Question

If $$2{\tan ^2}x - 5\sec x$$    is equal to 1 for exactly 7 distinct values of $$x \in \left[ {0,\frac{{n\pi }}{2}} \right],n \in N,$$     then the greatest value of $$n$$ is

A. 6
B. 12
C. 13
D. 15  
Answer :   15
Solution :
Here, $$\left( {2\sec x + 1} \right)\left( {\sec x - 3} \right) = 0;$$      but $$\left| {\sec x} \right| \geqslant 1.$$   So, $$\sec x = 3,$$   which gives two values of $$\theta $$ in each of $$\left[ {0,2\pi } \right],\left( {2\pi ,4\pi } \right],\left( {4\pi ,6\pi } \right],$$     and one value in $$\left( {6\pi ,6\pi + \frac{{3\pi }}{2}} \right].$$

Releted MCQ Question on
Trigonometry >> Trignometric Equations

Releted Question 1

The equation $$2\,{\cos ^2}\frac{x}{2}{\sin ^2}x = {x^2} + {x^{ - 2}};0 < x \leqslant \frac{\pi }{2}$$        has

A. no real solution
B. one real solution
C. more than one solution
D. none of these
Releted Question 2

The general solution of the trigonometric equation $$\sin x + \cos x = 1$$    is given by:

A. $$x = 2n\pi ;\,\,n = 0,\,\, \pm 1,\,\, \pm 2\,.....$$
B. $$x = 2n\pi + \frac{\pi }{2};\,\,n = 0,\,\, \pm 1,\,\, \pm 2\,.....$$
C. $$x = n\pi + {\left( { - 1} \right)^n}\,\,\frac{\pi }{4} - \frac{\pi }{4}$$
D. none of these
Releted Question 3

The general solution of $$\sin \,x - 3\,\sin \,2x\, + \sin \,3x\, = \cos x - 3\,\cos \,\,2x + \cos \,3x$$           is

A. $$n\pi + \frac{\pi }{8}$$
B. $$\frac{{n\pi }}{2} + \frac{\pi }{8}$$
C. $${\left( { - 1} \right)^n}\frac{{n\pi }}{2} + \frac{\pi }{8}$$
D. $$2n\pi + {\cos ^{ - 1}}\frac{3}{2}$$
Releted Question 4

Number of solutions of the equation $$\tan x + \sec x = 2\cos x$$     lying in the interval $$\left[ {0,2\pi } \right]$$  is:

A. 0
B. 1
C. 2
D. 3

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