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RBSE Solutions for Class 12 Maths Chapter 2 Inverse Circular Functions Ex 2.1

May 23, 2019 by Fazal Leave a Comment

Rajasthan Board RBSE Class 12 Maths Chapter 2 Inverse Circular Functions Ex 2.1

Question 1.
Find the principal value of the following angles:
RBSE Solutions for Class 12 Maths Chapter 2 Inverse Circular Functions Ex 2.1
Solution:
(i) sin-1(1)
RBSE Solutions for Class 12 Maths Chapter 2 Inverse Circular Functions Ex 2.1
RBSE Solutions for Class 12 Maths Chapter 2 Inverse Circular Functions Ex 2.1
RBSE Solutions for Class 12 Maths Chapter 2 Inverse Circular Functions Ex 2.1
RBSE Solutions for Class 12 Maths Chapter 2 Inverse Circular Functions Ex 2.1

Prove the following : (Q. 2 to 8)
Question 2.
RBSE Solutions for Class 12 Maths Chapter 2 Inverse Circular Functions Ex 2.1
Solution:
RBSE Solutions for Class 12 Maths Chapter 2 Inverse Circular Functions Ex 2.1

Question 3.
RBSE Solutions for Class 12 Maths Chapter 2 Inverse Circular Functions Ex 2.1
Solution:
RBSE Solutions for Class 12 Maths Chapter 2 Inverse Circular Functions Ex 2.1

Question 4.
RBSE Solutions for Class 12 Maths Chapter 2 Inverse Circular Functions Ex 2.1
Solution:
RBSE Solutions for Class 12 Maths Chapter 2 Inverse Circular Functions Ex 2.1
RBSE Solutions for Class 12 Maths Chapter 2 Inverse Circular Functions Ex 2.1

Question 5.
sec2 (tan-1 2) + cosec2 (cot-1 3) = 15
Solution:
Let tan-12 = θ
⇒ tan θ = 2
sec2 θ = 1 + tan2 θ
= 1 + (2)2
= 1 + 4 = 5
∴ sec2 (tan-12) = 5
Let cot-13 = Φ = cot Φ = 3
∴ cosec2 Φ = 1 + cot2 Φ
= 1 + (3)2 = 1 + 9 = 10
∴ cosec2(cot-1 3) = 10 …..(ii)
Adding eq. (i) and (ii), we get
sec2 (tan-12) + cosec2 (cot-13) = 5 + 10
⇒ sec2 (tan-12) + cosec2(cot-13) = 15
Hence proved.

Question 6.
RBSE Solutions for Class 12 Maths Chapter 2 Inverse Circular Functions Ex 2.1
Solution:
RBSE Solutions for Class 12 Maths Chapter 2 Inverse Circular Functions Ex 2.1
RBSE Solutions for Class 12 Maths Chapter 2 Inverse Circular Functions Ex 2.1

Question 7.
RBSE Solutions for Class 12 Maths Chapter 2 Inverse Circular Functions Ex 2.1
Solution:
RBSE Solutions for Class 12 Maths Chapter 2 Inverse Circular Functions Ex 2.1

Question 8.
RBSE Solutions for Class 12 Maths Chapter 2 Inverse Circular Functions Ex 2.1
Solution:
RBSE Solutions for Class 12 Maths Chapter 2 Inverse Circular Functions Ex 2.1
RBSE Solutions for Class 12 Maths Chapter 2 Inverse Circular Functions Ex 2.1

Question 9.
If cos-1x + cos-1y + cos-1z = π, then prove that x2 + y2 + x2 + 2xyz = 1. Solution:
cos-1x + cos-1y + cos-1z = π
(According to question)
cos-1x + cos-1 y = π – cos-1z
RBSE Solutions for Class 12 Maths Chapter 2 Inverse Circular Functions Ex 2.1

Question 10.
If sin-1x + sin-1 y + sin-1z = π, then prove that:
RBSE Solutions for Class 12 Maths Chapter 2 Inverse Circular Functions Ex 2.1
Solution:
Let sin-1 x = A ⇒ x = sin A
sin-1 y = B ⇒ y = sin B
sin-1 z = C ⇒ z = sin C
∵ sin-1 x + sin-1 y + sin-1 z = π
∴ A + B + C = π
∴ To prove
RBSE Solutions for Class 12 Maths Chapter 2 Inverse Circular Functions Ex 2.1
RBSE Solutions for Class 12 Maths Chapter 2 Inverse Circular Functions Ex 2.1

Question 11.
If tan-1x + tan-1y + tan-1z = π/2, then prove that xy + yz + zx = 1.
Solution:
According to question,
RBSE Solutions for Class 12 Maths Chapter 2 Inverse Circular Functions Ex 2.1

Question 12.
If
RBSE Solutions for Class 12 Maths Chapter 2 Inverse Circular Functions Ex 2.1
then prove that x + y + z = xyz.
Solution:
Let x = tan A, y = tan B, z = tan C
RBSE Solutions for Class 12 Maths Chapter 2 Inverse Circular Functions Ex 2.1
RBSE Solutions for Class 12 Maths Chapter 2 Inverse Circular Functions Ex 2.1

Question 13.
If
RBSE Solutions for Class 12 Maths Chapter 2 Inverse Circular Functions Ex 2.1
then prove that x + y + z = xyz.
Solution:
According to question
RBSE Solutions for Class 12 Maths Chapter 2 Inverse Circular Functions Ex 2.1

Question 14.
Prove that :
tan-1x + cot-1(x + 1) = tan-1(x2 + x + 1).
Solution:
L.H.S. = tan-1x + cot-1(x + 1)
RBSE Solutions for Class 12 Maths Chapter 2 Inverse Circular Functions Ex 2.1

Question 15.
If tan-1x, tan-1y, tan-1z are in arithmetic progression, then prove that
y2(x +z) + 2y(1 – xz) – x – z = 0.
Solution:
tan-1x, tan-1y, tan-1z are in A.P.
RBSE Solutions for Class 12 Maths Chapter 2 Inverse Circular Functions Ex 2.1

Question 16.
If α, β, γ be the roots of equation x3 + px2 + qx + p = 0, then prove that, except a special condition,
tan-1α + tan-1 β + tan-1γ = nπ also find the special condition, when it does so on.
Solution:
Given, α, β, γ are root of equation x3+ px2 + qx + p = 0
RBSE Solutions for Class 12 Maths Chapter 2 Inverse Circular Functions Ex 2.1
RBSE Solutions for Class 12 Maths Chapter 2 Inverse Circular Functions Ex 2.1
Special Situation: When sum and product of roots is not equal, then tan-1 α + tan-1 β + tan-1γ ≠ nπ

Solve the following euqations : (Q. 17 to 25)

Question 17.
RBSE Solutions for Class 12 Maths Chapter 2 Inverse Circular Functions Ex 2.1
Solution:
RBSE Solutions for Class 12 Maths Chapter 2 Inverse Circular Functions Ex 2.1
RBSE Solutions for Class 12 Maths Chapter 2 Inverse Circular Functions Ex 2.1

Question 18.
RBSE Solutions for Class 12 Maths Chapter 2 Inverse Circular Functions Ex 2.1
Solution:
RBSE Solutions for Class 12 Maths Chapter 2 Inverse Circular Functions Ex 2.1

Question 19.
RBSE Solutions for Class 12 Maths Chapter 2 Inverse Circular Functions Ex 2.1
Solution:
RBSE Solutions for Class 12 Maths Chapter 2 Inverse Circular Functions Ex 2.1
RBSE Solutions for Class 12 Maths Chapter 2 Inverse Circular Functions Ex 2.1

Question 20.
RBSE Solutions for Class 12 Maths Chapter 2 Inverse Circular Functions Ex 2.1
Solution:
RBSE Solutions for Class 12 Maths Chapter 2 Inverse Circular Functions Ex 2.1
RBSE Solutions for Class 12 Maths Chapter 2 Inverse Circular Functions Ex 2.1

Question 21.
RBSE Solutions for Class 12 Maths Chapter 2 Inverse Circular Functions Ex 2.1
Solution:
RBSE Solutions for Class 12 Maths Chapter 2 Inverse Circular Functions Ex 2.1
RBSE Solutions for Class 12 Maths Chapter 2 Inverse Circular Functions Ex 2.1

Question 22.
RBSE Solutions for Class 12 Maths Chapter 2 Inverse Circular Functions Ex 2.1
Solution:
RBSE Solutions for Class 12 Maths Chapter 2 Inverse Circular Functions Ex 2.1
RBSE Solutions for Class 12 Maths Chapter 2 Inverse Circular Functions Ex 2.1

Question 23.
sin 2 (cos-1 {cot (2 tan-1x)} = 0
Solution:
Given equation
RBSE Solutions for Class 12 Maths Chapter 2 Inverse Circular Functions Ex 2.1
RBSE Solutions for Class 12 Maths Chapter 2 Inverse Circular Functions Ex 2.1

Question 24.
RBSE Solutions for Class 12 Maths Chapter 2 Inverse Circular Functions Ex 2.1
Solution:
RBSE Solutions for Class 12 Maths Chapter 2 Inverse Circular Functions Ex 2.1
RBSE Solutions for Class 12 Maths Chapter 2 Inverse Circular Functions Ex 2.1

Question 25.
RBSE Solutions for Class 12 Maths Chapter 2 Inverse Circular Functions Ex 2.1
Solution:
According to question,
RBSE Solutions for Class 12 Maths Chapter 2 Inverse Circular Functions Ex 2.1
RBSE Solutions for Class 12 Maths Chapter 2 Inverse Circular Functions Ex 2.1

RBSE Solutions for Class 12 Maths

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