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 :

1. tan ^–1(-1).
2. cos ^–1 (√3/2).

Solution :

1. tan^–1(–1) = – tan ^–1(1) = – π/4. [Ans.]
2. 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