Equations with Solutions at Boundary Values

Q.

Find the principal solution of $ta{n}^{2}x+(\sqrt{3}-1)tanx-\sqrt{3}=0$

1. $x=\frac{\pi }{4}$

2. $x=\frac{\pi }{3}$

3. $x=-\frac{\pi }{4}$

4. $x=-\frac{\pi }{3}$

Q.

Find the principal solution of $\frac{1+cosx}{cosx}=2$

1. none of these

2. $x=\pi$

3. $x=0$

4. $x=\frac{\pi }{2}$

Q.

Find the principal solutions of sin 2x + cos x = 0

1. $x=-\frac{\pi }{3}$, $x=\frac{\pi }{3}$

2. $x=-\frac{\pi }{6}$, $x=\frac{\pi }{2}$

3. $x=\frac{\pi }{2}$

4. $x=\frac{\pi }{3}$

Q. If equation in variable $\theta$, $3\tan (\theta -\alpha )=\tan (\theta +\alpha )$, (where $\alpha$ is constant), has no real solution, then $\alpha$ can be (wherever $\tan (\theta -\alpha )$ and $\tan (\theta +\alpha )$ both are defined)

1. $\frac{\pi }{15}$
2. $\frac{5\pi }{18}$
3. $\frac{5\pi }{12}$
4. $\frac{17\pi }{18}$

Q. Number of solutions of the equation $(2\text{cosec}x-1{)}^{\frac{1}{3}}+(\text{cosec}x-1{)}^{\frac{1}{3}}=1$ in $(-k\pi ,k\pi )$ is $16$, then the possible value of '$k$' is

1. $2$
2. $4$
3. $8$
4. $16$

Q. The number of solution(s) of the system of equations $x+y=\frac{2\pi }{3}$ and $\cos x+\cos y=\frac{3}{2},$ where $x$ and $y$ are real, is

Q. The number of solution of ${\sin }^4\theta -2{\sin }^2\theta -1=0$ for $\theta \in [0,2\pi ]$ is

Q. The number of solution(s) of the equation $e^{\sin x}-e^{-\sin x}-4=0$ is

Q.

$\text{Write the number of values of}\theta in[\theta ,2\pi ]thatsatisfytheequationsi{n}^2\theta -cos\theta =\frac{1}{4}.$

Q.

$Ifsecxcos5x+1=0,where0<x\le \frac{\pi }{2},\text{find the value of}$