Equations with Solutions at Boundary Values

Q.

If the product of the roots of equation 2$x^2$ + ax + 4 sin a = 0 is 1, then roots will be imaginary if :

1. 

2. 

3. 

4. 

Q.

Find the principal and general solutions of the following equation.
cosec x = - 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.

$\underset{x\to 0}{lim}\frac{tan2x-x}{3x-sinx}$

1. 2

2. 

3. -1

4. 

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.

The value of ${lim}_{x\to \infty }x[ta{n}^{-1}\left(\frac{x+2}{x+2}\right)-ta{n}^{-1}\left(\frac{x}{x+2}\right)]is$

1. 1

2. -1

3. 1/2

4. -1/2

Q. Find the general solution:
$cos3x+cosx-cos2x=0$

Q.

Prove that $\frac{\text{sin}3x-\text{sin}x}{\text{cos}2x}=2\text{sin}x$

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

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. $16$