Trigonometric EquationMedium
Question
The sum of all the integral values of $p$ such that the equation $3\sin^{2}x + 12cosx - 3 = p,x \in R$, has at least one solution, is :
Options
A.-54
B.-60
C.-75
D.-84
Solution
Sol. [Key Concept: Write $p = - 3(cos\ x{- 2)}^{2} + 9$ and vary $cos\ x \in \lbrack - 1,1\rbrack$ to get the range of $p$; sum the integers in it.]
$${P = 12cosx - 3\cos^{2}x }{P = - 3\left( \cos^{2}x - 4cosx \right) }{P = - 3\left( (cosx - 2)^{2} - 2 \right) }$$put $cosx = - 1 \Rightarrow P = - 15$
put $cosx = 1 \Rightarrow P = 9$
$$- 15 \leq P \leq 9 $$Sum of all integers $= - (10 + 11 + \ldots + 15) = - 75$
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