CBSE Class 12 Maths 2024-25: Chapter 2 Inverse Trigonometric Functions Important Competency-Based Questions with Answers; Download Free PDF

As the CBSE Class 12 board exams get closer, it’s important for students to understand the new exam pattern. Starting in the 2024-25 school year, CBSE will include 50% more competency-based questions. These questions will be both multiple choice and written, focusing on how to use what students have learned in real-life situations.

This article explores Chapter 2 Inverse Trigonometric Functions It highlights key competency-based questions and provides answers to help students succeed.

Understanding Competency-Based Questions in Chapter 2 Inverse Trigonometric Functions

Competency-based questions are designed to see how well students can apply their knowledge in everyday life. They can come in different forms, such as case studies, true-false questions, gap-filling tasks, and long or short answer questions.

These questions are different from regular memorization. They encourage students to think critically and solve problems, helping them understand the concepts in Chapter 2 Inverse Trigonometric Functions better.

CBSE Class 12 Maths Chapter 2 Inverse Trigonometric Functions Important Competency-Based Questions

Multiple Choice Questions :

Q1. What is the domain of the function y = sec-¹ x + sin-¹ x ?

1. -1 and 1

2. [-1, 1]

3. (-∞, -1] U [1, ∞)

4. Φ

Ans. 1. -1 and 1

Q2.

1. Expression 1

2. Expression 2

3. Expression 3

4. Expression 4

Ans. 3. Expression 3

Q3. Which of the following is the domain of the function given below?

y = cos^-1(1/x-3)

1. [-1, 1]

2. [2, 4]

3. (-∞, 2] U [4, ∞)

4. (-∞, -1] U [1, ∞)

Ans. 3. (-∞, 2] U [4, ∞)

Q4. The domain of f ( x ) = sin¹ x - cos¹ x is [-1, 1] while its range is [-3π/2, π/2].

If g ( x ) is the inverse of f ( x ), which of the following is true about the domain of the function g ( x )?

1. It is [-1, 1].

2. It is [-3п/2, п/2].

3. It is independent of the domain and range of f (x).

4. (cannot be said without knowing g(x))

Ans. 2. It is [-3п/2, п/2].

Q5. If sec^-1 (- x ) = п/8, which of the following could be the value of sec-1 ( x )?

1. (-п/8)

2. 7п/8

3. 9п/8

4. (cannot be determined without knowing the value of x )

Ans. 2. 7п/8

Q6. Which of the following is equal to -tan^-1( 2/3π )?

1. cot^-1 ( 3π/2 )

2. -cot^-1 ( 3π/2 )

3. π/2 - cot^-1 ( 3π/2 )

4. π/2 + cot^-1 ( 3π/2 )

Ans. 2. -cot^-1 ( 3π/2 )

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CBSE Class 12 Maths Chapter 1: Relations and Functions Important Competency-Based Questions	Click Here
CBSE Class 12 Maths Chapter 2 : Inverse Trigonometric Functions Important Competency-Based Questions	Click Here

Free Response Questions :

Q7. Akash says that, since the domain of the sine function is (-∞, ∞), sin^-1 x is well defined in the domain (-∞, ∞).

Is Akash right or wrong? Justify your answer.

Ans. Writes that Akash is wrong.

Writes that sin^-1 x cannot be defined in the domain (-∞, ∞) as sin x is not one-one in that domain.

(Award full marks for any other valid reason.)

Q8. Simplify:

cos( π/2 + sin^-1 1/√3)

Show your work.

Ans. Simplifies the above expression as:

cos( π/2 + sin^-1 1/√3) = -sin(sin^-1 1/√3)

Simplifies the above expression as (- 1/√3).

Q9. While solving an inverse trigonometry problem on the blackboard, Satish wrote the following as part of his solution:

His teacher stopped him and said that he must have made a mistake in the solution.

How did the teacher recognise that Satish had made a mistake? Justify your answer.

Ans. Writes that the cos^-1 function is not defined for 5/3 as it is outside the domain of the inverse cosine function. Therefore, the teacher stopped Satish here.

Q10. cosec π/6 = cosec 5π/6 = 2, but π/6 ≠ 5π/6·

Since the cosecant function is not one-one, how can it be made invertible? Give a reason for your answer.

Ans. Writes that the cosecant function can be made invertible when its domain is restricted to [n π- π/2‚ п π + π/2 ] - {0}, where n is an integer.

Q11. Prove that:

Ans. Rewrites the LHS of the given equation as:

Uses the property of inverse trigonometric functions and writes:

Q12. Chirag asked his students to find the value of:

tan^-1 (- x ) - tan^-1 ( 1/x )

Rahul said that the value of the above expression can be found ONLY if x is known.

Is Rahul correct? Justify your answer.

Ans. Writes that Rahul is incorrect.

Writes that tan^-1 (- x ) - tan^-1 1/x can be written as -(tan^-1 x + cot^-1 x ).

Finds the value of the given expression as (- π/2 ) as tan^-1 x + cot^-1 x = π/2.

Q13. Considering the principal value branch, prove that the property below is true ONLY for xy > (-1).

tan^-1 x - tan^-1 y = tan^-1 x-y/1+xy

Ans. Assumes xy < (-1), x = tan θ and y = tan Θ.

Rewrites the above inequality as tan θ < tan ( Θ - π/2 ).

Writes that, since tangent is an increasing function in the principal value branch, 0 < ( Θ - π/2 ).

Uses steps 1 and 3 to write tan^-1 x - tan^-1 y < - π/2.

Uses the above step to conclude that:

If xy < (-1), the value of tan^-1 x - tan^-1 y ∉(-π/2, π/2 ).

Hence, proves that the given property is true only for xy > (-1).

Case-Based Type Questions :

Q14. Answer the questions based on the given information.

Madhu created a design on his floor using a combination of graphs of inverse trigonometric functions in the domain [-4, 4]. He also represented the coordinate axes for reference. He asked his friends, Kulsum and Snehal, to choose a path to walk on. Kulsum chose path ABCDEF, while Snehal chose path PQCRES. They both started walking at the same time and with the same speed.

1. Write the range of each of the two functions that Kulsum chose as her path to walk on.

Ans. Identifies two functions that Kulsum chose to walk on as cos^-1 ( x ) and sec^-1 ( x ).

Writes that the range of cos^-1 ( x ) is [0, π].

Writes that the range of sec^-1 ( x ) is [0, π] - π/2.

2. The graphs of which of the two functions are combined to form the path that Snehal chose?

Ans. Identifies the trigonometric functions that Snehal chose to walk on as sin^-1 ( x ) and cosec^-1 ( x ).

3. What is the x -coordinate of Kulsum's and Snehal's first meeting point? Show your steps.

Ans. Mentions that both friends met at point C which is the point of intersection of two 0.5 functions, sin^-1 ( x ) and cos^-1 ( x ) and writes:

cos^-1 ( x ) = sin^-1 ( x )

Simplifies the above equation as:

cos(cos^-1 ( x )) = cos(sin^-1 ( x ))

=> x = cos(sin^-1( x ))

Substitutes sin^-1 ( x ) as u and simplifies the above equation as:

sin u = cos u

=> sin u = sin( π/2 - u )

=> u = π/4

(Award full marks if cos^-1 ( x ) and sin^-1 ( x ) are shown to be equal at y = π/4 directly.)

Finds the x -coordinate of Kulsum's and Snehal's first meeting point as:

x = sin u = sin ( π/4 ) = 1/√2

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👉 CBSE Class 12 Study Materials

CBSE Class 12 Syllabus 2024-25	CBSE Class 12 Previous Year Papers
NCERT Books For Class 12 Books	NCERT Class 12 Solutions
CBSE Class 12 Full Study Material	CBSE Class 12 Sample Paper 2023-24