Show that the relation R in the set A of points in a plane given by R = {(P, Q): distance of the point P from the origin is same as the distance of the point Q from the origin}, is an equivalence relation. Further, show that the set of all point related to a point P ≠ (0, 0) is the circle passing through P with origin as centre.
Find the zeroes of the following quadratic polynomials and verify the relationship between the zeroes and the coefficients.
(i) \(x^2 – 2x – 8\) (ii) \(4s^2 – 4s + 1\) (iii) \(6x^2 – 3 – 7x\) (iv) \(4u^2 + 8u\) (v) \( t^2 – 15\) (vi) \(3x^2 – x – 4\)
Let \(S=\left\{0∈(0,\frac{π}{2}) : \sum^{9}_{m=1} \sec(θ+(m-1)\frac{π}{6})\sec(θ+\frac{mπ}{6}) = -\frac{8}{\sqrt3}\right\}\)Then,
Factorise each of the following:
(i) 8a 3 + b 3 + 12a 2b + 6ab2
(ii) 8a 3 – b 3 – 12a 2b + 6ab2
(iii) 27 – 125a 3 – 135a + 225a 2
(iv) 64a 3 – 27b 3 – 144a 2b + 108ab2
(v) 27p 3 – \(\frac{1}{ 216}\) – \(\frac{9 }{ 2}\) p2 + \(\frac{1 }{4}\) p
Find the value of the polynomial 5x – 4x 2 + 3 at
(i) x = 0 (ii) x = –1 (iii) x = 2
If (x-a)2+(y-b)2=c2, for some c>0 prove that[1+(\(\frac{dy}{dx}\))2]\(^{\frac{3}{2}}\)/\(\frac{d^2y}{dx^2}\) is a constant independent of a and b
