Equations with Solutions at Boundary Values

Q. If an angle $A$ of a $△ABC$ satisfies $5\cos A+3=0,$ then the roots of the quadratic equation, $9x^2+27x+20=0$ are:

1. $\sec A,\cot A$
2. $\sin A,\sec A$
3. $\sec A,\tan A$
4. $\tan A,\cos A$

Q.

If $A=2si{n}^2\theta -cos2\theta$, then A lies in the interval

1. [-2, 4]

2. none of these

3. [1, 2]

4. [-1, 3]

Q.

Write the number of solutions of the equation 4 sin x-3 cos x =7.

Q. Solve $\cos x=\frac{1}{2}$

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 sum of all solutions of the equation $\cos 3\theta =\sin 2\theta$ in the interval $[-\frac{\pi }{2},\frac{\pi }{2}]$ is

1. $-\frac{\pi }{3}$
2. $\frac{\pi }{4}$
3. $-\frac{\pi }{5}$
4. $\frac{3\pi }{10}$

Q. Let $x,y,z$ be non-zero real numbers such that $\frac{x}{y}+\frac{y}{z}+\frac{z}{x}=7$ and $\frac{y}{x}+\frac{z}{y}+\frac{x}{z}=9$, then $\frac{x^3}{y^3}+\frac{y^3}{z^3}+\frac{z^3}{x^3}-3$ is equal to

1. $152$
2. $154$
3. $155$
4. $153$

Q. The value of ` $\theta$' for which the system of linear equation in $x,y,z$ given as $\left(\sin 3\theta \right)x-y+z=0$

1. $\frac{n\pi }{2}$
2. $\frac{n\pi }{3}$
3. $n\pi +(-1{)}^n\frac{\pi }{6}$
4. $\frac{\pi }{12}$

Q.

$If2si{n}^2\theta =3cos\theta ,where0\le \theta \le 2\pi ,thenfindthevalueof\theta .$

Q.

Find $\frac{dy}{dx}$, if x and y are connected parametrically by the equations given in questions without eliminating the parameter.

$x=cos\theta -cos2\theta ,y=sin\theta -sin2\theta .$

Q.

$\text{Write the values of x in}[0,\pi ]forwhichsin2x,\frac{1}{2}andcos2xareinA.P.$

Q.

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

Q. The number of solution of ${\sin }^4\theta -2{\sin }^2\theta -1=0$ for $\theta \in [0,2\pi ]$ 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. If the system of equations $2x+3y=7$ and $(a+b)x+(2a-b)y=21$ has infinitely many solutions, then

1. $a=-1,b=+5$
2. $a=1,b=5$
3. $a=5,b=1$
4. $a=5,b=-1$

Q.

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

Q.

$\text{Write the solution set of the equation}(2cos\theta +1)(4cos\theta +5)=0intheinterval[0,2\pi ].$

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(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.

Write the number of sulutions of the equation tan x +sec x = 2 cos x in the interval $[0,2\pi ]$.

Q. The principal solution(s) of the trigonometric equation
$\sec \theta =\frac{2}{\sqrt{3}}\text{is/are:}$

1. $\theta =\frac{\pi }{6}$
2. $\theta =\frac{7\pi }{6}$
3. $\theta =\frac{11\pi }{6}$
4. $\theta =\frac{5\pi }{6}$

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

Q. If $|1-\frac{|sinx|}{1+|sinx|}|\ge \frac{2}{3}$, then sin x lies in

1. $(-\frac{1}{2},\frac{1}{2})$
2. $(-\infty ,\frac{-1}{2}]\cup [\frac{1}{2},\infty )$
3. $[-\frac{1}{2},\frac{1}{2}]$
4. none of these

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

Q. The solution of the equation $\cos 3\theta =4\cos \theta \cos (\theta +x)\cos (\theta -x)(0<\theta <\pi )$ is

1. $k\pi +\frac{\pi }{3}$
2. $2k\pi$
3. $k\pi -\frac{\pi }{3}$
4. None of these

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

Q. If $cos3\theta -cos4\theta =cos5\theta -cos6\theta then\theta =$

1. 
2. 
3. 
4. 

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. A solution of the equation $(1-tanx)(1+tanx)se{c}^{2}x+2^{ta{n}^2}=0$, where x lies in the interval $[\frac{-\pi }{2},\frac{\pi }{2}]$ is given by

1. $x=0$
2. $x=\frac{\pi }{3}$
3. $x=\frac{-\pi }{3}$
4. $x=\frac{\pi }{6}$