Q. For the function f(x) = x^2 - 4x + 5, find the vertex.
A.
(2, 1)
B.
(2, 5)
C.
(4, 1)
D.
(4, 5)
Show solution
Solution
The vertex is at x = -b/(2a) = 4/2 = 2. f(2) = 2^2 - 4(2) + 5 = 1, so the vertex is (2, 1).
Correct Answer:
A
— (2, 1)
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Q. For the function f(x) = x^2 - 6x + 8, find the x-coordinate of the vertex.
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Solution
The x-coordinate of the vertex is given by x = -b/(2a) = 6/(2*1) = 3.
Correct Answer:
B
— 3
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Q. For the function f(x) = x^3 - 3x^2 + 2, find the points where it is not differentiable.
A.
None
B.
x = 0
C.
x = 1
D.
x = 2
Show solution
Solution
As a polynomial, f(x) is differentiable everywhere, hence no points of non-differentiability.
Correct Answer:
A
— None
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Q. For the function f(x) = x^3 - 3x^2 + 4, find the points where it is not differentiable.
A.
None
B.
x = 0
C.
x = 1
D.
x = 2
Show solution
Solution
The function is a polynomial and is differentiable everywhere, hence there are no points where it is not differentiable.
Correct Answer:
A
— None
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Q. For the function f(x) = x^3 - 3x^2 + 4, find the value of x where f is not differentiable.
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Solution
The function is a polynomial and is differentiable everywhere, so there is no such x.
Correct Answer:
A
— 0
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Q. For the function f(x) = x^3 - 3x^2 + 4, find the x-coordinate of the point where f is differentiable.
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Solution
f(x) is a polynomial and is differentiable everywhere. The x-coordinate can be any real number.
Correct Answer:
C
— 1
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Q. For the function f(x) = x^3 - 6x^2 + 9x, find the critical points.
A.
x = 0, 3
B.
x = 1, 2
C.
x = 2, 3
D.
x = 3, 4
Show solution
Solution
First, find f'(x) = 3x^2 - 12x + 9. Setting f'(x) = 0 gives (x - 3)(x - 1) = 0, so critical points are x = 1 and x = 3.
Correct Answer:
A
— x = 0, 3
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Q. For the function f(x) = x^3 - 6x^2 + 9x, find the intervals where the function is increasing.
A.
(-∞, 0)
B.
(0, 3)
C.
(3, ∞)
D.
(0, 6)
Show solution
Solution
f'(x) = 3x^2 - 12x + 9. The critical points are x = 1 and x = 3. The function is increasing on (1, 3) and (3, ∞).
Correct Answer:
B
— (0, 3)
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Q. For the function f(x) = x^4 - 8x^2 + 16, find the coordinates of the inflection point.
A.
(0, 16)
B.
(2, 0)
C.
(4, 0)
D.
(2, 4)
Show solution
Solution
Find f''(x) = 12x^2 - 16. Setting f''(x) = 0 gives x^2 = 4, so x = ±2. f(2) = 0, thus the inflection point is (2, 0).
Correct Answer:
B
— (2, 0)
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Q. For the function f(x) = x^4 - 8x^2 + 16, find the intervals where the function is increasing.
A.
(-∞, -2)
B.
(-2, 2)
C.
(2, ∞)
D.
(-2, ∞)
Show solution
Solution
f'(x) = 4x^3 - 16x. Setting f'(x) = 0 gives x(x^2 - 4) = 0, so x = -2, 0, 2. Test intervals: f' is positive in (-2, ∞).
Correct Answer:
D
— (-2, ∞)
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Q. For the function f(x) = { x^2 + 1, x < 0; 2x + b, x = 0; 3 - x, x > 0 to be continuous at x = 0, what is b?
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Solution
Setting the left limit (0 + 1 = 1) equal to the right limit (3 - 0 = 3), we find b = 1.
Correct Answer:
B
— 0
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Q. For the function f(x) = { x^2, x < 1; 3, x = 1; 2x, x > 1 }, what is the value of f(1)?
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Solution
By definition, f(1) = 3, as given in the piecewise function.
Correct Answer:
C
— 3
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Q. For the function f(x) = { x^2, x < 1; kx + 1, x >= 1 }, find k such that f is differentiable at x = 1.
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Solution
Setting f(1-) = f(1+) and f'(1-) = f'(1+) gives k = 2 for differentiability.
Correct Answer:
B
— 1
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Q. For the function f(x) = { x^2, x < 3; 9, x = 3; 3x, x > 3 } to be continuous at x = 3, the value of f(3) must be:
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Solution
For continuity, f(3) must equal the limit as x approaches 3, which is 9.
Correct Answer:
B
— 9
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Q. For the function f(x) = |x - 2| + |x + 3|, find the point where it is not differentiable.
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Solution
The function is not differentiable at x = -3 and x = 2, but the first point of interest is -3.
Correct Answer:
A
— -3
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Q. For the hyperbola x^2/25 - y^2/16 = 1, what is the distance between the foci?
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Solution
The distance between the foci of the hyperbola is 2c, where c = √(a^2 + b^2) = √(25 + 16) = √41, so the distance is 2√41.
Correct Answer:
A
— 10
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Q. For the lines represented by the equation 2x^2 + 3xy + y^2 = 0, find the slopes of the lines.
A.
-3/2, -1
B.
1, -1/3
C.
0, -1
D.
1, 1
Show solution
Solution
The slopes can be found by solving the quadratic equation derived from the given equation.
Correct Answer:
A
— -3/2, -1
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Q. For the lines represented by the equation 2x^2 + 3xy + y^2 = 0, what is the product of the slopes?
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Solution
The product of the slopes of the lines can be found from the equation, which gives -1.
Correct Answer:
A
— -1
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Q. For the lines represented by the equation 2x^2 + 3xy + y^2 = 0, what is the sum of the slopes?
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Solution
The sum of the slopes can be found using the relationship between the coefficients of the quadratic.
Correct Answer:
A
— -3
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Q. For the lines represented by the equation 3x^2 - 2xy + y^2 = 0 to be parallel, the condition is:
A.
3 + 1 = 0
B.
3 - 1 = 0
C.
2 = 0
D.
None of the above
Show solution
Solution
The condition for parallel lines is that the determinant of the coefficients must equal zero.
Correct Answer:
A
— 3 + 1 = 0
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Q. For the lines represented by the equation 4x^2 - 12xy + 9y^2 = 0, find the slopes of the lines.
A.
1, 3
B.
2, 4
C.
3, 1
D.
0, 0
Show solution
Solution
Factoring the equation gives the slopes as m1 = 1 and m2 = 3.
Correct Answer:
A
— 1, 3
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Q. For the lines represented by the equation 4x^2 - 4xy + y^2 = 0, the angle between them is:
A.
0 degrees
B.
45 degrees
C.
90 degrees
D.
180 degrees
Show solution
Solution
The lines are at an angle of 45 degrees as the determinant of the coefficients gives a non-zero value.
Correct Answer:
B
— 45 degrees
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Q. For the lines represented by the equation 5x^2 + 6xy + 5y^2 = 0, what is the sum of the slopes?
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Solution
The sum of the slopes is given by - (coefficient of xy)/(coefficient of x^2) = -6/5.
Correct Answer:
A
— -6/5
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Q. For the lines represented by the equation 6x^2 + 5xy + y^2 = 0, what is the sum of the slopes?
Show solution
Solution
The sum of the slopes of the lines is given by -b/a, which is -5/6.
Correct Answer:
A
— -5/6
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Q. For the lines represented by the equation x^2 - 2xy + y^2 = 0, find the slopes of the lines.
A.
1, -1
B.
2, -2
C.
0, 0
D.
1, 1
Show solution
Solution
The slopes can be found by solving the quadratic equation formed by the coefficients of x^2, xy, and y^2.
Correct Answer:
A
— 1, -1
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Q. For the lines represented by the equation x^2 - 2xy + y^2 = 0, the angle between them is:
A.
0 degrees
B.
45 degrees
C.
90 degrees
D.
180 degrees
Show solution
Solution
The angle can be calculated using the slopes derived from the equation.
Correct Answer:
B
— 45 degrees
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Q. For the matrix \( B = \begin{pmatrix} 1 & 2 \\ 2 & 4 \end{pmatrix} \), what is the determinant \( |B| \)?
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Solution
The determinant is 0 because the rows are linearly dependent.
Correct Answer:
A
— 0
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Q. For the parabola defined by the equation y^2 = 20x, what is the coordinates of the vertex?
A.
(0, 0)
B.
(5, 0)
C.
(0, 5)
D.
(10, 0)
Show solution
Solution
The vertex of the parabola y^2 = 4px is at (0, 0).
Correct Answer:
A
— (0, 0)
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Q. For the parabola y = x^2 - 4x + 3, find the coordinates of the vertex.
A.
(2, -1)
B.
(1, 2)
C.
(2, 1)
D.
(1, -1)
Show solution
Solution
To find the vertex, use x = -b/(2a). Here, a = 1, b = -4, so x = 2. Substitute x = 2 into the equation to find y = -1. Thus, the vertex is (2, -1).
Correct Answer:
A
— (2, -1)
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Q. For the parabola y^2 = 16x, what is the coordinates of the vertex?
A.
(0, 0)
B.
(4, 0)
C.
(0, 4)
D.
(0, -4)
Show solution
Solution
The vertex of the parabola y^2 = 4px is at (0, 0).
Correct Answer:
A
— (0, 0)
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Showing 841 to 870 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!
