution
Solution
The discriminant must be non-negative: 4^2 - 4*1*k >= 0, which simplifies to k <= 4.
Correct Answer:
A
— k >= 4
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Q. For the quadratic equation x^2 + 6x + 8 = 0, what are the roots?
A.
-2 and -4
B.
-4 and -2
C.
2 and 4
D.
0 and 8
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Solution
Factoring gives (x+2)(x+4) = 0, hence the roots are -2 and -4.
Correct Answer:
B
— -4 and -2
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Q. For the quadratic equation x^2 + 6x + 9 = 0, what is the nature of the roots?
A.
Two distinct real roots
B.
One real root
C.
No real roots
D.
Complex roots
Show solution
Solution
The discriminant is 0, indicating one real root (a repeated root).
Correct Answer:
B
— One real root
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Q. For the quadratic equation x^2 + mx + n = 0, if the roots are 2 and 3, what is the value of n?
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Solution
The product of the roots is n = 2 * 3 = 6.
Correct Answer:
B
— 6
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Q. For the quadratic equation x^2 + px + q = 0, if the roots are 1 and -3, what is the value of p?
Show solution
Solution
The sum of the roots is 1 + (-3) = -2, hence p = -2.
Correct Answer:
A
— 2
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Q. For the quadratic equation x^2 - 10x + 25 = 0, what is the double root?
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Solution
The equation can be factored as (x-5)^2 = 0, hence the double root is x = 5.
Correct Answer:
A
— 5
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Q. For the quadratic equation x^2 - 6x + k = 0 to have equal roots, what must be the value of k?
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Solution
Setting the discriminant to zero: (-6)^2 - 4*1*k = 0 gives k = 9.
Correct Answer:
B
— 9
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Q. For the set E = {1, 2, 3, 4}, how many subsets contain the element 1?
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Solution
If 1 is included, we can choose from the remaining elements {2, 3, 4}, which has 2^3 = 8 subsets.
Correct Answer:
C
— 12
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Q. For the set E = {1, 2, 3, 4}, how many subsets have exactly 2 elements?
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Solution
The number of ways to choose 2 elements from 4 is given by the combination formula C(4,2) = 6.
Correct Answer:
B
— 6
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Q. For the set F = {a, b, c}, how many subsets have exactly 2 elements?
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Solution
The subsets with exactly 2 elements are {a, b}, {a, c}, and {b, c}. Total = 3.
Correct Answer:
C
— 3
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Q. For the vectors A = (1, 0, 0) and B = (0, 1, 0), what is the scalar product A · B?
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Solution
A · B = 1*0 + 0*1 + 0*0 = 0.
Correct Answer:
A
— 0
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Q. For vectors A = (2, 3) and B = (4, 5), find the scalar product A · B.
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Solution
A · B = 2*4 + 3*5 = 8 + 15 = 23.
Correct Answer:
A
— 23
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Q. For vectors A = (3, -2, 1) and B = (1, 4, -2), find A · B.
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Solution
A · B = 3*1 + (-2)*4 + 1*(-2) = 3 - 8 - 2 = -7.
Correct Answer:
A
— -1
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Q. For what value of b is the function f(x) = { x^3 - 3x + b, x < 1; 2x + 1, x >= 1 continuous at x = 1?
Show solution
Solution
Setting 1^3 - 3(1) + b = 2(1) + 1 gives b = 2.
Correct Answer:
B
— 1
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Q. For which value of a is the function f(x) = x^2 + ax + 1 differentiable at x = -1?
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Solution
To ensure differentiability at x = -1, we find f'(-1) exists. Setting a = 0 ensures the derivative is defined.
Correct Answer:
B
— 0
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Q. For which value of a is the function f(x) = x^2 + ax + 1 differentiable everywhere?
Show solution
Solution
The function is a polynomial and is differentiable for all real numbers, hence any value of a works.
Correct Answer:
B
— 0
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Q. For which value of a is the function f(x) = x^2 - ax + 2 differentiable at x = 1?
Show solution
Solution
Setting the derivative f'(1) = 0 gives a = 1 for differentiability.
Correct Answer:
B
— 1
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Q. For which value of a is the function f(x) = x^2 - ax + 4 differentiable at x = 2?
Show solution
Solution
f(x) is a polynomial and is differentiable for all a, hence any value of a works.
Correct Answer:
A
— 0
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Q. For which value of a is the function f(x) = x^3 - 3ax + 2 differentiable at x = 1?
Show solution
Solution
Setting f'(1) = 0 gives a = 1, ensuring differentiability at that point.
Correct Answer:
B
— 1
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Q. For which value of a is the function f(x) = x^3 - 3ax^2 + 3a^2x + 1 differentiable at x = 1?
Show solution
Solution
Setting f'(1) = 0 gives a = 1 for differentiability at x = 1.
Correct Answer:
B
— 1
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Q. For which value of a is the function f(x) = { 2x + a, x < 0; x^2 + 1, x >= 0 continuous at x = 0?
Show solution
Solution
Setting a = 1 gives continuity at x = 0.
Correct Answer:
B
— 0
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Q. For which value of a is the function f(x) = { 3x + a, x < 2; 4x - 1, x >= 2 continuous at x = 2?
Show solution
Solution
Setting 3(2) + a = 4(2) - 1 gives a = 1.
Correct Answer:
A
— -1
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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!
