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RBSE Solutions for Class 12 Maths Chapter 2 प्रतिलोम वृत्तीय फलन Miscellaneous Exercise

May 3, 2019 by Fazal Leave a Comment

Rajasthan Board RBSE Class 12 Maths Chapter 2 प्रतिलोम वृत्तीय फलन Miscellaneous Exercise

प्रश्न 1.
tan-1 (-1) का मुख्य मान है
(a) 45°
(b) 135°
(c) – 45°
(d) – 60°
हल :
∵ tan-1 (-x) = – tan-1 x
∴ tan-1 (-1) = – tan-1 (1)
माना tan-1 1 = θ
∴ tan θ = 1
tan θ = tan 45°
∴ θ = 45°
∴ tan-1 (-1) = – 45°
अत: सही विकल्प (c) है।

प्रश्न 2.
2 tan-1 (1/2) बराबर है
RBSE Solutions for Class 12 Maths Chapter 2 Additional Questions 1
हल :
RBSE Solutions for Class 12 Maths Chapter 2 Additional Questions 2
अत: सही विकल्प (a) है।

प्रश्न 3.
यदि tan-1 (3/4) = θ, तो sin θ का मान है
RBSE Solutions for Class 12 Maths Chapter 2 Additional Questions 3
हल :
प्रश्नानूसार,
RBSE Solutions for Class 12 Maths Chapter 2 Additional Questions 4
अतः सही उत्तर का विकल्प (b) है।

प्रश्न 4.
cot [tan-1 α + cot-1 α] का मान है
(a) 1
(b) ∞
(c) 0
(d) इनमें से कोई नहीं
हल :
cos (tan-1 α + cot-1 α).
= cot \(\frac { \pi }{ 2 } \) (∵ tan-1 x + cot-1 x = \(\frac { \pi }{ 2 } \))
= 0
अत: सही उत्तर का विकल्प (c) है।

प्रश्न 5.
यदि \({ sin }^{ -1 }\frac { 1 }{ 2 } =x\) तो का व्यापक मान है
RBSE Solutions for Class 12 Maths Chapter 2 Additional Questions 5
हल :
दिया है,
RBSE Solutions for Class 12 Maths Chapter 2 Additional Questions 6
अत: सही उत्तर का विकल्प (d) है।

प्रश्न 6.
2 tan (tan-1 x + tan-1 x³) का मान है
(a) \(\frac { 2x }{ 1-{ x }^{ 2 } } \)
(b) 1 + x²
(c) 2x
(d) इनमें से कोई नहीं
हल :
RBSE Solutions for Class 12 Maths Chapter 2 Additional Questions 7
अतः सही विकल्प (a) है।

प्रश्न 7.
यदि tan-1 (3x) + tan-1 (2x) = \(\frac { \pi }{ 2 } \). तो x का मान
RBSE Solutions for Class 12 Maths Chapter 2 Additional Questions 8
हल :
tan-1 (3x) + tan-1 (2x) = \(\frac { \pi }{ 2 } \)
RBSE Solutions for Class 12 Maths Chapter 2 Additional Questions 9
⇒ \(\frac { 5x }{ 1-{ 6x }^{ 2 } } =1\)
⇒ 1 – 6x² = 5x
⇒ 6x² + 5x – 1 = 0
⇒ 6x² + 6x – x – 1 = 0
⇒ 6x(x + 1) – 1(x + 1) = 0
⇒ (x + 1) (6x – 1) = 0
⇒ x = – 1,x = \(\frac { 1 }{ 6 }\)
अतः सही विकल्प (a) है।

प्रश्न 8.
RBSE Solutions for Class 12 Maths Chapter 2 Additional Questions 10
हल :
RBSE Solutions for Class 12 Maths Chapter 2 Additional Questions 11
अतः सही विकल्प (c) है।

प्रश्न 9.
यदि tan-1 (1) + cos-1 \(\left( \frac { 1 }{ \sqrt { 2 } } \right) \) = sin-1 x, तो x का मान है
(a) -1
(b) 0
(c) 1
(d) \(-\frac { 1 }{ 2 }\)
हल :
RBSE Solutions for Class 12 Maths Chapter 2 Additional Questions 12
अतः सही विकल्प (c) है।

प्रश्न 10.
यदि cot-1 x + tan-1 \(\left( \frac { 1 }{ 3 } \right) =\frac { \pi }{ 2 } \) तो x का मान है
(a) 1
(b) 3
(c) \(\frac { 1 }{ 3 }\)
(d) इनमें से कोई नहीं
हल :
RBSE Solutions for Class 12 Maths Chapter 2 Additional Questions 13
RBSE Solutions for Class 12 Maths Chapter 2 Additional Questions 14
अत: सही विकल्प (c) है।

प्रश्न 11.
यदि 4 sin-1 x + cos-1 x = π, तो x का मान कीजिए।
हल :
4 sin-1 x + cos-1 x = π
RBSE Solutions for Class 12 Maths Chapter 2 Additional Questions 15

प्रश्न 12.
RBSE Solutions for Class 12 Maths Chapter 2 Additional Questions 16
का मान कीजिए।
हल :
RBSE Solutions for Class 12 Maths Chapter 2 Additional Questions 17

प्रश्न 13.
यदि
RBSE Solutions for Class 12 Maths Chapter 2 Additional Questions 18
तो x का मान कीजिए।
हल :
RBSE Solutions for Class 12 Maths Chapter 2 Additional Questions 19

प्रश्न 14.
RBSE Solutions for Class 12 Maths Chapter 2 Additional Questions 20
का मान कीजिए।
हल :
RBSE Solutions for Class 12 Maths Chapter 2 Additional Questions 21
RBSE Solutions for Class 12 Maths Chapter 2 Additional Questions 22

प्रश्न 15.
यदि
RBSE Solutions for Class 12 Maths Chapter 2 Additional Questions 23
तो x का मान कीजिए।
हल :
RBSE Solutions for Class 12 Maths Chapter 2 Additional Questions 24

प्रश्न 16.
सिद्ध कीजिए कि :
RBSE Solutions for Class 12 Maths Chapter 2 Additional Questions 25
हल :
RBSE Solutions for Class 12 Maths Chapter 2 Additional Questions 26

प्रश्न 17.
यदि tan-1 x + tan-1 y + tan-1 z = π, तो सिद्ध कीजिए : x + y + z = xyz
हल :
tan-1 x + tan-1 y + tan-1 z = π
RBSE Solutions for Class 12 Maths Chapter 2 Additional Questions 27
x + y + z – xyz = 0 × (1 – xy – yz + xyz)
x + y + z – xyz = 0
अत: x + y + z = xyz
इति सिद्धम्।

प्रश्न 18.
सिद्ध कीजिए कि :
tan-1 [\(\frac { 1 }{ 2 }\) tan 2A] + tan-1 (cot A) + tan-1 (cot A) = 0.
हल :
tan-1 [\(\frac { 1 }{ 2 }\) tan 2A] + tan-1 (cot A) + tan-1 (cot² A) = 0.
RBSE Solutions for Class 12 Maths Chapter 2 Additional Questions 28

= 0
= R.H.S
इति सिद्धम्।

प्रश्न 19.
सिद्ध कीजिए कि :
tan-1 x = 2 tan-1 [cosec (tan-1 x) – tan (cot-1 x)]
हल :
माना tan-1 θ
RBSE Solutions for Class 12 Maths Chapter 2 Additional Questions 29
= tan-1 x
= RHS
इति सिद्धम्।

प्रश्न 20.
यदि \(\Phi ={ ta }n^{ -1 }\frac { x\sqrt { 3 } }{ 2K-x } \) और \(\theta ={ tan }^{ -1 }\frac { 2x-K }{ K\sqrt { 3 } } \) तो सिद्ध कीजिए Φ – θ का मान 30° है।
हल :
RBSE Solutions for Class 12 Maths Chapter 2 Additional Questions 30
RBSE Solutions for Class 12 Maths Chapter 2 Additional Questions 31
tan (Φ – θ) = tan 30°
∴ Φ – θ = 30°
इति सिद्धम्।

प्रश्न 21.
सिद्ध कीजिए कि :
RBSE Solutions for Class 12 Maths Chapter 2 Additional Questions 32
हल :
LHS
= 2 tan-1 [tan (45° – α) tan β/2]
RBSE Solutions for Class 12 Maths Chapter 2 Additional Questions 33
RBSE Solutions for Class 12 Maths Chapter 2 Additional Questions 34

RBSE Solutions for Class 12 Maths

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