Solving Simultaneous Trigonometric Equations

Q.

The number of integral value of K for which the equation 7cosx+5sinx=2K+1 has a solution is

1. 4

2. 8

3. 10

4. 12

Q.

If $\sin (x+y)+\cos (x+y)=\log (x+y)$ then $\frac{d^{2}y}{d{x}^2}$ is equal to

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

2. $0$

3. $-1$

4. $1$

Q.

Find the general solution of the equation sin 2x+sin 4x +sin 6x =0.

Q. The number of solutions of the equation $16(si{n}^{5}x+co{s}^{5}x)=11(sinx+cosx)$ in the interval $[0,2\pi ]$ is

1. 6
2. 7
3. 8
4. 9

Q.

Number of ordered pairs (x, y) satisfying |y|=cosx, y=$si{n}^{-1}\left(sinx\right),x\in [-2\pi ,3\pi ]$ is

1. 5

2. 3

3. 6

4. 7

Q. The most general value of $\theta$ satisfying both the equations $sin\theta =\frac{1}{2},tan\theta =\frac{1}{\sqrt{3}}is\left(nϵZ\right)$

1. 
2. 
3. 
4. 

Q.

If the sum of all the solutions of the equation $8\cos \left(x\right)\begin{array}{cl}\left(\cos \left(\left(\frac{\mathrm{\pi }}{6}\right)+x\right)\right).& \left(\cos \left(\left(\frac{\pi }{6}\right)-\left(x\right)\right)-\frac{1}{2}\right)\end{array}=1\mathrm{in}\left[0,\mathrm{\pi }\right]\mathrm{is}\mathrm{k\pi }$, then $k$ is equal to

1. $\frac{8}{9}$

2. $\frac{20}{9}$

3. $\frac{2}{3}$

4. $\frac{13}{9}$

Q.

The number of distinct solutions of sin5$\theta$.cos3$\theta$ = sin9$\theta$.cos7$\theta$ in [0, $\frac{\pi }{2}$] is

1. 5

2. 8

3. 9

4. 4

Q.

The minimum value of $2cos\theta +\frac{1}{sin\theta }+\sqrt{2}tan\theta$ in the interval $(0,\frac{\pi }{2})$ is

1. 

2. 

3. 

4. 

Q.

The total number of solutions of cosx = $\sqrt{1-sin2x}$ in (0, 2$\pi$ ) is equal to

1. 2

2. 5

3. 1

4. 3

Q.

Find the general solution for the following equation:

cos 4x = cos 2x

Q.

Find the general solutions of sin 2x - sin 4x + sin 6x = 0.

1. $x=\frac{n\pi }{4}$

2. $x=\frac{n\pi }{2}$

3. $x=n\pi$

4. $x=n\pi \pm \frac{\pi }{6}$

Q.

Find the angle made by the line x cos ${30}^{\circ }$ + y sin ${30}^{\circ }$ + sin ${120}^{\circ }$ = 0 with the positive direction of the x-axis

Q.

Find the general solution for the following equation:

$se{c}^2$ 2x = 1 - tan 2x

Q.

If |z - 3i| = 3, (where i = $\sqrt{-1}$) and argz $\in (0,\frac{\pi }{2})$, then cot(arg(z)) - $\frac{6}{z}$ is equal to

1. 0

2. -i

3. 

4. i

Q.

If $\alpha$ is the fifth root of unity then which of the following is false

1. 

2. 

3. 

4. 

Q.

$\frac{1}{4}$$[\sqrt{3}cos{23}^{\circ }-sin{23}^{\circ }]$ =

1. cos43⁰

2. cos7⁰

3. cos53⁰

4. None of these

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

1. 
2. 
3. 
4. 

Q. The least difference between the roots of the equation $4cosx(2-3si{n}^{2}x)+(cos2x+1)=0(0\le x\le \frac{\pi }{2})$ is

1. 
2. 
3. 
4. 

Q.

The number of solutions of the equation 5 sec$\theta$ - 13 = 12 tan$\theta$ in [0 , 2$\pi$] is

1. 2

2. 1

3. 4

4. 0

Q.

The value of x and y which satisfies the equation is $4^{sinx}+3^{\frac{1}{cosy}}=11and5.{16}^{sinx}-2.3^{secy}=2$

1. 

2. 

3. 

4. 

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. The most general solution of the equations $\tan \theta =-1,\cos \theta =\frac{1}{\sqrt{2}}$ is

1. none of these
2. 
3. 
4. 

Q.

Find the principal solution of sin x + sin 3x +sin 5x = 0

1. $x=0$

2. $x=\frac{\pi }{6}$

3. $x=\frac{\pi }{3}$

4. $x=\frac{\pi }{2}$

Q. sin x + 2 sin 2x = 3 + sin 3x, 0 $\le$ x $\le 2\pi$ has

1. 2 solutions in I quadrant
2. One solution in II quadrant
3. no solution in any quadrant
4. one solution in each quadrant

Q.

If x $\in [\frac{\pi }{2},\frac{5\pi }{2}]$, the greatest positive solution of 1 + $si{n}^{4}x=co{s}^{2}3x$ is

1. 2π

2. 

3. 

4. π

Q.

If $\alpha$ and $\beta$ are the solutions of the sequation $a\tan \theta +b\sec \theta =c,$ show that tan $(\alpha +\beta )=\frac{2ac}{a^2-c^2}.$

Q.

No. of real roots of the equation $x^3+x^2$ + 10x + sin x = 0, is :

1. 

2. 

3. 

4. 

Q.

Consider the system of linear equations in x, y and z ;
(sin 3$\theta$) x - y + z = 0 ......(i)
(cos 2$\theta$)x + 4y + 3z = 0 ......(ii)
2x + 7y+ 7z = 0 ......(iii)

The value of $\theta$ for which the system has nontrivial solution is

1. θ = nπ, nπ + (-1)ⁿ (π/6), where n ϵ I.

2. θ= nπ+(π/6), , nπ + (-1)ⁿ (π/4), where n ϵ I.

3. θ= nπ-π/6, nπ + (-1)ⁿ (π/3), where n ϵ I.

4. θ=2nπ , where n ϵ I.

Q.

If $\sin (A+B)=1,\cos (A-B)=1,0^{\circ }\le A+B\le {90}^{\circ }$, Find $A$ and $B$.

1. ${45}^{\circ },{45}^{\circ }$

2. ${60}^{\circ },{30}^{\circ }$

3. ${30}^{\circ },{60}^{\circ }$

4. ${15}^{\circ },{75}^{\circ }$