Q. In triangle ABC, if the lengths of the sides are a = 8, b = 15, and c = 17, what is the value of cos A?
A.
0.5
B.
0.6
C.
0.8
D.
0.9
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Solution
Using the cosine rule, cos A = (b² + c² - a²) / (2bc) = (15² + 17² - 8²) / (2 * 15 * 17) = 0.8.
Correct Answer:
C
— 0.8
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Q. In triangle ABC, if the lengths of the sides are in the ratio 3:4:5, what type of triangle is it?
A.
Acute
B.
Obtuse
C.
Right
D.
Equilateral
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Solution
Since the sides are in the ratio of a Pythagorean triplet (3, 4, 5), triangle ABC is a right triangle.
Correct Answer:
C
— Right
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Q. In triangle ABC, if the sides are in the ratio 3:4:5, what is the nature of the triangle?
A.
Equilateral
B.
Isosceles
C.
Right
D.
Scalene
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Solution
The sides satisfy the Pythagorean theorem, hence it is a right triangle.
Correct Answer:
C
— Right
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Q. In triangle ABC, if the sides are in the ratio 3:4:5, what type of triangle is it?
A.
Acute
B.
Obtuse
C.
Right
D.
Equilateral
Show solution
Solution
A triangle with sides in the ratio 3:4:5 is a right triangle, as it satisfies the Pythagorean theorem.
Correct Answer:
C
— Right
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Q. In triangle MNO, if angle M = 45 degrees and angle N = 45 degrees, what is angle O?
A.
90 degrees
B.
45 degrees
C.
60 degrees
D.
30 degrees
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Solution
Angle O = 180 - (angle M + angle N) = 180 - (45 + 45) = 90 degrees.
Correct Answer:
A
— 90 degrees
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Q. In triangle PQR, if PQ = 10 cm, QR = 24 cm, and PR = 26 cm, what is the area of the triangle?
A.
120 cm²
B.
120√3 cm²
C.
240 cm²
D.
48 cm²
Show solution
Solution
Using Heron's formula, s = (10 + 24 + 26)/2 = 30. Area = √(30(30-10)(30-24)(30-26)) = √(30*20*6*4) = 120 cm².
Correct Answer:
A
— 120 cm²
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Q. In triangle XYZ, if XY = 8 cm, YZ = 15 cm, and XZ = 17 cm, is it a right triangle?
A.
Yes
B.
No
C.
Cannot be determined
D.
Only if XY is the hypotenuse
Show solution
Solution
Since 8^2 + 15^2 = 17^2, triangle XYZ is a right triangle.
Correct Answer:
A
— Yes
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Q. Is the function f(x) = x^2 - 2x + 1 differentiable at x = 1?
A.
Yes
B.
No
C.
Only from the left
D.
Only from the right
Show solution
Solution
f(x) is a polynomial function, which is differentiable everywhere, including at x = 1.
Correct Answer:
A
— Yes
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Q. Is the function f(x) = x^2 - 4x + 4 differentiable at x = 2?
A.
Yes
B.
No
C.
Only from the left
D.
Only from the right
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Solution
The function is a polynomial and is differentiable everywhere, hence yes.
Correct Answer:
A
— Yes
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Q. Is the function f(x) = x^2 - 4x + 4 differentiable everywhere?
A.
Yes
B.
No
C.
Only at x = 0
D.
Only at x = 2
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Solution
This is a polynomial function, which is differentiable everywhere on its domain.
Correct Answer:
A
— Yes
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Q. Is the function f(x) = x^2 sin(1/x) differentiable at x = 0?
A.
Yes
B.
No
C.
Only from the left
D.
Only from the right
Show solution
Solution
Using the limit definition, f'(0) = lim (h -> 0) [(h^2 sin(1/h) - 0)/h] = 0. Thus, f(x) is differentiable at x = 0.
Correct Answer:
A
— Yes
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Q. Is the function f(x) = x^3 - 3x + 2 differentiable at x = 1?
A.
Yes
B.
No
C.
Only left differentiable
D.
Only right differentiable
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Solution
The function is a polynomial and hence differentiable everywhere, including at x = 1.
Correct Answer:
A
— Yes
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Q. Is the function f(x) = { e^x, x < 0; ln(x + 1), x >= 0 } continuous at x = 0?
A.
Yes
B.
No
C.
Only left continuous
D.
Only right continuous
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Solution
Both limits as x approaches 0 from the left and right are equal to 1, hence f(x) is continuous at x = 0.
Correct Answer:
A
— Yes
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Q. Is the function f(x) = { sin(x), x < 0; x^2, x >= 0 } continuous at x = 0?
A.
Yes
B.
No
C.
Depends on x
D.
Not defined
Show solution
Solution
Both limits as x approaches 0 from the left and right are equal to 0, hence f(x) is continuous at x = 0.
Correct Answer:
A
— Yes
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Q. Is the function f(x) = { x^3, x < 1; 2x + 1, x >= 1 } continuous at x = 1?
A.
Yes
B.
No
C.
Only left continuous
D.
Only right continuous
Show solution
Solution
Both limits as x approaches 1 from the left and right are equal to 2, hence f(x) is continuous at x = 1.
Correct Answer:
A
— Yes
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Q. Is the function f(x) = |x|/x continuous at x = 0?
A.
Yes
B.
No
C.
Depends on direction
D.
None of the above
Show solution
Solution
The left limit is -1 and the right limit is 1, which are not equal. Therefore, f(x) is not continuous at x = 0.
Correct Answer:
B
— No
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Q. Let A = {1, 2, 3, 4} and R be the relation defined by R = {(a, b) | a < b}. How many ordered pairs are in R?
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Solution
The pairs are (1,2), (1,3), (1,4), (2,3), (2,4), (3,4). Thus, there are 6 ordered pairs.
Correct Answer:
B
— 6
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Q. Let A = {1, 2, 3, 4} and R be the relation defined by R = {(x, y) | x < y}. How many ordered pairs are in R?
Show solution
Solution
The ordered pairs are (1,2), (1,3), (1,4), (2,3), (2,4), (3,4). Thus, there are 6 ordered pairs.
Correct Answer:
B
— 6
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Q. Let R be a relation on the set of natural numbers defined by R = {(m, n) | m divides n}. Is R a partial order?
A.
Yes
B.
No
C.
Only reflexive
D.
Only transitive
Show solution
Solution
R is reflexive, antisymmetric, and transitive, thus it is a partial order.
Correct Answer:
A
— Yes
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Q. Solve for x: 3(x - 2) = 2(x + 1).
Show solution
Solution
Expanding both sides gives 3x - 6 = 2x + 2. Rearranging gives x = 8.
Correct Answer:
B
— 0
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Q. Solve for x: log_3(x + 1) - log_3(x - 1) = 1.
Show solution
Solution
Using properties of logarithms, log_3((x + 1)/(x - 1)) = 1 => (x + 1)/(x - 1) = 3 => x + 1 = 3(x - 1) => x = 2.
Correct Answer:
A
— 2
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Q. Solve for x: log_3(x) = 2.
Show solution
Solution
log_3(x) = 2 implies x = 3^2 = 9.
Correct Answer:
B
— 9
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Q. Solve for x: log_5(x + 1) - log_5(x - 1) = 1.
Show solution
Solution
Using properties of logarithms: log_5((x + 1)/(x - 1)) = 1 => (x + 1)/(x - 1) = 5 => x + 1 = 5(x - 1) => 4x = 6 => x = 2.
Correct Answer:
A
— 2
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Q. Solve for x: log_5(x) = 2.
Show solution
Solution
log_5(x) = 2 implies x = 5^2 = 25.
Correct Answer:
C
— 25
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Q. Solve the differential equation dy/dx + 2y = 4.
A.
y = 2 - Ce^(-2x)
B.
y = 2 + Ce^(-2x)
C.
y = 4 - Ce^(-2x)
D.
y = 4 + Ce^(2x)
Show solution
Solution
This is a linear first-order differential equation. The integrating factor is e^(2x). Solving gives y = 2 - Ce^(-2x).
Correct Answer:
A
— y = 2 - Ce^(-2x)
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Q. Solve the differential equation dy/dx = 3x^2.
A.
y = x^3 + C
B.
y = 3x^3 + C
C.
y = x^2 + C
D.
y = 3x + C
Show solution
Solution
Integrating both sides gives y = x^3 + C.
Correct Answer:
A
— y = x^3 + C
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Q. Solve the differential equation dy/dx = x^2 + y^2.
A.
y = x^3/3 + C
B.
y = x^2 + C
C.
y = x^2 + x + C
D.
y = Cx^2 + C
Show solution
Solution
This is a non-linear differential equation. The solution can be found using substitution methods.
Correct Answer:
A
— y = x^3/3 + C
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Q. Solve the differential equation y' = 3y + 6.
A.
y = Ce^(3x) - 2
B.
y = Ce^(3x) + 2
C.
y = 2e^(3x)
D.
y = 3e^(3x) + 2
Show solution
Solution
Using the integrating factor method, we find y = Ce^(3x) + 2.
Correct Answer:
B
— y = Ce^(3x) + 2
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Q. Solve the differential equation y'' + 4y = 0.
A.
y = C1 cos(2x) + C2 sin(2x)
B.
y = C1 e^(2x) + C2 e^(-2x)
C.
y = C1 cos(x) + C2 sin(x)
D.
y = C1 e^(x) + C2 e^(-x)
Show solution
Solution
The characteristic equation is r^2 + 4 = 0, giving complex roots. The solution is y = C1 cos(2x) + C2 sin(2x).
Correct Answer:
A
— y = C1 cos(2x) + C2 sin(2x)
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Q. Solve the differential equation y'' - 5y' + 6y = 0.
A.
y = C1 e^(2x) + C2 e^(3x)
B.
y = C1 e^(3x) + C2 e^(2x)
C.
y = C1 e^(x) + C2 e^(2x)
D.
y = C1 e^(2x) + C2 e^(x)
Show solution
Solution
The characteristic equation is r^2 - 5r + 6 = 0, which factors to (r - 2)(r - 3) = 0, giving the solution y = C1 e^(2x) + C2 e^(3x).
Correct Answer:
B
— y = C1 e^(3x) + C2 e^(2x)
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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!
