CBSE Guess > Papers > Important Questions > Class XII > 2010 > Maths > Mathematics By Mr. M.P.Keshari
CBSE CLASS XII
2.1 Basic Concept.
Q.1. Find the principal value of the following :
- tan –1(-1).
- cos –1 (√3/2).
Solution :
- tan–1(–1) = – tan –1(1) = – π/4. [Ans.]
- cos –1(√3/2) = cos –1(cos π/6) = π/6. [Ans.]
Q.2. Find the value of the following :
Using principal value, evaluate the following :
cos –1(cos2π/3) + sin–1(sin2π/3).
Solution :
We have, cos –1[cos(2π/3)] + sin –1[sin(2π/3)]
= cos –1[cos(π – π/3)] + sin–1[sin(π – π/3)]
= – π/3 + π/3
= 0 [Ans.]
2.2. Properties of Inverse Trigonometric Function.
Q.1. Find the value of each of the following :
sin[π/3 – sin –1(–1/2)]
Solution :
sin[π/3 – sin –1(–1/2)] = sin[π/3 + sin –1(1/2)]
= sin[π/3 + π/6]
= sin π/2 = 1. [Ans.]
Q.2. Prove the following :
tan –1 1/3 + tan –1 1/5 + tan –1 1/7 + tan –1 1/8 = π/4.
Proof :
[tan –1 1/3 + tan –1 1/5] + [tan –1 1/7 + tan–1 1/8]
= [tan–1 (1/3 + 1/5)/(1 – 1/3×1/5)] + [tan –1 (1/7 + 1/8)/(1 – 1/7×1/8)]
= [tan –1 (8/15)/(1 – 1/15)] + [tan–1 (15/56)/(1 – 1/56)]
= tan–1 (8/14) + tan –1 (15/55)
= tan–1 4/7 + tan –1 3/11
= tan–1 (4/7 + 3/11)/(1 – 4/7×3/11)
= tan –1 (65/77)/(1 – 12/77)
= tan –1 (65/65)
= tan –1 (1)
= π/4. [Proved.]
Q.3. Prove the following :
tan –1(1/2) + tan –1(1/5) + tan –1(1/8) = π/4.
Solution :
L.H.S. = tan –1 (1/2) + tan–1 (1/5) + tan –1 (1/8)
= tan –1[(1/2 + 1/5)/{1 – (1/2)(1/5)}] + tan –1(1/8)
= tan –1[{(5 + 2)/10}/{(10 – 1)/10}] + tan –1 (1/8)
= tan –1[7/10×10/9] + tan – 1(1/8)
= tan –1(7/9) + tan – 1(1/8)
= tan –1[(7/9 + 1/8)/{1 – (7/9)(1/8)}]
= tan –1 [{(56 + 9)/72}/{(72 – 7)/72}]
= tan –1[65/72×72/65]
= tan –1(1)
= π/4. [Proved.]
Paper By Mr. M.P.Keshari
Email Id : [email protected]
Ph No. : 09434150289
