Q. Find the derivative of f(x) = sin(x) + cos(x) at x = π/4.
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Solution
f'(x) = cos(x) - sin(x), thus f'(π/4) = √2/2 - √2/2 = 0.
Correct Answer:
C
— √2
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Q. Find the derivative of f(x) = sin(x) at x = π/2.
A.
0
B.
1
C.
-1
D.
undefined
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Solution
f'(x) = cos(x); f'(π/2) = cos(π/2) = 0.
Correct Answer:
B
— 1
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Q. Find the derivative of f(x) = tan(x) at x = 0.
A.
0
B.
1
C.
undefined
D.
1/2
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Solution
f'(x) = sec^2(x); f'(0) = sec^2(0) = 1.
Correct Answer:
B
— 1
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Q. Find the derivative of f(x) = tan(x) at x = π/4.
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Solution
f'(x) = sec^2(x). At x = π/4, f'(π/4) = sec^2(π/4) = 2.
Correct Answer:
A
— 1
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Q. Find the derivative of f(x) = tan(x).
A.
sec^2(x)
B.
csc^2(x)
C.
sin^2(x)
D.
cos^2(x)
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Solution
The derivative f'(x) = d/dx(tan(x)) = sec^2(x).
Correct Answer:
A
— sec^2(x)
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Q. Find the derivative of f(x) = x^2 * e^x.
A.
e^x(x^2 + 2x)
B.
e^x(x^2 - 2x)
C.
2xe^x
D.
x^2e^x
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Solution
Using the product rule: f'(x) = x^2 * e^x + 2x * e^x = e^x(x^2 + 2x).
Correct Answer:
A
— e^x(x^2 + 2x)
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Q. Find the derivative of f(x) = x^2 sin(1/x) at x = 0.
A.
0
B.
1
C.
undefined
D.
does not exist
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Solution
Using the limit definition of the derivative, we find that f'(0) = 0, hence it is differentiable at x = 0.
Correct Answer:
A
— 0
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Q. Find the derivative of f(x) = x^3 - 3x^2 + 4 at x = 2.
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Solution
f'(x) = 3x^2 - 6x. At x = 2, f'(2) = 3(2^2) - 6(2) = 12 - 12 = 0.
Correct Answer:
B
— 8
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Q. Find the derivative of f(x) = x^3 - 3x^2 + 4x - 5.
A.
3x^2 - 6x + 4
B.
3x^2 - 3x + 4
C.
3x^2 - 6x + 5
D.
3x^2 + 6x - 4
Show solution
Solution
Using the power rule, f'(x) = 3x^2 - 6x + 4.
Correct Answer:
A
— 3x^2 - 6x + 4
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Q. Find the derivative of f(x) = x^3 - 4x^2 + 6x.
A.
3x^2 - 8x + 6
B.
3x^2 - 4x + 6
C.
3x^2 - 8x
D.
x^2 - 4x + 6
Show solution
Solution
Using the power rule, f'(x) = 3x^2 - 8x + 6.
Correct Answer:
A
— 3x^2 - 8x + 6
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Q. Find the determinant of the matrix \( D = \begin{pmatrix} 1 & 0 & 2 \\ 0 & 1 & 3 \\ 0 & 0 & 1 \end{pmatrix} \).
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Solution
The determinant of an upper triangular matrix is the product of its diagonal elements, which is 1.
Correct Answer:
B
— 1
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Q. Find the determinant of the matrix \( I = \begin{pmatrix} 3 & 2 \\ 1 & 4 \end{pmatrix} \).
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Solution
The determinant is calculated as \( 3*4 - 2*1 = 12 - 2 = 10 \).
Correct Answer:
A
— 10
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Q. Find the determinant of the matrix \( \begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix} \).
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Solution
The determinant of the identity matrix is always 1.
Correct Answer:
B
— 1
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Q. Find the determinant of the matrix \( \begin{pmatrix} 2 & 1 \\ 3 & 4 \end{pmatrix} \).
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Solution
The determinant is calculated as \( 2*4 - 1*3 = 8 - 3 = 5 \).
Correct Answer:
A
— 5
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Q. Find the determinant of the matrix \( \begin{pmatrix} 2 & 3 & 1 \\ 1 & 0 & 4 \\ 5 & 2 & 1 \end{pmatrix} \).
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Solution
The determinant evaluates to 0.
Correct Answer:
A
— -1
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Q. Find the determinant of the matrix \( \begin{pmatrix} 2 & 3 & 1 \\ 1 & 0 & 2 \\ 4 & 1 & 0 \end{pmatrix} \).
Show solution
Solution
Using the determinant formula, we find it equals 10.
Correct Answer:
A
— -10
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Q. Find the determinant of the matrix \( \begin{pmatrix} 2 & 3 \\ 1 & 4 \end{pmatrix} \).
Show solution
Solution
The determinant is \( 2*4 - 3*1 = 8 - 3 = 5 \).
Correct Answer:
A
— 5
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Q. Find the determinant of the matrix \( \begin{pmatrix} a & b \\ c & d \end{pmatrix} \).
A.
ad - bc
B.
bc - ad
C.
a + b + c + d
D.
a^2 + b^2
Show solution
Solution
The determinant is given by the formula \( ad - bc \).
Correct Answer:
A
— ad - bc
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Q. Find the determinant of the matrix | 1 0 0 | | 0 1 0 | | 0 0 1 |.
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Solution
This is the identity matrix, and its determinant is 1.
Correct Answer:
B
— 1
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Q. Find the determinant of the matrix | 1 2 3 | | 0 1 4 | | 5 6 0 |.
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Solution
The determinant evaluates to 0 as the third row can be expressed as a linear combination of the first two.
Correct Answer:
A
— -12
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Q. Find the determinant of the matrix: | 1 2 | | 3 5 |.
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Solution
det = (1*5) - (2*3) = 5 - 6 = -1.
Correct Answer:
A
— -1
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Q. Find the directrix of the parabola y^2 = -8x.
A.
x = 2
B.
x = -2
C.
x = 4
D.
x = -4
Show solution
Solution
For the parabola y^2 = 4px, here 4p = -8, so p = -2. The directrix is given by x = -p, which is x = 2.
Correct Answer:
B
— x = -2
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Q. Find the distance between the points (3, 4) and (7, 1).
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Solution
Distance = √[(7-3)² + (1-4)²] = √[4 + 9] = √13 ≈ 3.6, closest option is 4.
Correct Answer:
A
— 5
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Q. Find the distance between the points A(2, 3) and B(5, 7).
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Solution
Distance = √[(5-2)² + (7-3)²] = √[3² + 4²] = √[9 + 16] = √25 = 5.
Correct Answer:
C
— 5
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Q. Find the distance from the point (1, 2) to the line 3x + 4y - 12 = 0.
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Solution
Distance = |Ax1 + By1 + C| / sqrt(A^2 + B^2) = |3(1) + 4(2) - 12| / sqrt(3^2 + 4^2) = |3 + 8 - 12| / 5 = 1.
Correct Answer:
A
— 2
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Q. Find the distance from the point (3, 4) to the line 2x + 3y - 6 = 0.
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Solution
Distance = |Ax1 + By1 + C| / sqrt(A^2 + B^2) = |2*3 + 3*4 - 6| / sqrt(2^2 + 3^2) = |6 + 12 - 6| / sqrt(13) = 12 / sqrt(13).
Correct Answer:
B
— 3
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Q. Find the eigenvalues of the matrix A = [[2, 1], [1, 2]].
A.
1, 3
B.
2, 2
C.
3, 1
D.
0, 4
Show solution
Solution
The characteristic polynomial is det(A - λI) = (2-λ)(2-λ) - 1 = λ^2 - 4λ + 3 = 0, giving eigenvalues 1 and 3.
Correct Answer:
A
— 1, 3
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Q. Find the equation of the circle with center (2, -3) and radius 5.
A.
(x-2)² + (y+3)² = 25
B.
(x+2)² + (y-3)² = 25
C.
(x-2)² + (y-3)² = 25
D.
(x+2)² + (y+3)² = 25
Show solution
Solution
Equation of circle: (x-h)² + (y-k)² = r² => (x-2)² + (y+3)² = 5².
Correct Answer:
A
— (x-2)² + (y+3)² = 25
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Q. Find the equation of the family of curves represented by y = mx + c, where m and c are constants.
A.
y = mx + c
B.
y = mx^2 + c
C.
y = c/x + m
D.
y = m^2x + c
Show solution
Solution
The equation y = mx + c represents a family of straight lines where m is the slope and c is the y-intercept.
Correct Answer:
A
— y = mx + c
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Q. Find the equation of the line passing through the points (1, 2) and (3, 4).
A.
y = x + 1
B.
y = 2x
C.
y = x + 3
D.
y = 2x - 1
Show solution
Solution
The slope m = (4-2)/(3-1) = 1. Using point-slope form: y - 2 = 1(x - 1) gives y = x + 1.
Correct Answer:
A
— y = x + 1
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Showing 511 to 540 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!
