If cos x y = cos y x, find dydx.
If sin y = x sin (a + y), prove that dydx = sin2 a + ysin a.
Let A = R – {3} and B = R – {1}. Consider the function f : A →B defined by f ( x ) = x - 2x - 3 . Show that f is one-one and onto and hence find f - 1.
Prove that tan- 1 cos x 1 + sin x = π4 - π2, x ∈ - π2, π2
Prove that sin- 1 817 + sin- 1 35 = cos- 1 3685 .
Let sin- 1 817 = x.Then, sin x =817; cos x = 1 - x2 ⇒ cos x = 1 - 817 2⇒ cos x = 225289⇒ cos x = 1517
∴ tan x = sin xcos x⇒ tan x = 8171517⇒ tan x = 815⇒ = x = tan- 1 815 ...........( i )Let sin- 1 35 = y ...........( ii )Then, sin y = 35; cos y = 1 - y2
⇒ cos y = 1 -352 ⇒ cos y = 1625 ⇒ cos y = 45∴ tan y = sin ycos y⇒ tan y = 3545⇒ tan y = 34⇒ y = tan- 1 34 ...............( iii )
From equation ( ii ) and ( iii ), we have,
sin- 1 35 = tan- 1 34 Now consider sin- 1 817 + sin- 1 35 :
From equation ( i ) and ( iii ), we have,
sin- 1 817 + sin- 1 35 = tan- 1 815 + tan- 1 34 = tan- 1 815 + 341 - 815 ×34 ....... ∵ tan- 1 x + tan- 1 y = tan- 1 x + y1 - xy = tan- 1 32 + 4560 - 24 sin- 1 817 + sin- 1 35 = tan- 1 7736 ........( iv )
Now, we have:
Let tan- 1 7736 = z.Then tan z = 7736⇒ sec z = 1 + 77362 ....... ∵ sec θ = 1 + tan2 θ ⇒ sec z = 1296 + 59291296⇒ sec z = 72251296⇒ sec z = 8536
We know that cos z = 1sec zThus, sec z = 8536, cos z = 3685⇒ z = cos- 1 3685 ⇒ tan- 1 7736 = cos- 1 3685⇒ sin- 1 817 + sin- 1 35 = cos- 1 3685 .......[ ∵ From equation ( iv ) ]
Hence proved.
Find the point on the curve y = x3 – 11x + 5 at which the equation of tangent is y = x – 11.
Using differentials, find the approximate value of 49.5.
Using properties of determinants prove the following:
1 1 1a b ca3 b3 c3 = a - b b - c c - a a + b +c
If y = 3 cos ( log x ) + 4 sin ( log x ), show that
x2 d2ydx2 + x dydx + y = 0
Using matrices solve the following system of linear equations:
x - y + 2 z = 7
3 x + 4 y - 5 z = - 5
2 x - y + 3 z = 12