Mastering Relations & Functions and Inverse Trigonometric Functions for Class 12: Key MCQs and Insights
Master the Relations, Functions & Inverse Trigonometric Functions (Class 12 Mathematics)
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Class 12 Mathematics plays a pivotal role in shaping students' academic paths, especially for those preparing for the West Bengal Higher Secondary (WBHS) Test. Among the key chapters, Relations and Functions and Inverse Trigonometric Functions form the backbone of understanding advanced mathematical concepts. This blog provides a concise yet comprehensive guide, including 30 carefully curated MCQs to help students excel.
Understanding Relations and Functions
Relations and Functions deal with the correlation between elements of two sets. Here’s what you need to know:
Definition:
- Relation: A subset of the Cartesian product of two sets.
- Function: A relation where each element of the domain has a unique image in the codomain.
Types of Functions:
- One-to-One (Injective)
- Onto (Surjective)
- One-to-One and Onto (Bijective)
- Constant Function
Important Properties:
- Domain: All possible input values.
- Range: All output values.
- Codomain: Set containing possible outputs.
Special Functions:
- Identity Function
- Polynomial Function
- Exponential and Logarithmic Functions
A Glimpse into Inverse Trigonometric Functions
Inverse Trigonometric Functions are critical for solving equations involving trigonometric functions. They provide a way to determine angles from given trigonometric values.
Key Functions:
- arcsin(x), arccos(x), arctan(x), arccot(x), arcsec(x), arccosec(x)
Principal Values: Each function has a restricted domain to ensure it is one-to-one and invertible. For example:
- arcsin(x): [-1, 1] → [-π/2, π/2]
- arccos(x): [-1, 1] → [0, π]
Important Properties:
- sin⁻¹(sin x) = x, if x ∈ [-π/2, π/2]
- tan⁻¹(tan x) = x, if x ∈ (-π/2, π/2)
MCQs on Relations and Functions
A relation R is defined as R = {(x, y) | x² + y² = 1}. What is the domain of R?
- (a) [-1, 1]
- (b) [0, 1]
- (c) All real numbers
- (d) None of the above
Answer: (a) [-1, 1]
If f(x) = 2x + 3 and g(x) = x², find (f ∘ g)(x).
- (a) 2x² + 3
- (b) x² + 2x
- (c) 2x + 3
- (d) x² + 3
Answer: (a) 2x² + 3
Which of the following is not a function?
- (a) y = x³
- (b) y² = x
- (c) y = |x|
- (d) y = 1/x
Answer: (b) y² = x
What type of function is f(x) = e^x?
- (a) One-to-One
- (b) Onto
- (c) Bijective
- (d) None of the above
Answer: (a) One-to-One
Determine whether f(x) = x³ is one-to-one, onto, or bijective.
- (a) One-to-One
- (b) Onto
- (c) Bijective
- (d) None of the above
Answer: (c) Bijective
If f(x) = x + 3 and g(x) = x² + 1, find f(g(2)).
- (a) 8
- (b) 9
- (c) 7
- (d) 12
Answer: (b) 9
The range of f(x) = |x| is:
- (a) (-∞, ∞)
- (b) [0, ∞)
- (c) (-∞, 0)
- (d) None of the above
Answer: (b) [0, ∞)
If f(x) = x² and g(x) = √x, find g(f(2)).
- (a) 2
- (b) 4
- (c) √2
- (d) None of the above
Answer: (a) 2
Which of the following is a bijective function?
- (a) f(x) = x²
- (b) f(x) = e^x
- (c) f(x) = sin(x)
- (d) f(x) = cos(x)
Answer: (b) f(x) = e^x
The domain of the function f(x) = √(x + 2) is:
- (a) (-∞, ∞)
- (b) [2, ∞)
- (c) [-2, ∞)
- (d) None of the above
Answer: (c) [-2, ∞)
A relation R is defined as R = {(x, y) | x² + y² = 1}. What is the domain of R?
- (a) [-1, 1]
- (b) [0, 1]
- (c) All real numbers
- (d) None of the above
Answer: (a) [-1, 1]
If f(x) = 2x + 3 and g(x) = x², find (f ∘ g)(x).
- (a) 2x² + 3
- (b) x² + 2x
- (c) 2x + 3
- (d) x² + 3
Answer: (a) 2x² + 3
Which of the following is not a function?
- (a) y = x³
- (b) y² = x
- (c) y = |x|
- (d) y = 1/x
Answer: (b) y² = x
What type of function is f(x) = e^x?
- (a) One-to-One
- (b) Onto
- (c) Bijective
- (d) None of the above
Answer: (a) One-to-One
Determine whether f(x) = x³ is one-to-one, onto, or bijective.
- (a) One-to-One
- (b) Onto
- (c) Bijective
- (d) None of the above
Answer: (c) Bijective
If f(x) = x + 3 and g(x) = x² + 1, find f(g(2)).
- (a) 8
- (b) 9
- (c) 7
- (d) 12
Answer: (b) 9
The range of f(x) = |x| is:
- (a) (-∞, ∞)
- (b) [0, ∞)
- (c) (-∞, 0)
- (d) None of the above
Answer: (b) [0, ∞)
If f(x) = x² and g(x) = √x, find g(f(2)).
- (a) 2
- (b) 4
- (c) √2
- (d) None of the above
Answer: (a) 2
Which of the following is a bijective function?
- (a) f(x) = x²
- (b) f(x) = e^x
- (c) f(x) = sin(x)
- (d) f(x) = cos(x)
Answer: (b) f(x) = e^x
The domain of the function f(x) = √(x + 2) is:
- (a) (-∞, ∞)
- (b) [2, ∞)
- (c) [-2, ∞)
- (d) None of the above
Answer: (c) [-2, ∞)
MCQs on Inverse Trigonometric Functions
Find the value of arcsin(1/2).
- (a) π/4
- (b) π/3
- (c) π/6
- (d) None of the above
Answer: (c) π/6
What is the domain of arctan(x)?
- (a) [-1, 1]
- (b) (-∞, ∞)
- (c) [0, π]
- (d) None of the above
Answer: (b) (-∞, ∞)
Simplify sin⁻¹(sin π/3).
- (a) π/3
- (b) -π/3
- (c) 2π/3
- (d) None of the above
Answer: (a) π/3
Evaluate cos⁻¹(-1).
- (a) 0
- (b) π/2
- (c) π
- (d) None of the above
Answer: (c) π
What is tan(tan⁻¹ x) if x ∈ ℝ?
- (a) x
- (b) -x
- (c) |x|
- (d) None of the above
Answer: (a) x
Find the range of sin⁻¹(x).
- (a) [-π/2, π/2]
- (b) [0, π]
- (c) [-1, 1]
- (d) None of the above
Answer: (a) [-π/2, π/2]
What is the value of arcsin(-1)?
- (a) -π/2
- (b) π/2
- (c) 0
- (d) None of the above
Answer: (a) -π/2
Evaluate sin⁻¹(0.5) + cos⁻¹(0.5).
- (a) π/2
- (b) π
- (c) 0
- (d) None of the above
Answer: (a) π/2
If arcsin(x) = π/6, what is x?
- (a) 1/2
- (b) √3/2
- (c) 1
- (d) None of the above
Answer: (a) 1/2
What is the principal value of cos⁻¹(1)?
- (a) 0
- (b) π
- (c) π/2
- (d) None of the above
Answer: (a) 0
Find the value of arcsin(1/2).
- (a) π/4
- (b) π/3
- (c) π/6
- (d) None of the above
Answer: (c) π/6
What is the domain of arctan(x)?
- (a) [-1, 1]
- (b) (-∞, ∞)
- (c) [0, π]
- (d) None of the above
Answer: (b) (-∞, ∞)
Simplify sin⁻¹(sin π/3).
- (a) π/3
- (b) -π/3
- (c) 2π/3
- (d) None of the above
Answer: (a) π/3
Evaluate cos⁻¹(-1).
- (a) 0
- (b) π/2
- (c) π
- (d) None of the above
Answer: (c) π
What is tan(tan⁻¹ x) if x ∈ ℝ?
- (a) x
- (b) -x
- (c) |x|
- (d) None of the above
Answer: (a) x
Find the range of sin⁻¹(x).
- (a) [-π/2, π/2]
- (b) [0, π]
- (c) [-1, 1]
- (d) None of the above
Answer: (a) [-π/2, π/2]
What is the value of arcsin(-1)?
- (a) -π/2
- (b) π/2
- (c) 0
- (d) None of the above
Answer: (a) -π/2
Evaluate sin⁻¹(0.5) + cos⁻¹(0.5).
- (a) π/2
- (b) π
- (c) 0
- (d) None of the above
Answer: (a) π/2
If arcsin(x) = π/6, what is x?
- (a) 1/2
- (b) √3/2
- (c) 1
- (d) None of the above
Answer: (a) 1/2
What is the principal value of cos⁻¹(1)?
- (a) 0
- (b) π
- (c) π/2
- (d) None of the above
Answer: (a) 0
Mixed MCQs
The value of sin⁻¹(sin(3π/4)) is:
- (a) π/4
- (b) -π/4
- (c) π/2
- (d) None of the above
Answer: (a) π/4
If cos⁻¹(x) = π/3, find x.
- (a) 1/2
- (b) √3/2
- (c) -1/2
- (d) None of the above
Answer: (b) √3/2
Find the domain of sin⁻¹(x) + cos⁻¹(x).
- (a) [0, π]
- (b) [-1, 1]
- (c) (-∞, ∞)
- (d) None of the above
Answer: (b) [-1, 1]
Evaluate tan⁻¹(√3).
- (a) π/6
- (b) π/3
- (c) π/4
- (d) None of the above
Answer: (b) π/3
Solve: sin⁻¹(1) + cos⁻¹(1).
- (a) π
- (b) π/2
- (c) 0
- (d) None of the above
Answer: (c) 0
If arcsin(x) + arccos(x) = π/2, then x equals:
- (a) 0
- (b) 1
- (c) -1
- (d) Any real number
Answer: (b) 1
What is the principal value of tan⁻¹(1)?
- (a) π/2
- (b) π/4
- (c) 0
- (d) None of the above
Answer: (b) π/4
The value of cos⁻¹(0) is:
- (a) π/4
- (b) π/2
- (c) π
- (d) None of the above
Answer: (b) π/2
The domain of tan⁻¹(x) is:
- (a) [-1, 1]
- (b) [0, π]
- (c) (-∞, ∞)
- (d) None of the above
Answer: (c) (-∞, ∞)
If y = arcsin(sin(2π/3)), what is y?
- (a) π/3
- (b) -π/3
- (c) 2π/3
- (d) None of the above
Answer: (a) π/3
These 30 MCQs, along with their answers, will help students prepare thoroughly for the West Bengal Higher Secondary Test in the chapters "Relations and Functions" and "Inverse Trigonometric Functions."
The value of sin⁻¹(sin(3π/4)) is:
- (a) π/4
- (b) -π/4
- (c) π/2
- (d) None of the above
Answer: (a) π/4
If cos⁻¹(x) = π/3, find x.
- (a) 1/2
- (b) √3/2
- (c) -1/2
- (d) None of the above
Answer: (b) √3/2
Find the domain of sin⁻¹(x) + cos⁻¹(x).
- (a) [0, π]
- (b) [-1, 1]
- (c) (-∞, ∞)
- (d) None of the above
Answer: (b) [-1, 1]
Evaluate tan⁻¹(√3).
- (a) π/6
- (b) π/3
- (c) π/4
- (d) None of the above
Answer: (b) π/3
Solve: sin⁻¹(1) + cos⁻¹(1).
- (a) π
- (b) π/2
- (c) 0
- (d) None of the above
Answer: (c) 0
If arcsin(x) + arccos(x) = π/2, then x equals:
- (a) 0
- (b) 1
- (c) -1
- (d) Any real number
Answer: (b) 1
What is the principal value of tan⁻¹(1)?
- (a) π/2
- (b) π/4
- (c) 0
- (d) None of the above
Answer: (b) π/4
The value of cos⁻¹(0) is:
- (a) π/4
- (b) π/2
- (c) π
- (d) None of the above
Answer: (b) π/2
The domain of tan⁻¹(x) is:
- (a) [-1, 1]
- (b) [0, π]
- (c) (-∞, ∞)
- (d) None of the above
Answer: (c) (-∞, ∞)
If y = arcsin(sin(2π/3)), what is y?
- (a) π/3
- (b) -π/3
- (c) 2π/3
- (d) None of the above
Answer: (a) π/3
These 30 MCQs, along with their answers, will help students prepare thoroughly for the West Bengal Higher Secondary Test in the chapters "Relations and Functions" and "Inverse Trigonometric Functions."
Preparation Tips for WBHS Test
- Focus on Concepts: Ensure clarity on definitions and properties.
- Practice Problems: Solve numerical problems and previous years' MCQs.
- Revise Regularly: Dedicate time to memorizing key formulas and values.
Conclusion
The chapters on Relations and Functions and Inverse Trigonometric Functions are fundamental to Class 12 Mathematics. Mastering them ensures a strong foundation for competitive exams like WBHS. Leverage this guide, practice rigorously, and ace your tests confidently!
~ For more tips and resources, visit MightyKnowledge and enhance your learning journey today!
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