Q. Find the limit: lim (x -> 0) (x^2)/(sin(x)) (2023)
A.
0
B.
1
C.
2
D.
Undefined
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Solution
As x approaches 0, sin(x) approaches x, thus lim (x -> 0) (x^2/sin(x)) = lim (x -> 0) (x^2/x) = lim (x -> 0) x = 0.
Correct Answer:
A
— 0
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Q. Find the limit: lim (x -> 1) (x^4 - 1)/(x - 1) (2023)
A.
0
B.
1
C.
4
D.
Undefined
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Solution
Factoring gives ((x - 1)(x^3 + x^2 + x + 1))/(x - 1). For x ≠ 1, this simplifies to x^3 + x^2 + x + 1. Thus, lim (x -> 1) = 4.
Correct Answer:
A
— 0
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Q. Find the limit: lim (x -> 2) (x^2 + 3x - 10)/(x - 2) (2021)
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Solution
Factoring gives (x - 2)(x + 5)/(x - 2). For x ≠ 2, this simplifies to x + 5. Evaluating at x = 2 gives 7.
Correct Answer:
D
— 7
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Q. Find the limit: lim (x -> 2) (x^2 - 3x + 2)/(x - 2) (2021)
A.
1
B.
2
C.
0
D.
Undefined
Show solution
Solution
The expression is undefined at x=2. The limit does not exist as the function approaches infinity.
Correct Answer:
D
— Undefined
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Q. Find the limit: lim (x -> 3) (x^2 - 9)/(x - 3) (2023)
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Solution
The expression can be factored as ((x - 3)(x + 3))/(x - 3). For x ≠ 3, this simplifies to x + 3. Thus, lim (x -> 3) (x + 3) = 6.
Correct Answer:
A
— 0
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Q. Find the local maxima of f(x) = -x^2 + 4x + 1. (2020)
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Solution
The maximum occurs at x = -b/(2a) = -4/(2*-1) = 2. f(2) = -2^2 + 4(2) + 1 = 5.
Correct Answer:
B
— 5
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Q. Find the local maxima of f(x) = -x^3 + 3x^2 + 1. (2020)
A.
(0, 1)
B.
(1, 3)
C.
(2, 5)
D.
(3, 1)
Show solution
Solution
f'(x) = -3x^2 + 6x. Setting f'(x) = 0 gives x(3x - 6) = 0, so x = 0 or x = 2. f(2) = 5.
Correct Answer:
B
— (1, 3)
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Q. Find the local maximum of f(x) = -x^3 + 3x^2 + 4. (2020)
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Solution
Set f'(x) = 0 to find critical points. The local maximum occurs at x = 2. f(2) = 5.
Correct Answer:
B
— 5
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Q. Find the magnitude of the vector A = 3i - 4j. (2020)
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Solution
|A| = √(3^2 + (-4)^2) = √(9 + 16) = √25 = 5.
Correct Answer:
A
— 5
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Q. Find the maximum area of a triangle with a base of 10 m and height varying. (2020)
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Solution
Area = 1/2 * base * height. Max area occurs when height is maximized, thus Area = 1/2 * 10 * 10 = 50.
Correct Answer:
B
— 50
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Q. Find the maximum area of a triangle with a base of 10 units and height as a function of x. (2022)
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Solution
Area = 1/2 * base * height. Max area occurs when height is maximized, which is 10 units, giving Area = 50.
Correct Answer:
B
— 50
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Q. Find the maximum area of a triangle with a base of 10 units and height as a function of the base. (2021)
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Solution
Area = 1/2 * base * height. Max area occurs when height is maximized at 10 units, giving Area = 50.
Correct Answer:
B
— 50
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Q. Find the maximum area of a triangle with a fixed perimeter of 30 cm. (2022)
A.
75 cm²
B.
100 cm²
C.
50 cm²
D.
60 cm²
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Solution
For maximum area, the triangle should be equilateral. Area = (sqrt(3)/4) * (10)^2 = 75 cm².
Correct Answer:
A
— 75 cm²
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Q. Find the maximum height of the projectile modeled by h(t) = -16t^2 + 32t + 48. (2020)
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Solution
The maximum occurs at t = -b/(2a) = -32/(2*-16) = 1. h(1) = 64.
Correct Answer:
A
— 48
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Q. Find the maximum height of the projectile modeled by h(t) = -16t^2 + 64t + 48. (2020)
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Solution
The maximum occurs at t = -b/(2a) = 64/(2*16) = 2. h(2) = -16(2^2) + 64(2) + 48 = 80.
Correct Answer:
B
— 64
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Q. Find the maximum value of the function f(x) = -2x^2 + 8x - 3. (2021) 2021
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Solution
The function is a downward-opening parabola. The maximum occurs at x = -b/(2a) = -8/(2*-2) = 2. f(2) = -2(2^2) + 8(2) - 3 = 8.
Correct Answer:
B
— 8
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Q. Find the midpoint of the line segment joining the points (2, 3) and (4, 7). (2022) 2022
A.
(3, 5)
B.
(2, 5)
C.
(4, 5)
D.
(3, 4)
Show solution
Solution
Midpoint = ((2+4)/2, (3+7)/2) = (3, 5).
Correct Answer:
A
— (3, 5)
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Q. Find the minimum value of f(x) = 4x^2 - 16x + 20. (2022)
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Solution
The vertex gives the minimum at x = 2. f(2) = 4(2^2) - 16(2) + 20 = 4.
Correct Answer:
A
— 4
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Q. Find the minimum value of f(x) = x^2 - 4x + 6. (2021)
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Solution
The vertex form gives the minimum at x = 2. f(2) = 2.
Correct Answer:
A
— 2
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Q. Find the minimum value of f(x) = x^2 - 4x + 7. (2021)
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Solution
The vertex form gives the minimum at x = 2. f(2) = 2^2 - 4*2 + 7 = 3.
Correct Answer:
A
— 3
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Q. Find the minimum value of f(x) = x^2 - 4x + 7. (2021) 2021
Show solution
Solution
The vertex form gives the minimum at x = 2. f(2) = 2^2 - 4*2 + 7 = 3.
Correct Answer:
A
— 3
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Q. Find the minimum value of the function f(x) = 2x^2 - 8x + 10. (2022)
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Solution
The minimum occurs at x = 2. f(2) = 2(2^2) - 8(2) + 10 = 6.
Correct Answer:
B
— 4
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Q. Find the particular solution of dy/dx = 4y with the initial condition y(0) = 2.
A.
y = 2e^(4x)
B.
y = e^(4x)
C.
y = 4e^(x)
D.
y = 2e^(x)
Show solution
Solution
The general solution is y = Ce^(4x). Using the initial condition y(0) = 2, we find C = 2, thus y = 2e^(4x).
Correct Answer:
A
— y = 2e^(4x)
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Q. Find the particular solution of dy/dx = 4y, given y(0) = 2.
A.
y = 2e^(4x)
B.
y = e^(4x)
C.
y = 4e^(2x)
D.
y = 2e^(x/4)
Show solution
Solution
The general solution is y = Ce^(4x). Using the initial condition y(0) = 2, we find C = 2, thus y = 2e^(4x).
Correct Answer:
A
— y = 2e^(4x)
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Q. Find the point of inflection for f(x) = x^3 - 6x^2 + 9x. (2022)
A.
(1, 4)
B.
(2, 3)
C.
(3, 0)
D.
(0, 0)
Show solution
Solution
f''(x) = 6x - 12. Setting f''(x) = 0 gives x = 2. f(2) = 3.
Correct Answer:
C
— (3, 0)
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Q. Find the point of intersection of the lines y = x + 2 and y = -x + 4. (2023)
A.
(1, 3)
B.
(2, 4)
C.
(3, 5)
D.
(0, 2)
Show solution
Solution
Setting x + 2 = -x + 4 gives 2x = 2, so x = 1. Substituting x back gives y = 3. Thus, the point is (1, 3).
Correct Answer:
A
— (1, 3)
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Q. Find the point on the curve y = x^3 - 3x^2 + 4 that has a horizontal tangent. (2023)
A.
(0, 4)
B.
(1, 2)
C.
(2, 2)
D.
(3, 4)
Show solution
Solution
To find horizontal tangents, set the derivative y' = 3x^2 - 6x = 0. This gives x = 0 and x = 2. The point (1, 2) has a horizontal tangent.
Correct Answer:
B
— (1, 2)
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Q. Find the point on the curve y = x^3 - 3x^2 + 4 where the tangent is horizontal. (2023)
A.
(0, 4)
B.
(1, 2)
C.
(2, 2)
D.
(3, 4)
Show solution
Solution
To find horizontal tangents, set dy/dx = 0. dy/dx = 3x^2 - 6x = 0 gives x = 0 and x = 2. At x = 1, y = 2.
Correct Answer:
B
— (1, 2)
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Q. Find the real part of the complex number 4 + 5i. (2023)
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Solution
The real part of the complex number 4 + 5i is 4.
Correct Answer:
A
— 4
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Q. Find the roots of the equation 3x² - 12x + 12 = 0. (2021)
Show solution
Solution
Dividing by 3 gives x² - 4x + 4 = 0, which factors to (x - 2)² = 0, hence the root is 2.
Correct Answer:
A
— 2
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Showing 241 to 270 of 973 (33 Pages)
Mathematics (MHT-CET) MCQ & Objective Questions
Mathematics plays a crucial role in the MHT-CET exams, serving as a foundation for various scientific and engineering disciplines. Practicing MCQs and objective questions not only enhances your problem-solving skills but also boosts your confidence in tackling important questions during exams. Engaging with practice questions is essential for effective exam preparation, helping you identify your strengths and areas that need improvement.
What You Will Practise Here
Algebra: Understanding equations, inequalities, and functions.
Geometry: Key concepts of shapes, theorems, and properties.
Trigonometry: Ratios, identities, and applications in problems.
Calculus: Basics of differentiation and integration.
Statistics: Data interpretation, mean, median, and mode.
Probability: Fundamental principles and problem-solving techniques.
Coordinate Geometry: Graphing lines, circles, and conic sections.
Exam Relevance
Mathematics is a significant component of various examinations including CBSE, State Boards, NEET, and JEE. In these exams, you can expect a mix of direct application questions and conceptual problems. Common question patterns include multiple-choice questions that test your understanding of formulas, definitions, and theorems, making it imperative to be well-versed in the subject matter.
Common Mistakes Students Make
Misinterpreting the question, leading to incorrect answers.
Overlooking the importance of units in calculations.
Rushing through problems without checking for calculation errors.
Neglecting to review fundamental concepts before advanced topics.
FAQs
Question: What types of questions can I expect in Mathematics (MHT-CET)?Answer: You can expect a variety of MCQs that cover theoretical concepts, problem-solving, and application-based questions.
Question: How can I improve my performance in Mathematics (MHT-CET)?Answer: Regular practice of Mathematics (MHT-CET) MCQ questions and understanding the underlying concepts will significantly enhance your performance.
Start solving practice MCQs today to test your understanding and sharpen your skills. Remember, consistent practice is the key to success in Mathematics (MHT-CET) and achieving your academic goals!
