Q. Evaluate the determinant \( \begin{vmatrix} 3 & 1 & 2 \\ 1 & 2 & 3 \\ 2 & 3 & 1 \end{vmatrix} \).
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Solution
The determinant is 0 because the rows are linearly dependent.
Correct Answer:
A
— 0
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Q. Evaluate the determinant \( \det \begin{pmatrix} 2 & 1 & 3 \\ 1 & 0 & 2 \\ 3 & 4 & 1 \end{pmatrix} \).
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Solution
The determinant is calculated as \( 2(0*1 - 2*4) - 1(1*1 - 2*3) + 3(1*4 - 0*3) = -10 \).
Correct Answer:
A
— -10
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Q. Evaluate the determinant \( |C| \) where \( C = \begin{pmatrix} 2 & 1 & 3 \\ 1 & 0 & 2 \\ 3 & 2 & 1 \end{pmatrix} \).
A.
-12
B.
-10
C.
-8
D.
-6
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Solution
The determinant is calculated as 2(0*1 - 2*2) - 1(1*1 - 2*3) + 3(1*2 - 0*3) = 2(0 - 4) - 1(1 - 6) + 3(2) = -8 + 5 + 6 = 3.
Correct Answer:
A
— -12
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Q. Evaluate the determinant | 1 1 1 | | 1 2 3 | | 1 3 6 |.
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Solution
The rows are linearly dependent, hence the determinant is 0.
Correct Answer:
A
— 0
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Q. Evaluate the determinant | 1 1 1 | | 2 2 2 | | 3 3 3 |.
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Solution
The determinant is 0 because the rows are linearly dependent.
Correct Answer:
A
— 0
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Q. Evaluate the determinant: | 1 0 0 | | 0 1 0 | | 0 0 1 |
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Solution
The determinant of the identity matrix is 1.
Correct Answer:
A
— 1
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Q. Evaluate the determinant: | 1 2 3 | | 4 5 6 | | 7 8 9 |
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Solution
The determinant of a matrix with linearly dependent rows is 0. Here, the rows are linearly dependent.
Correct Answer:
A
— 0
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Q. Evaluate the expression sin^(-1)(1) + cos^(-1)(0).
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Solution
sin^(-1)(1) = π/2 and cos^(-1)(0) = π/2. Therefore, π/2 + π/2 = π.
Correct Answer:
A
— π/2
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Q. Evaluate the expression sin^(-1)(x) + cos^(-1)(x).
A.
0
B.
π/2
C.
π
D.
undefined
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Solution
sin^(-1)(x) + cos^(-1)(x) = π/2 for all x in the domain [-1, 1].
Correct Answer:
B
— π/2
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Q. Evaluate the expression sin^(-1)(x) + sin^(-1)(√(1-x^2)).
A.
π/2
B.
π/4
C.
π/3
D.
0
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Solution
Using the identity sin^(-1)(x) + sin^(-1)(√(1-x^2)) = π/2 for x in [0, 1], the value is π/2.
Correct Answer:
A
— π/2
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Q. Evaluate the expression: 2sin^(-1)(1/2) + 2cos^(-1)(1/2).
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Solution
2sin^(-1)(1/2) = 2(π/6) = π/3 and 2cos^(-1)(1/2) = 2(π/3) = 2π/3. Therefore, the total is π/3 + 2π/3 = π.
Correct Answer:
A
— π
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Q. Evaluate the expression: tan^(-1)(1) + tan^(-1)(1) + tan^(-1)(0).
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Solution
tan^(-1)(1) = π/4, so the expression becomes π/4 + π/4 + 0 = π/2.
Correct Answer:
A
— π/2
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Q. Evaluate the expression: tan^(-1)(1) + tan^(-1)(1) = ?
A.
π/2
B.
π/4
C.
π/3
D.
π/6
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Solution
tan^(-1)(1) = π/4, thus tan^(-1)(1) + tan^(-1)(1) = π/4 + π/4 = π/2.
Correct Answer:
A
— π/2
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Q. Evaluate the expression: tan^(-1)(1) + tan^(-1)(√3).
A.
π/3
B.
π/2
C.
2π/3
D.
π
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Solution
tan^(-1)(1) = π/4 and tan^(-1)(√3) = π/3. Therefore, π/4 + π/3 = 7π/12.
Correct Answer:
A
— π/3
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Q. Evaluate the integral ∫ (1/x) dx.
A.
ln
B.
x
C.
+ C
D.
ln(x) + C
.
1/x + C
.
x + C
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Solution
The integral of 1/x is ln|x|. Therefore, ∫ (1/x) dx = ln|x| + C.
Correct Answer:
A
— ln
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Q. Evaluate the integral ∫ (2x + 1)/(x^2 + x) dx.
A.
ln
B.
x^2 + x
C.
+ C
D.
ln
.
x
.
+ C
.
ln
.
x^2 + x
.
+ 1 + C
.
ln
.
x^2 + x
.
+ 1
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Solution
Using partial fraction decomposition, we can integrate to find that ∫ (2x + 1)/(x^2 + x) dx = ln|x^2 + x| + C.
Correct Answer:
A
— ln
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Q. Evaluate the integral ∫ (3x^2 + 2x + 1) dx.
A.
x^3 + x^2 + x + C
B.
x^3 + x^2 + C
C.
x^3 + x^2 + x
D.
3x^3 + 2x^2 + x + C
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Solution
The integral of 3x^2 is x^3, the integral of 2x is x^2, and the integral of 1 is x. Therefore, ∫ (3x^2 + 2x + 1) dx = x^3 + x^2 + x + C.
Correct Answer:
A
— x^3 + x^2 + x + C
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Q. Evaluate the integral ∫ (sec^2(x)) dx.
A.
tan(x) + C
B.
sec(x) + C
C.
sin(x) + C
D.
cos(x) + C
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Solution
The integral of sec^2(x) is tan(x) + C.
Correct Answer:
A
— tan(x) + C
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Q. Evaluate the integral ∫ (x^2 + 2x + 1)/(x + 1) dx.
A.
(1/3)x^3 + x^2 + C
B.
x^2 + x + C
C.
(1/3)x^3 + (1/2)x^2 + C
D.
x^2 + 2x + C
Show solution
Solution
By simplifying the integrand, we can integrate to find that ∫ (x^2 + 2x + 1)/(x + 1) dx = (1/3)x^3 + x^2 + C.
Correct Answer:
A
— (1/3)x^3 + x^2 + C
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Q. Evaluate the integral ∫ cos(3x) dx.
A.
(1/3)sin(3x) + C
B.
sin(3x) + C
C.
(1/3)cos(3x) + C
D.
-(1/3)sin(3x) + C
Show solution
Solution
The integral of cos(kx) is (1/k)sin(kx). Here, k = 3, so ∫ cos(3x) dx = (1/3)sin(3x) + C.
Correct Answer:
A
— (1/3)sin(3x) + C
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Q. Evaluate the integral ∫ cos(5x) dx.
A.
1/5 sin(5x) + C
B.
-1/5 sin(5x) + C
C.
5 sin(5x) + C
D.
sin(5x) + C
Show solution
Solution
The integral of cos(kx) is (1/k)sin(kx). Here, k = 5, so ∫ cos(5x) dx = (1/5)sin(5x) + C.
Correct Answer:
A
— 1/5 sin(5x) + C
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Q. Evaluate the integral ∫ e^(3x) dx.
A.
(1/3)e^(3x) + C
B.
(1/3)e^(3x)
C.
3e^(3x) + C
D.
e^(3x) + C
Show solution
Solution
The integral of e^(kx) is (1/k)e^(kx). Here, k = 3, so ∫ e^(3x) dx = (1/3)e^(3x) + C.
Correct Answer:
A
— (1/3)e^(3x) + C
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Q. Evaluate the integral ∫ from 0 to 1 of (x^2 + 2x) dx.
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Solution
The integral evaluates to [x^3/3 + x^2] from 0 to 1 = (1/3 + 1) - (0) = 4/3.
Correct Answer:
B
— 2
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Q. Evaluate the integral ∫ from 0 to 1 of e^x dx.
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Solution
The integral evaluates to [e^x] from 0 to 1 = e - 1.
Correct Answer:
A
— e - 1
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Q. Evaluate the integral ∫ from 1 to 3 of (2x + 1) dx.
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Solution
The integral evaluates to [x^2 + x] from 1 to 3 = (9 + 3) - (1 + 1) = 10.
Correct Answer:
B
— 8
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Q. Evaluate the integral ∫(0 to 1) (x^3 + 2x^2)dx.
A.
1/4
B.
1/3
C.
1/2
D.
1
Show solution
Solution
The integral evaluates to [x^4/4 + 2x^3/3] from 0 to 1 = 1/4 + 2/3 = 11/12.
Correct Answer:
B
— 1/3
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Q. Evaluate the integral ∫(1 to 2) (2x + 3)dx.
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Solution
∫(2x + 3)dx = [x^2 + 3x] from 1 to 2 = (4 + 6) - (1 + 3) = 10 - 4 = 6.
Correct Answer:
B
— 8
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Q. Evaluate the integral ∫(1 to 2) (3x^2 - 2)dx.
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Solution
The integral evaluates to [(x^3 - 2x)] from 1 to 2 = (8 - 4) - (1 - 2) = 5.
Correct Answer:
A
— 3
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Q. Evaluate the integral ∫(1 to 2) (x^2 + 2x)dx.
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Solution
The integral ∫(x^2 + 2x)dx = [(1/3)x^3 + x^2] from 1 to 2 = 8.
Correct Answer:
B
— 8
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Q. Evaluate the integral ∫(2x^3 - 4x)dx.
A.
(1/2)x^4 - 2x^2 + C
B.
(1/4)x^4 - 2x^2 + C
C.
(1/2)x^4 - 4x^2 + C
D.
(1/3)x^4 - 2x^2 + C
Show solution
Solution
The integral ∫(2x^3 - 4x)dx = (1/2)x^4 - 2x^2 + C.
Correct Answer:
A
— (1/2)x^4 - 2x^2 + C
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Showing 361 to 390 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!
