Roots of quadratic equation x2 + 5x + 3 = 0
are 4 cos2 α + a and 4 sin2 α + a
\(\therefore\) sum of roots = \(\cfrac{-b}a\) = -5
= (4 cos2 α + a) + (4 sin2 α +a) = - 5
4 (sin2 α + cos2 α) + 2a = - 5
= 2a + 4 = -5
= 2a = -5 - 4 = -9
= a = \(\cfrac{-9}2\)....(1)
product of roots = \(\cfrac{c}a\) = 3
= (4cos2 α + a) (4 sin2 α + a) = 3
= 16 sin2 α cos2 α + 4a (sin2 α + cos2 α) + a2 = 3
= 4(2 sin α cos α)2 + 4 x \(\cfrac{-9}2\) + \(\left(\cfrac{-9}2\right)^2\) = 3 (From (1))
4 sin2 2 α - 18 + \(\cfrac{81}4\) = 3
= 4 sin2 2 α = 21 - \(\cfrac{81}4\) = \(\cfrac{84-81}4\)
= \(\cfrac34\)
= sin2 2 α = \(\cfrac{3}{16}\)
= sin 2 α = \(\cfrac{\sqrt3}{4}\)....(2)
Second quadratic equation is x2 + px + q = 0,
p, q ε N and p,q ε (1,10)
Total possible value for p and q = 8 x 8 = 64
sum of roots = (4 cos4 α + b) + (4 sin4 α + b) = -p
= -p = 4(sin4 α + cos4 α) + 2b
= -p = 4 ((sin2 α)2 + (cos2 α)2 + 2sin2 cos2 α - 2 sin2 α cos2 α) + 2b
= - p = 4 ((sin2 α + cos2 α)2 - 2 sin2 cos2 α) + 2b
= - p = 4 (1 - \(\cfrac12\)(2sin α cos α)2) + 2b
= - p = 4 (1 - \(\cfrac12\)sin2 2 α) + 2b
= - p = 4 (1 - \(\cfrac12\) x \(\cfrac{3}{16}\)) + 2b (From (2))
= - p = 4 - \(\cfrac{3}{8}\) + 2b
= 2b = \(\cfrac{3}{8}\) - 4 - p
= 2b = \(\cfrac{3-32-8p}8\)
= b = \(\cfrac{-29-8p}{16}\)....(3)
product of roots = (4cos4 α + b)(4sin4 α + b) = q
= q = 16 cos4 α sin4 α + 4b(sin4 α + cos4 α) + b2
= q = (2 sin α co α)4 + 4b ((sin2 α + cos2 α)2 - 2sin2 α cos2 α) +b2
= (sin 2α)4 + 4b (1 - \(\cfrac12\)(sin 2α)2) + b2
= \(\left(\cfrac{\sqrt3}4\right)^4\) + 4b (1 - \(\cfrac12\) x \(\cfrac{3}{16}\)) + b2 (From (3))
= \(\cfrac{9}{256}\) + \(\cfrac{29\,b}8\) + b2
= \(\cfrac{9}{256}\) + \(\cfrac{29}8\) x - \(\cfrac{(29+8p)}{16}\) + \(\left(\cfrac{-(29+8p)}{16}\right)^2\)
= \(\cfrac{9}{256}\) - \(\cfrac{841+232p)}{128}\) + \(\cfrac{841+464p+64p^2}{256}\)
= \(\cfrac{9-1682-464p+841+464p+64p^2}{256}\)
= \(\cfrac{64p^2-832}{256}\) = \(\cfrac{64(p^2-13)}{256}\)
= q = \(\cfrac{p^2-13}{4}\)....(4)
\(\because\) p and q are natural number such that p,q ε [1,10]
For possible values of p and q are
(p,q) = (5,3), (7,9)
\(\therefore\) Favorable outcomes = 2
\(\therefore\) Required probability = \(\cfrac{2}{64}\) = \(\cfrac1{32}\)