1)No of ordered triplets (x,y,z) satisfying
(1+sin^{4}x)(2+cot^{2}y)(4+sin4z)\leq 12sin^{2}x
2)Let P be a point on the hyperbola x2-y2=a2,wher a is parameter,such that P is nearest to the line y=2a.The locus of P is _____
3)suppose a sequence {bn} is defined as b1=1 and bn=cosbn-1
and another sequence {pn} p1=1 and pn=pn-1+cospn-1
then prove that the two sequences were non monotonic.
113)
Monotonicity of \left\{P_n \right\} & \left\{B_n \right\} both depend on the inequality cos\theta<\theta or the reverse....now from the graph itself we can cocnclude that the line y=x and y=cosθ have no specific inequality-relation....i mean we can have both cosθ>θ as well as the reverse....From which we can cocnclude that the sequences are not monotonic.
11I don't get the 2nd qsn...there's a point of intersection of that line and the given hyperbola....i.e. \left(a\sqrt{5},2a \right)....what is the locus for?
3soumi i thot of the same for the second and the third,and i am clueless in the first.tx for the confirmation.
1(1 + sin4xsin2x)(1 + (1 + cot2y))(4 + sin4z) ≤ 12
(cosec2x + sin2x)(1 + cosec2y)(4 + sin4z) ≤ 12 ...... (1)
now, cosec2x + sin2x2 ≥ (cosec2x sin2x)1/2 = 1
cosec2x + sin2x ≥ 2 .... (a)
1 + cosec2y ≥ 2 (since min of cosec2y is 1) .... (b)
4 + sin4z ≥ 3 (since min of sin4z is -1) ......... (c)
multiplying (a), (b), (c) :
(cosec2x + sin2x)(1 + cosec2y)(4 + sin4z) ≥ 12 ...... (2)
from (1) and (2) :
(cosec2x + sin2x)(1 + cosec2y)(4 + sin4z) = 12
for which : sin2x = 1, cosec2y = 1, sin4z = -1
