Mathematics

Calculus

100

Explanation: From the fig (1) ω=∅+θ…….. (triangle rule & opt^ = ∅)
tan ⁡ω=tan(⁡θ+∅)
tanω=tan∅+tanθ1−tan∅tanθ ……(1) further more w.k.t x=rcos⁡θ, y=rsin⁡θ
But tanω=dydx……(slope of the tangent TT’)
tanω=dydθdxdθ……..since x & y are functions of θ
i.e tanω=d(rsinθ)dθd(rcosθ)dθ=rcosθ+drdθsinθ−rsinθ+drdθcosθ
tanω=rcosθdrdθcosθ+drdθsinθdrdθcosθ−rsinθdrdθcosθ+drdθcosθdrdθcosθ=rdrdθ+tanθrdrdθtanθ…………(2)
from (1) &(2)
tan∅=rdrdθ=r(dθdr).

1. For the below mentioned figure the angle between radius vector (op) ⃗ and tangent to the polar curve where r=f(θ) has the one among the following relation?

200

rn+1=pan

The pedal Equation of the polar curve rn=an cos⁡nθ is given by ______

Integral Calculus

100

Explanation: r= a(1-cos⁡θ)
taking logarithms on both sides we get,
log⁡r = log⁡a + log⁡(1-cos⁡θ)
differentiating w.r.t θ we get,
1rdrdθ=0+sinθ1−cosθ
1rdrdθ=2sinθ2cosθ22sin2θ2=cotθ2 ..(1),
but cot∅=1rdrdθ….(2)
From (1)&(2)
∅=π2.

The angle between Radius vector r=a(1-cos⁡θ)and tangent to the curve is ∅ given by _______

200

p=2a2–√

The length of the perpendicular from the pole to the tangent at the point θ=π2 on the curve. r=a sec2(π2) is _____

Differential Equation

100

tan⁡φ1 . tan⁡φ3=-1

Angle of intersection of two polar curves is equal to the angle between the tangents drawn at the point of intersection of the two curves then What is the condition for the two curves intersecting orthogonally for the below mentioned figure?

200

x = a+ r cos⁡θ, y = b + r sin⁡θ

Polar equations of the circle for the given coordinate (x,y) which satisfies the equation given by (x-a)2+(y-b)2=r2 where (a,b) is the coordinates of the centre of the circle &r is the radius.

PDE

100

π2

Angle of intersection between two polar curves given by r=a(1+sin⁡θ) & r=a(1-sin⁡θ) is given by ________

200

(1+sec2 θ) = y2x2

In an polar curve r=f (θ) what is the relation between θ & the coordinates (x,y)?

Fs,Ft

100

 It is expressed in terms of p& r only

 One among the following is the correct explanation of pedal equation of an polar curve, r=f (θ), p=r sin(∅) (where p is the length of the perpendicular from the pole to the tangent & ∅ is the angle made by tangent to the curve with vector drawn to curve from pole)is _______

200

dsdx=1+(dydx)2−−−−−−−−√

For the cartesian curve y=f(x) with ‘s’ as arc length which of the following condition holds good?