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Show that if `p,q,r` and `s` are real numbers and `pr=2(q+s)`, then atleast one of the equations `x^(2)+px+q=0` and `x^(2)+rx+s=0` has real roots.

Show that if `p,q,r` and `s` are real numbers and `pr=2(q+s)`, then atleast one of the equations `x^(2)+px+q=0` and `x^(2)+rx+s=0` has real roots.

Show that if `p,q,r` and `s` are real numbers and `pr=2(q+s)`, then atleast one of the equations `x^(2)+px+q=0` and `x^(2)=rx+s=0` has real roots.

In which of the following cases the given equations has atleast one root in the indicated interval `x-cos x = 0` in `(0, pi/2)`

**(i) Prove that general solution of `sintheta=0` is given by `theta=npi; n in Z` (ii) Prove that general solution of `costheta=0` is `theta=((2n+1)pi/2); n in Z`**

**(iii)Prove that general solution of `tan theta=0` is `theta=npi; n in Z` (iv)Prove that the general solution of `cot theta=0` is `theta=(2n+1)pi/2, n in Z`**

**Find the general solution of the equations (i)`sin theta= sqrt3/2` (ii) `2sintheta+1=0` (iii)`cosectheta=2`**

**Prove that the general solution of `costheta=cosalpha`is given by `theta=2npipmalpha`; where `n inZ`**

**Find the general solution of the equation (i)`costheta=1/2 (ii) cos3theta=-1/2 (iii)sqrt3sec2theta=2`**

**General solution of `(i) sin^2theta = sin^2alpha; (ii) cos^2 theta = cos^2 alpha (iii) tan^2 theta = tan^2 alpha`**

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