A geometry problem by Akshat Sharda

Geometry Level 4

If $\cos^2 x-a\sin x+b=0$ has only one solution in $[0,\pi]$ , then which are correct?

$A. \ \ a\in (-\infty,-2]\cup (-1,\infty)$

$B. \ \ a≠b$

$C. \ \ a=b$

$D. \ \ b\in (-\infty,-2]\cup (-1,\infty)$

(A),(B)

(A),(C),(D)

None

(B),(D)

1 solution

Prakhar Bindal
Sep 4, 2016

Its a nice one.

For exactly one solution 1 should be the root. plug in to get a = b

Now if a = b on solving the quadratic in sinx you will get

sinx = 1 or -a-1.

Now so either the second root should be greater than 1 or should be less than zero for no solution in 0 to pi .

Solving above you will get The range of values of a , b which are given in choices A And D.

Nicely done!

Akshat Sharda - 4 years, 9 months ago

For exactly one solution why 1 should be a solution ??

Kushal Bose - 4 years, 9 months ago

Because there is only one angle between 0 and 180 ie 90 for which sinx = 1

rest all will have two angles between 0 and 180

Prakhar Bindal - 4 years, 9 months ago

yes understood

Kushal Bose - 4 years, 9 months ago