General Solution of Trigonometric Equation

Q.

Prove that: $\tan \left(3x\right)\tan \left(2x\right)\tan \left(x\right)=\tan \left(3x\right)-\tan \left(2x\right)-\tan \left(x\right)$

Q.

If $p=\left[\frac{2\sin \theta }{(1+\cos \theta +\sin \theta )}\right]$and $q=\left[\frac{\cos \theta }{(1+\sin \theta )}\right]$, then

1. $pq=1$

2. $\frac{q}{p}=1$

3. $q–p=1$

4. $q+p=1$

Q.

The maximum value of $f\left(x\right)=\left|\begin{array}{ccc}{\sin }^{2}x& 1+{\cos }^{2}x& \cos 2x\\ 1+{\sin }^{2}x& {\cos }^{2}x& \cos 2x\\ {\sin }^{2}x& {\cos }^{2}x& \sin 2x\end{array}\right|$$,x\in R$ is:

1. $\sqrt{7}$

2. $\sqrt{5}$

3. $5$

4. $\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$4$}\right.$

Q.

If ${\tan }^{-1}x-{\tan }^{-1}y={\tan }^{-1}A$, then $A=$

1. $x-y$

2. $x+y$

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

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

Q. General solutions of the equation $\cos 3x=\sin 2x$ is

1. $\{\frac{2n\pi }{5}-\frac{\pi }{10}\}\cup \{2n\pi +\frac{\pi }{2}\},n\in Z$
2. $\{\frac{2n\pi }{5}-\frac{\pi }{10}\}\cup \{2n\pi -\frac{\pi }{2}\},n\in Z$
3. $\{\frac{2n\pi }{5}+\frac{\pi }{10}\}\cup \{2n\pi +\frac{\pi }{2}\},n\in Z$
4. $\{\frac{2n\pi }{5}+\frac{\pi }{10}\}\cup \{2n\pi -\frac{\pi }{2}\},n\in Z$

Q.

The number of solution of $2\sin \left(x\right)+\cos \left(x\right)=3$ is

1. $1$

2. $2$

3. Infinite

4. No solution

Q.

The number of values of $x$ satisfying the equation $3\cos \left(2x\right)-10\cos \left(x\right)+7=0$ is

1. $1$

2. $2$

3. $3$

4. $4$

Q.

If $\alpha$ and $\beta$ are the roots of $x^2-2x\cos \varnothing +1=0$, then the equation, whose roots are ${\alpha }^n$and ${\beta }^n$

1. $x^2–2x\cos n\varnothing –1=0$

2. $x^2–2x\cos n\varnothing +1=0$

3. $x^2–2x\sin n\varnothing +1=0$

4. $x^2+2x\sin n\varnothing -1=0$

Q.

The complete solution set of the inequality $[co{t}^{-1}x{]}^2-6[co{t}^{-1}x]+9\le 0,$ where $[.]$ denotes greatest integer function is

1. $(-\infty ,cot2)$
2. $(-\infty ,cot3)$
3. $(cot2,cot3)$
4. $(cot2,\infty )$

Q. In a $△$ ABC, if $c^2+a^2-b^2=ac,$ then $\angle B=$
[MP PET 1983, 89, 90]

1. $\frac{\pi }{6}$
2. $\frac{\pi }{4}$
3. $\frac{\pi }{3}$
4. None of these

Q. General solution of the equation $\tan 5x=\cot 5x$ is :

1. $n\pi \pm \frac{\pi }{20},n\in Z$
2. $\frac{n\pi }{5}+\frac{\pi }{20},n\in Z$
3. $\frac{n\pi }{5}\pm \frac{\pi }{4},n\in Z$
4. $\frac{n\pi }{5}\pm \frac{\pi }{20},n\in Z$

Q. The value of $x$ in the interval $[0,2\pi ]$ for which $4{\sin }^{2}x-8\sin x+3\le 0$ is

1. $[\frac{\pi }{6},\frac{\pi }{2}]$
2. $[\frac{\pi }{3},\frac{\pi }{2}]$
3. $[\frac{\pi }{6},\frac{3\pi }{2}]$
4. $[\frac{\pi }{6},\frac{5\pi }{6}]$

Q.

If $f\left(x\right)=kx-\cos x$ is monotonically increasing for all $x\in R$, then

1. $K>–1$

2. $K<1$

3. $K>1$

4. None of these

Q. The general solution(s) of the equation $\sec 4\theta -\sec 2\theta =2$ can be

1. $(2n+1)\frac{\pi }{12},n\in Z$
2. $(2n+1)\frac{\pi }{10},n\in Z$
3. $(2n+1)\frac{\pi }{2},n\in Z$
4. $(2n+1)\frac{\pi }{4},n\in Z$

Q.

If $\tan \left(A+B\right)=p$and $\tan \left(A-B\right)=q$, then the value of $\tan \left(2A\right)$ in terms of $p$and $q$is:

1. $\frac{p+q}{p-q}$

2. $\frac{p-q}{1+pq}$

3. $\frac{p+q}{1-pq}$

4. $\frac{1+pq}{1-p}$

Q. The solution set of the equation $\sin 3x+\cos 2x=-2$ is
(where $n\in Z$)

1. $\frac{(2n+1)\pi }{2}$
2. $\frac{(4n-1)\pi }{2}$
3. $\frac{(2n-1)\pi }{2}$
4. $\frac{(4n+1)\pi }{2}$

Q.

Solve $(1+y){\tan }^{2}x+\tan x\left(\frac{dy}{dx}\right)+y=0$

Q. If ${\sin }^{-1}\left(\sin p\right)=3\pi -p$ and the point of intersection of the lines $x+y=6$ and $px-y=3$ will have integral co-ordinates (both abscissa and ordinate), then the number of values of $p$ is

Q. The number of solution(s) of the equation $e^{\sin x}-e^{-\sin x}=4,(e\approx 2.72)$ is

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

Q. General solution of the equation $\tan x+\tan 2x+\sqrt{3}\tan x\tan 2x=\sqrt{3}$ is :

1. $\frac{n\pi }{3}+\frac{\pi }{9},n\in Z$
2. $n\pi +\frac{\pi }{9},n\in Z$
3. $n\pi -\frac{\pi }{9},n\in Z$
4. $\frac{n\pi }{2}+\frac{\pi }{3},n\in Z$

Q. The range of $k$ for which the equation $k\cos x-3\sin x=k+1$ has a solution is

1. $(-\infty ,4]$
2. $(-\infty ,4)$
3. $(4,\infty )$
4. $[4,\infty )$

Q. The general solution(s) of $\theta$ which satisfy $3-2\cos \theta –4\sin \theta -\cos 2\theta +\sin 2\theta =0$ is/are (where $n\in Z$)

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

Q. The general solutions for the equation $\cos 4x=\frac{\sqrt{5}+1}{4}$ is

1. $x=n\pi \pm \frac{\pi }{20},n\in Z$
2. $x=\frac{n\pi }{2}\pm \frac{\pi }{40},n\in Z$
3. $x=\frac{n\pi }{4}\pm \frac{\pi }{20},n\in Z$
4. $x=\frac{n\pi }{2}\pm \frac{\pi }{20},n\in Z$

Q. The number of ordered pairs $(x,y)$ satisfying the equation $x^2+2x\sin \left(xy\right)+1=0$ is
$($where $y\in [0,2\pi ])$

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

Q. Consider the trigonometric equation $\tan x(\sin 2x+1)=\sin x(2+\tan x)$. The number of solution(s) of the equation in $(0,4\pi )$ is

Q. General solutions of $x$ for which $2\sin x+1=0$ is :

1. $n\pi +(-1{)}^n\left(\frac{-\pi }{6}\right),n\in Z$
2. $n\pi \pm (-1{)}^n\frac{2\pi }{3},n\in Z$
3. $n\pi \pm \frac{\pi }{3},n\in Z$
4. $n\pi \pm (-1{)}^n\left(\frac{-\pi }{6}\right),n\in Z$

Q. If $\tan a,\tan b,\tan c,\tan d$ are the roots of the eqn $\tan (45°+\theta )=3\tan 3\theta$, then
$\frac{1}{\tan a}+\frac{1}{\tan b}+\frac{1}{\tan c}+\frac{1}{\tan d}$ is

Q. If $\left[\sin x\right]+\left[\sqrt{2}\cos x\right]=-3$, (where $[.]$ represents the greatest integer function), then $x$ belongs to

1. $[\pi ,\frac{5\pi }{4})$
2. $(\pi ,\frac{5\pi }{4})$
3. $[\pi ,\frac{5\pi }{4}]$
4. no solution

Q. If $\cos x+\cos y+\cos z=0=\sin x+\sin y+\sin z,$ then

1. $\cos \frac{x-y}{2}=\pm \frac{\sqrt{3}}{2}$
2. $\cos \frac{x-y}{2}=\pm \frac{1}{2}$
3. ${\sin }^2\frac{x-y}{2}=\frac{3}{4}$
4. $\tan \frac{x-y}{2}=\pm \frac{1}{\sqrt{3}}$

Q.

If $3\sin x+4\cos ax=7$ has atleast one solution, then 'a' has to be necessarily _______.

1. irrational number
2. rational number
3. even integer
4. odd integer