CBSE Guess > Papers > Important Questions > Class XII > 2010 > Maths > Mathematics By Mr. M.P.Keshari

CBSE CLASS XII

Q.4. Prove the following :

2 tan ^–1(1/2) + tan ^–1(1/7) = tan ^–1(31/17).

Solution :

We have, L.H.S. = 2tan ^–1(1/2) + tan ^–1(1/7)
= tan ^–1[{2.(1/2)}/{1 – (1/2)2} + tan ^–1(1/7)
= tan^–1(4/3) + tan ^–1(1/7)
= tan ^–1[(4/3 + 1/7)/{1 – (4/3)(1/7)}]
= tan ^–1[{(28 + 3)/21}/{(21 – 4)/21}]
= tan ^–1[(31/21)(21/17)]
= tan ^–1(31/17) = R.H.S. [Proved.]

Q.6. Solve the following for x :

1. tan ^–1(2x) + tan ^–1(3x) = π/4.
   
   Solution:
   
   We have, tan ^–1(2x) + tan ^–1(3x) = π/4
   Or, tan ^–1 [(2x + 3x)/{1 – (2x)(3x)}] = π/4
   Or, 5x/(1 – 6x²) = tan π/4
   Or, 5x/(1 – 6x²) = 1
   Or, 5x = 1 – 6x²
   Or, 6x²+ 5x – 1 = 0
   Or, (6x – 1)(x + 1) = 0
   Or, x = 1/6, –1.
   As x = –1 does not satisfy the equation,
   x = 1/6 is the required solution. [Ans.]
   
   
2. tan ^–1[(x – 1)/(x – 2)] + tan ^–1[(x + 1)/(x + 2)] = π/4.
   
   Solution :
   
   We have, tan ^–1 [(x – 1)/(x – 2)] + tan ^–1 [(x + 1)/(x + 2)] = π/4
   Or, tan^–1[[{(x –1)/(x –2)} + {(x + 1)/(x + 2)}]/{1– (x –1)/(x –2).(x + 1)/(x + 2)}]
   = π/4
   Or, tan^–1{ [{(x –1)(x + 2) + (x + 1)(x –2)}/{(x –2)(x + 2)}]
   /[{(x –2)(x + 2) – (x –1)(x + 1)}/{(x –2)(x + 2)}]} = π/4
   Or, (x² + x – 2 + x2 – x – 2)/(x2 – 4 – x² + 1) = tan (π/4)
   Or, (2x² – 4)/ –3 = 1
   Or, 2x² – 4 = – 3
   Or, 2x² – 1 = 0
   Or, x² = 1/2
   Or, x = ±1/√2. [Ans.]
   
   
3. tan –1[(1 + x)/(1 – x)] = π/4 + tan –1x
   
   Solution :
   
   We have, tan ^–1[(1 + x)/(1 – x)] = π/4 + tan –1x
   Or, tan ^–1[(1 + x)/(1 – x)] – tan ^–1x = π/4
   Or, tan^–1[{(1 + x)/(1 – x)} – x]/[1 + {(1 + x)/(1 – x)}.x] = π/4
   Or, [{(1 + x) – x (1 – x)}/(1 – x)]/[{(1 – x) + x(1 + x)}= tan π/4
   Or, [(1 + x – x + x²)/(1 – x)]×[(1 – x)/(1 – x + x + x²)] = 1
   Or, (1 + x²)/(1 + x²) = 1
   Or, 1 + x² = 1 + x²
   Or, 1 = 1
   Therefore, equation has infinitely many solutions. [Ans.]
   
   
4. tan ^–1{(1 – x)/(1 + x)} – 1/2tan –1 x = 0; x > 0.
   
   Solution :
   
   Do yourself. [Ans. x = ±1/√3]
   

Paper By Mr. M.P.Keshari
Email Id : [email protected]
Ph No. : 09434150289