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World Biofuel Day

World Biofuel Day is observed on August 10 every year. It is observed to raise awareness about the importance of alternatives to conventional... View Article

Himalayan Ibex

The Himalayan Ibex (Capra sibirica hemalayanus) is a species of wild goat found in the trans-himalayan regions of Jammu and Kashmir and Ladakh.... View Article

If cos^-1 x – cos^-1 (y / 2) = ?, then 4x² – 4xy cos ? + y² =

1) 4 sin2 ɑ 2) - 4 sin2 ɑ 3) 2 sin 2ɑ 4) 4 Solution: (1) 4 sin2 ɑ If cos-1 (x / a) + cos-1 (y / b) = θ (x2 / a2) - (2xy / ab) cos... View Article

cos^-1 [3 + 5 cos x] / [5 + 3 cos x] =

1) tan-1 [(1 / 2) tan (x / 2)] 2) 2 tan-1 [2 tan (x / 2)] 3) (1 / 2) tan-1 (2 tan (x / 2)) 4) 2 tan-1 [(1 / 2) tan (x / 2)] 5) tan-1... View Article

sin [(1 / 2) cos^-1 (4 / 5)] =

1) 1 / √10 2) - 1 / √10 3) 1 / 10 4) - 1 / 10 Solution: (1) 1 / √10 Let cos-1 (4 / 5) = x cos x = 4 / 5 ---- (i) sin [(1 / 2)... View Article

The value of sin [2 tan^-1 (1 / 3)] + cos (tan^-1 2?2) =

1) 16 / 15 2) 14 / 15 3) 12 / 15 4) 11 / 15 Solution: (2) 14 / 15 sin [2 tan-1 (1 / 3)] + cos (tan-1 2√2) sin [2 tan-1 (1 / 3) ]... View Article

If 3 sin^-1 [2x] / [1 – x²] – 4 cos^-1 [1 – x²] / [1 + x²] + 2 tan^-1 [2x] / [1 – x²] = ? / 3, then x =

1) √3 2) 1 / √3 3) 1 4) None of these Solution: (2) 1 / √3 3 sin-1 [2x] / [1 - x2] - 4 cos-1 [1 - x2] / [1 + x2] + 2 tan-1 [2x] /... View Article

The value of cos^-1 (cos 12) – sin^-1 (sin 14) is

1) - 2 2) 8π - 26 3) 4π + 2 4) None of these Solution: (2) 8π - 26 cos-1 (cos 12) - sin-1 (sin 14) = 4π - 12 - (14 - 4π) = 8π - 26... View Article

If tan^-1 x + tan^-1 y = ? / 4, then

1) x + y + xy = 1 2) x + y - xy = 1 3) x + y + xy + 1 = 0 4) x + y - xy + 1 = 0 Solution: (1) x + y + xy = 1 Let tan-1 x + tan-1... View Article

4 tan^-1 (1 / 5) – tan^-1 (1 / 239) =

1) π 2) π / 2 3) π / 3 4) π / 4 Solution: (4) π / 4 2 tan-1 x = tan-1 [2x] / [1 - x2] 4 tan-1 (1 / 5) = 2 [2 tan-1 (1 / 5)] = 2... View Article

3 tan^-1 a =

1) tan-1 [3a + a3] / [1 + 3a2] 2) tan-1 [3a - a3] / [1 + 3a2] 3) tan-1 [3a + a3] / [1 - 3a2] 4) tan-1 [3a - a3] / [1 - 3a2] Solution:... View Article

sin (4 tan^-1 1 / 3) =

1) 12 / 25 2) 24 / 25 3) 1 / 5 4) None of these Solution: (2) 24 / 25 sin (4 tan-1 1 / 3) = sin [2 tan-1 (2 / 3) / (1 - (1 / 9))]... View Article

(1 / 2) cos^-1 [1 – x] / [1 + x] =

1) cot-1 √x 2) tan-1 √x 3) tan-1 x 4) cot-1 x Solution: (2) tan-1 √x Let x = tan2 θ θ = tan-1 √x (1 / 2) cos-1 [1 - x] / [1 + x]... View Article

tan [(1 / 2) cos^-1 (?5 / 3)] =

(a) [3 - √5] / 2 (b) [3 + √5] / 2 (c) 2 / [3 - √5] (d) 2 / [3 + √5] 1) a and d 2) d only 3) b only 4) None of these... View Article

If 2 cos^-1 ?(1 + x) / 2 = ? / 2, then x =

1) 1 2) 0 3) -1/2 4) 1/2 Solution: (2) 0 2 cos-1 √(1 + x) / 2 = π / 2 cos-1 √(1 + x) / 2 = π / 4 cos (π / 4) = √(1 + x) / 2 (1... View Article

If sin^-1 x + cot^-1 (1 / 2) = ? / 2, then x is

1) 0 2) 1 / √5 3) 2 / √5 4) √3 / 2 Solution: (2) 1 / √5 sin-1 x + cot-1 (1 / 2) = π / 2 sin-1 x = (π / 2) - cot-1 (1 / 2) =... View Article

If tan^-1 x + 2 cot^-1 x = 2? / 3, then x =

1) √2 2) 3 3) √3 4) [√3 - 1] / [√3 + 1] Solution: (3) √3 tan-1 x + 2 cot-1 x = 2π / 3 tan-1 x + 2 [(π / 2) - tan-1 x] = 2π / 3 -... View Article

The solution of sin^-1 x – sin^-1 2x = ± (? / 3) is

1) ± 1 / 3 2) ± 1 / 4 3) ± √3 / 2 4) ± 1 / 2 Solution: (4) ± 1 / 2 sin-1 x - sin-1 2x = ± (π / 3) sin-1 2x = sin-1 x - sin-1 (√3... View Article

tan^-1 (1 / 2) + tan^-1 (1 / 3) =

1) 0 2) π / 4 3) π / 2 4) π Solution: (2) π / 4 tan-1 (1 / 2) + tan-1 (1 / 3) = tan-1 [(1 / 2) + (1 / 3)] / [1 - (1 / 2) (1 /... View Article

For the equation cos^-1 x + cos^-1 2x + ? = 0, the number of real solution is

1) 1 2) 2 3) 0 4) Infinity Solution: (3) 0 cos-1 x + cos-1 2x + π = 0 cos-1 x + cos-1 2x = - π cos-1 (2x) = - π - cos-1 x 2x =... View Article