Solving Simultaneous Trigonometric Equations

Q. If $x=\sum _{n=0}^{\infty }(-1{)}^n{\tan }^{2n}\theta$ and $y=\sum _{n=0}^{\infty }{\cos }^{2n}\theta ,$ where $0<\theta <\frac{\pi }{4},$ then:

1. $y(1-x)=1$
2. $y(1+x)=1$
3. $x(1+y)=1$
4. $x(1-y)=1$

Q. Let $A=\{{\lambda }_1,{\lambda }_2,\dots ,{\lambda }_m\},B=\{{\mu }_1,{\mu }_2,\dots ,{\mu }_n\}$ be two sets of values of $\lambda$ and $\mu$, where $\lambda ,\mu \in (0,100\pi ]$. The equation $x^2(\sin \lambda x+\cos \mu x)-2x\left[\sin \right(\lambda +\mu )x+\sin (\lambda -\mu \left)x\right]+(\sin \lambda x+\cos \mu x)=0$ has a positive solution for the ordered pair ${\lambda }_i,{\mu }_j$. If $\sum _{i=1}^m{\lambda }_i+\sum _{j=1}^n{\mu }_j=25k\pi$, then the value of $k$ is

Q. If $si{n}^{-1}x+si{n}^{-1}y+si{n}^{-1}z=\frac{3\pi }{2}$
then $\frac{{\sum }_{r=1}^2(x^{100r}+y^{103r})}{\sum x^{201}{y}^{201}}$ =

1. 0
2. 2
3. 4
4. $\frac{4}{3}$

Q. Let $y=e^{\left\{\right({\sin }^{2}x+{\sin }^{4}x+{\sin }^{6}x+\dots \left){\log }_{e}2\right\}}$ satisfy the equation $x^2-17x+16=0,$ where $0<x<\frac{\pi }{2}.$ Then match the correct value of $\text{List I}$ from $\text{List II}.$

$\begin{array}{cccc}& \text{List I}& & \text{List II}\\ \text{(a)}& \frac{2\sin 2x}{1+{\cos }^{2}x}& \left(p\right)& 1\\ \text{(b)}& \frac{2\sin x}{\sin x+\cos x}& \left(q\right)& \frac{4}{9}\\ \text{(c)}& \sum _{n=1}^{\infty }(\cot x{)}^n& \left(r\right)& \frac{2}{3}\\ \text{(d)}& \sum _{n=1}^{\infty }n(\cot x{)}^{2n}& \left(s\right)& \frac{4}{3}\end{array}$

1. $\left(a\right)\to \left(p\right),\left(b\right)\to \left(q\right)\left(c\right)\to \left(r\right),\left(d\right)\to \left(s\right)$
2. $\left(a\right)\to \left(q\right),\left(b\right)\to \left(p\right)\left(c\right)\to \left(s\right),\left(d\right)\to \left(r\right)$
3. $\left(a\right)\to \left(r\right),\left(b\right)\to \left(s\right)\left(c\right)\to \left(q\right),\left(d\right)\to \left(p\right)$
4. $\left(a\right)\to \left(s\right),\left(b\right)\to \left(s\right)\left(c\right)\to \left(p\right),\left(d\right)\to \left(q\right)$

Q.

Total number of ordered pairs (x, y) satisfying $|x|+|y|=2,sin\left(\frac{\pi x^2}{3}\right)=1$ is

1. 2

2. 3

3. 4

4. 6

Q. If $3si{n}^{-1}\frac{2x}{1+x^2}-4co{s}^{-1}\frac{1-x^2}{1+x^2}+2ta{n}^{-1}\frac{2x}{1+x^2}=\frac{\pi }{3}$ then x=

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

Q. Let $k$ be a real number such that $\tan ({\tan }^{-1}2+{\tan }^{-1}20k)=k.$ Then the sum of all possible values of $k$ is

1. $\frac{1}{5}$
2. $0$
3. $-\frac{21}{40}$
4. $-\frac{19}{40}$

Q. The number of solution's of $2{\sin }^{2}x+{\sin }^{2}2x=2,x\in [0,2\pi ]$ are :

1. $4$
2. $5$
3. $7$
4. $6$

Q. $\begin{array}{cccc}& \text{List - I}& & \text{List - II}\\ \text{(I)}& \text{If A and B are two sets with elements 4 and 3},& \text{(P)}& 1\\ & \text{respectively. Then the number of onto function}& & \\ & \text{from A to B is}& & \\ \text{(II)}& \text{The number of points in the interval}[-\sqrt{13},\sqrt{13}]& \left(Q\right)& 3\\ & \text{at which}f\left(x\right)=\sin \left(x^2\right)+\cos \left(x^2\right)\text{attains its}& & \\ & \text{maximum value, is}& & \\ \text{(III)}& \text{In a}\Delta XYZ,y^2\sin \left(2Z\right)+z^2\sin \left(2Y\right)=2yz,& \text{(R)}& 4\\ & \text{where}y=15,z=8\text{. Then the length of inradius is}& & \\ \text{(IV)}& {\log }_{x+1}(4x^3+9x^2+6x+1)+{\log }_{4x+1}(x^2+2x+1)& \text{(S)}& 8\\ & =5,\text{then the number of solution is}& & \\ & & \text{(T)}& 24\\ & & \text{(U)}& 36\end{array}$

Which of the following has CORRECT pair of combination?

1. $\left(I\right)\to \left(T\right)$
2. $\left(II\right)\to \left(S\right)$
3. $\left(III\right)\to \left(R\right)$
4. $\left(IV\right)\to \left(P\right)$

Q. Let $f\left(x\right)=x^2$ and g(x) =sin x for all $x\in R.$ Then the set of all x satisfying (fogogof) (x) =(gogof) (x), where (fog) (x) =f(g(x)), is

1. $\pm \sqrt{n\pi },n\in \{0,1,2,....\}$
   
2. $\pm \sqrt{n\pi },n\in \{1,2,....\}$
   
3. $\frac{\pi }{2}+2n\pi ,nin\{....,-2,-1,0,1,2,....\}$
   
4. $2n\pi ,n\in \{....,-2,-1,0,1,2,....\}$
   

Q. The inequality $\left|x^2\sin x+{\cos }^{2}x{e}^x+{\ln }^{2}x\right|<x^2\left|\sin x\right|+{\cos }^{2}x{e}^x+{\ln }^{2}x$ is true for $x\in$

1. $\left(-\pi ,0\right)$
2. $\left(0,\frac{\pi }{2}\right)$
3. $\left(\frac{\pi }{2},\pi \right)$
4. $\left[2n\pi ,\left(2n+1\right)\pi \right]\forall n\in N$

Q. The solution set of the system of equations $x+y=\frac{2\pi }{3},cosx+cosy=\frac{3}{2}$, where x and y are real, is ___

Q.

Let $\theta ,\varphi \in [0,2\pi ]$ be such that $2cos\theta (1-sin\varphi )=si{n}^2\theta (tan\frac{\theta }{2}+cot\frac{\theta }{2})cos\theta -1,tan(2\pi -\theta )>0$ and -1 < $sin\theta <-\frac{\sqrt{3}}{2}$. Then $\varphi$ cannot satisfy

1. $0<\varphi <\frac{\pi }{2}$
   

2. $\frac{\pi }{2}<\varphi <\frac{4\pi }{3}$
   

3. $\frac{4\pi }{3}<\varphi <\frac{3\pi }{2}$
   

4. $\frac{3\pi }{2}<\varphi <2\pi$

Q. The smallest positive value of $x$ which satisfies the equation ${\log }_{\cos x}\sin x+{\log }_{\sin x}\cos x=2$ is

1. $\frac{\pi }{2}$
2. $\frac{\pi }{3}$
3. $\frac{\pi }{4}$
4. $\frac{\pi }{6}$

Q.

The range of $f\left(x\right)=ta{n}^{-1}(x^2+x+a)$ $\forall xϵ$ R is a subset of $[0,\frac{\pi }{2})$ then the range of a is -

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

Q. The inequality $\left|x^2\sin x+{\cos }^{2}x{e}^x+{\ln }^{2}x\right|<x^2\left|\sin x\right|+{\cos }^{2}x{e}^x+{\ln }^{2}x$ is true for $x\in$

1. $\left(-\pi ,0\right)$
2. $\left(0,\frac{\pi }{2}\right)$
3. $\left(\frac{\pi }{2},\pi \right)$
4. $\left[2n\pi ,\left(2n+1\right)\pi \right]\forall n\in N$

Q. Total number of solution of ${\cos }^{2}x+\frac{\sqrt{3}+1}{2}\sin x-\frac{\sqrt{3}}{4}-1=0$ in $x\in [-\pi ,\pi ]$ is :

1. $6$
2. $4$
3. $7$
4. $8$

Q.

The number of solutons of the equation tan x +sec x = 2cos x lying in the interval $[0,2\pi ]$ is

1. 0

2. 1

3. 2

4. 3

Q. If the angle of elevation of a cloud from a point $P$ which is $25\text{m}$ above a lake be ${30}^{\circ }$ and the angle of depression of reflection of the cloud in the lake from $P$ be ${60}^{\circ }$, then the height of the cloud (in meters) from the surface of the lake is :

1. $60$
2. $50$
3. $45$
4. $42$

Q. The smallest positive value of $x$ which satisfies the equation ${\log }_{\cos x}\sin x+{\log }_{\sin x}\cos x=2$ is

1. $\frac{\pi }{2}$
2. $\frac{\pi }{3}$
3. $\frac{\pi }{4}$
4. $\frac{\pi }{6}$

Q. If $4{\sec }^{2}x=7+{\tan }^{2}x$ , then $x$ is equal to

1. $n\pi \pm \frac{\pi }{4},n\in Z$
2. $n\pi \pm \frac{\pi }{6},n\in Z$
3. $n\pi \pm \frac{\pi }{2},n\in Z$
4. $n\pi \pm \frac{\pi }{3},n\in Z$

Q. A solution of the equation ${\log }_2(\sin x+\cos x)-{\log }_2\left(\cos x\right)+1=0$ in $(-\frac{\pi }{4},\frac{\pi }{4})$ is

1. $0$
2. ${\tan }^{-1}\left(\frac{1}{2}\right)$
3. $\frac{\pi }{6}$
4. ${\tan }^{-1}(-\frac{1}{2})$

Q. Suppose $si{n}^{3}sin3x={\sum }_{m=0}^n{C}_{m}cosnx$ is an identity in x,
Where $C_0,C_1,\dots C_n$ are constants and $C_n\ne 0$. Then the value of n is___

Q. For $x\in (0,\pi )$, then equation $sinx+sin2x-sin3x=3$ has

1. Infinitely many solutions
2. Three solutions
3. One solution

4. No solution

Q. The number of solutions of ${\cos }^{2}x+\frac{\sqrt{3}+1}{2}\sin x-\frac{\sqrt{3}}{4}-1=0$, where $x\in [-\pi ,\pi ]$ is

1. $6$
2. $4$
3. $7$
4. $8$

Q. If ${\cos }^{6}A+{\sin }^{6}A=1-k{\sin }^{2}2A$, then the value of $4k$ is

Q.

The number of all the possible triplets $(a_1,a_2,a_3)$ such that $a_1+a_{2}cos\left(2x\right)+a_{3}si{n}^2\left(x\right)=0$ for all x is

1. 0

2. 1

3. 3

4. infinite

Q. The value of $\theta \in (0,2\pi )$ for which the equation $2si{n}^2\theta -5sin\theta +2>0$ is

1. $(0,\frac{\pi }{6})\cup (\frac{5\pi }{6},2\pi )$
   
2. $(\frac{\pi }{8},\frac{5\pi }{6})$
   
3. $(0,\frac{\pi }{8})\cup (\frac{\pi }{6},\frac{5\pi }{6})$
   
4. $(\frac{41\pi }{48},\pi )$
   

Q. The equation $(cosp-1)x^2+\left(cosp\right)x+sinp=0$ in the variable x has real roots. Then p can take any value in the interval

1. $(0,2\pi )$
   
2. $(-\pi ,0)$
   
3. $(-\frac{\pi }{2},\frac{\pi }{2})$
   
4. $(0,\pi )$

Q.

The number of solutons of the equation tan x +sec x = 2cos x lying in the interval $[0,2\pi ]$ is

1. 0

2. 1

3. 2

4. 3