General Solution of tan theta = tan alpha

Q. Prove that equation sin x + cos x =2, has no solution.

Q. Solution set of the inequality ${\sin }^{-1}x\le {\cos }^{-1}x$ is

1. $(0,\frac{1}{\sqrt{2}}]$
2. $[-1,\frac{1}{\sqrt{3}}]$
3. $[-1,\frac{1}{\sqrt{2}}]$
4. $(-1,\frac{1}{\sqrt{3}})$

Q.

If $\tan \left(A\right)-\tan \left(B\right)=x$ and $\cot \left(B\right)-\cot \left(A\right)=y$, then $\cot \left(A-B\right)=?$

1. $\frac{1}{x}+y$

2. $\frac{1}{xy}$

3. $\frac{1}{x}-\frac{1}{y}$

4. $\frac{1}{x}+\frac{1}{y}$

Q. The general solution of $si{n}^{2}x-2cosx+\frac{1}{4}=0isx=\left(nϵZ\right)$

1. 
2. 
3. 
4. 

Q. Which of the following function fail to satisfy the condition of Rolle's theorem on interval $[-1,1]$.

1. $f\left(x\right)=|x|+|\sin x|$
2. $f\left(x\right)=\left\{x\right\}+\{-x\}$
3. $f\left(x\right)=\left[\begin{array}{c}\frac{\tan (x^2-1)}{1-x^2}+\frac{{\log }_e|x|}{|x|-1},x\ne 1,-1,0\\ 0,x=0\end{array}\right]$
4. $f\left(x\right)=|x|-|\sin x|$

Q. Find the general solution of the equation $\cos 3x+\cos x-\cos 2x=0$

Q.

If $\cos x+\cos y+\cos \alpha =0$ and $\sin x+\sin y+\sin \alpha =0$, then $cot\frac{(x+y)}{2}=$

1. $\sin \alpha$

2. $\cos \alpha$

3. $cot\alpha$

4. $\sin \frac{(x+y)}{2}$

Q. Find the principal and general solutions of $\text{cosec}x=-2$

Q. $\mathrm{If}\tan \left(\frac{p\pi }{4}\right)=cot\left(\frac{\mathrm{q\pi }}{4}\right),\mathrm{then}\mathrm{the}\mathrm{value}\mathrm{of}(p+q)\mathrm{is}$
.

1. $2(2n+1),n\in Z.$
2. $(2n-1),n\in Z.$
3. $2n+\frac{1}{2},n\in Z.$
4. $n+\frac{1}{2},n\in Z.$

Q. $\mathrm{The}\mathrm{number}\mathrm{of}real\mathrm{roots}\mathrm{of}\mathrm{the}\mathrm{equation}\tan \left(x+\frac{\pi }{6}\right)=2\tan x,\mathrm{for}x\in (0,3\pi )\mathrm{are}$
.

1. \N
2. 1
3. 2

Q. If $\alpha <\beta ,0\le \alpha ,\beta \le 2\pi$ and $\alpha ,\beta$ satisfy the equation $\sin 2x+3\sin x=1+3\cos x,$ then which of the following is/are true?

1. $\alpha +\beta =\frac{\pi }{2}$
2. $\alpha +\beta =\frac{3\pi }{2}$
3. $\cos \alpha +\cos \beta =0$
4. $\sin \alpha +\sin \beta =0$

Q. $\tan \frac{\pi }{10}+\tan \frac{3\pi }{10}+\tan \frac{7\pi }{10}+\tan \frac{9\pi }{10}=?$

1. $0$
2. $-1$
3. $1$
4. $2$

Q. Find the general solution of the equation $\sin x+\sin 3x+\sin 5x=0$

Q. If the equation ${\cos }^4\theta +{\sin }^4\theta +\lambda =0$ has real solutions for $\theta$, then $\lambda$ lies in the interval:

1. $(-\frac{1}{2},-\frac{1}{4}]$
2. $[-1,-\frac{1}{2}]$
3. $[-\frac{3}{2},-\frac{5}{4}]$
4. $(-\frac{5}{4},-1)$

Q.

How do you find the exact value of $\arctan 1$?

Q. The number of distinct real roots of the equation

$\left|\begin{array}{ccc}\cos x& \sin x& \sin x\\ \sin x& \cos x& \sin x\\ \sin x& \sin x& \cos x\end{array}\right|=0,$

in the interval $[-\frac{\pi }{4},\frac{\pi }{4}],$ is

Q. If $\frac{\tan 3\theta -1}{\tan 3\theta +1}=\sqrt{3}$, then the general value of $\theta$ is

1. $\frac{n\pi }{3}-\frac{\pi }{2}$
2. $n\pi +\frac{7\pi }{2}$
3. $\frac{n\pi }{3}-\frac{7\pi }{36}$
4. $n\pi +\frac{\pi }{12}$

Q. If $\exp \left[\right({\sin }^{2}x+{\sin }^{4}x+{\sin }^{6}x+....\infty ){\log }_{e}2$, satisfies the equation $x^2-9x+8=0$, then the value of $\frac{\cos x}{\cos x+\sin x},0<x<\frac{\pi }{2}$ is

1. $\frac{\sqrt{3}-1}{2}$
2. $\frac{\sqrt{3}+1}{2}$
3. $-\sqrt{3}$
4. $\sqrt{3}$

Q.

In a triangle ABC the value of $\angle A$ is given by 5 cos A+3=0, then the equation whose roots are sin A and tan A will be

1. 

2. 

3. 

4. 

Q. The general solution of the equation $\tan 5\theta =\cot 3\theta$ is given by

1. 
   $\theta =\frac{n\pi }{8}-\frac{\pi }{16},n\in Z.$
2. 
   $\theta =\frac{n\pi }{8}+\frac{\pi }{16},n\in Z.$
3. 
   $\theta =\frac{n\pi }{8}+\frac{\pi }{8},n\in Z.$
4. 
   $\theta =\frac{n\pi }{8}-\frac{\pi }{8},n\in Z.$

Q. The number of solutions of the equation $|5\tan 2\theta \tan \theta -12\tan \theta |+|3{\sin }^2\theta -\sin 2\theta |=0$ in the interval $[0,2\pi ]$ is

1. $3$
2. $5$
3. $6$
4. $2$

Q.

Log(3^x-8)to the base3=2-x solve the equation.

Q.

The value of $sech\left(i\mathrm{\pi }\right)$ is

1. $-1$

2. $i$

3. $0$

4. $1$

Q.

If tan $\alpha =2$, then the values of x which satisfy the relation

$tanx=\frac{1}{2}are$ $0<x<$$2\pi and$$0<\alpha <$$\frac{\pi }{2}$

1. 

2. 

3. 

4. 

Q. The solution of the differential equation $\frac{dy}{dx}=1+x+y^2+x{y}^2,y\left(0\right)=0$ is
(a) $y^2=\exp \left(x+\frac{x^2}{2}\right)-1$

(b) $y^2=1+C\exp \left(x+\frac{x^2}{2}\right)$

(c) y = tan (C + x + x²)

(d) $y=\tan \left(x+\frac{x^2}{2}\right)$

Q. If $\cot \theta -pq\tan \theta =p-q;p,q\ne 0$, then the general value of $\theta$ is

1. $m\pi +{\tan }^{-1}\left(\frac{1}{q}\right),m\in Z$
2. $m\pi +{\tan }^{-1}\left(\frac{-1}{p}\right),m\in Z$
3. $m\pi +{\tan }^{-1}\left(\frac{1}{pq}\right),m\in Z$
4. $m\pi +{\tan }^{-1}\left(\frac{1}{p}\right),m\in Z$

Q.

Which of the following statements are correct?

1 . if sin θ = sin α ⇒ θ = nπ + $(-1{)}^n$α where α ∈ [ - $\frac{\pi }{2}$ , $\frac{\pi }{2}$] n ∈ I

2 . if cos θ = cos α ⇒ 2nπ±α where α ∈ [0, π] n ∈​ I

3 . if tan θ = tan α ⇒ θ = nπ + α where α ∈ (- $\frac{\pi }{2}$ , $\frac{\pi }{2}$ ) n ∈​ I

1. Only 1 & 3

2. Only 2 & 3

3. Only 1 & 2

4. All 1, 2 & 3

Q. If $\exp \left[\right({\sin }^{2}x+{\sin }^{4}x+{\sin }^{6}x+....\infty ){\log }_{e}2$, satisfies the equation $x^2-9x+8=0$, then the value of $\frac{\cos x}{\cos x+\sin x},0<x<\frac{\pi }{2}$ is

1. $\frac{\sqrt{3}-1}{2}$
2. $\frac{\sqrt{3}+1}{2}$
3. $\sqrt{3}$
4. $-\sqrt{3}$

Q. The complete solution set of $\sin x-\sqrt{3}\cos x=0$ is

1. $n\pi -\frac{\pi }{3}$
2. $n\pi \pm \frac{\pi }{3}$
3. $n\pi +\frac{\pi }{3}$
4. $n\pi \pm \frac{\pi }{6}$

Q. One of the solutions of the equation $8{\sin }^3\theta -7\sin \theta +\sqrt{3}\cos \theta =0$ lies in the interval

1. $(0^{\circ },{10}^{\circ }]$
2. $({10}^{\circ },{20}^{\circ }]$
3. $({20}^{\circ },{30}^{\circ }]$
4. $({30}^{\circ },{40}^{\circ }]$