Equations Involving sinx + cosx and sinx.cosx

Q. If $\alpha +\beta +\gamma =2\pi ,$ then the system of equations
$x+\left(\cos \gamma \right)y+\left(\cos \beta \right)z=0$
$\left(\cos \gamma \right)x+y+\left(\cos \alpha \right)z=0$
$\left(\cos \beta \right)x+\left(\cos \alpha \right)y+z=0$ has

1. a unique solution
2. no solution
3. infinitely many solutions
4. exactly two solutions

Q.

If $\alpha ,\beta$ are solutions of $6\cos \theta +8\sin \theta =9$, then $\sin (\alpha +\beta )=$

1. $\frac{3}{5}$

2. $\frac{4}{5}$

3. $\frac{24}{25}$

4. $\frac{12}{13}$

Q. The domain of $f\left(x\right)={\sin }^{-1}x+{\cos }^{-1}x+{\tan }^{-1}x+{\cot }^{-1}x$ is

[1 mark]

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

Q. If $x$ and $y$ satisfies the equation $12\sin x+5\cos x=2y^2-8y+21$, then the value of $12\cot \left(\frac{xy}{2}\right)$ is

Q. Number of common solution(s) of the trigonometric equations $\cos 2x+(1-\sqrt{3})=(2-\sqrt{3})\cos x$ and $\sin 3x=2\sin x$ which satisfy the inequality $\sqrt{3}\tan x-1\ge 0$ in $[0,5\pi ]$ is

Q. The sine of the angle between the straight line $\frac{x-2}{3}=\frac{y-3}{4}=\frac{z-4}{5}$ and the plane $2x-2y+z=5$ is

1. $\frac{\sqrt{5}}{6}$
2. $\frac{2\sqrt{3}}{5}$
3. $\frac{\sqrt{2}}{10}$
4. $\frac{4}{5\sqrt{2}}$

Q.

Express each of the following as the sum or difference of sines and cosines :
(i) $2sin3\theta cos\theta$
(ii) $2cos3\theta sin2\theta$
(iii) $2sin4\theta sin3\theta$
(iv) $2cos7\theta cos3\theta$

Q. The number of integral values of n such that sinx(sinx+cosx)=n has atleast one real solution is

1. 4
2. 1
3. 3
4. 2

Q. If the sum of all solutions of the equation $\sin x+2\cos x=1+\sqrt{3}\cos x$ in $[0,2\pi ]$ is $\frac{k\pi }{6}$ then $\frac{k}{2}$ is `

Q. The general solution of $8\cos x\cdot \cos 2x\cdot \cos 4x=\frac{\sin 6x}{\sin x}$ is

1. $\frac{(2n+1)\pi }{14},n\in Z$
2. $\left\{n\pi \right\}\cup \left\{\frac{(2n+1)\pi }{14}\right\},n\in Z$
3. $\frac{(2n+1)\pi }{7},n\in Z$
4. $\frac{n\pi }{7},n\in Z$

Q. If $y=\cos ax,\text{then}\left|\begin{array}{ccc}y& y_1& y_2\\ y_3& y_4& y_5\\ y_6& y_7& y_8\end{array}\right|=$ (where $y_n=\frac{d^{n}y}{d{x}^n}$)

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

Q. Find the principal value of ${\cos }^{-1}(-\frac{1}{\sqrt{2}}).$

Q. Number of all pairs $(x,y)$ which satisfy $x^2+2x\sin \left(xy\right)+1=0,\text{if}yϵ(–4\pi ,4\pi )$ is

1. 4
2. 8
3. 10
4. 6

Q. If x and y are the two sides of a square and $x=(1+\sin A)(1+\sin B)(1+\sin C)=(1-\sin A)(1-\sin B)(1-\sin C),$
show that each side is equal to $\pm \cos A\cos B\cos C.$

Q. Solve the equation
$\left(vi\right)3{\cos }^{2}x-2\sqrt{3}\sin x\cos x-3{\sin }^{2}x=0$

Q. $\text{List I}$ has four entries and $\text{List II}$ has five entries. Each entry of $\text{List I}$ is to be matched with one or more than one entries of $\text{List II}.$

$\begin{array}{cccc}& \text{List I}& & \text{List II}\\ \text{(A)}& \text{The possible value(s) of}a\text{for which the largest}& \text{(P)}& 9\\ & \text{value of}{\sin }^{2}x-2a\sin x+a+3\text{is}7\text{is/are}& & \\ \text{(B)}& \text{The possible value(s) of}a\text{for which the smallest}& \text{(Q)}& 16\\ & \text{value of}{x}^4-a{x}^2+2a-1\text{for}x\in [-1,2]\text{is}& & \\ & -7\text{, is/are}& & \\ \text{(C)}& \text{If a relation}R\text{is defined on set of integers as}& \text{(R)}& -3\\ & R=\left\{\right(x,y):4x^2+9y^2\le 36\}\text{, then possible}& & \\ & \text{element(s) in the domain is/are}& & \\ \text{(D)}& \text{If}\sin x+\cos x=\frac{1}{5}\text{, then}|12\tan x|\text{is equal to}& \text{(S)}& 1\\ & & \text{(T)}& 11\end{array}$

Which of the following is the only CORRECT combination?

1. $\text{(A)}\to \text{(Q)},\text{(S)}$
2. $\text{(A)}\to \text{(Q)},\text{(R)}$
3. $\text{(B)}\to \text{(Q)},\text{(S)}$
4. $\text{(B)}\to \text{(R)},\text{(T)}$

Q. Solve the equation
$\left(vi\right)\sin x+\sin 2x+\sin 3x=0$

Q. For the given equations $\cos (4y-3x-2)-\cos (4y+3x+2)=2+2\ln (k^4-255)$ and $\cos (4y-3x-2)+\cos (4y+3x+2)=2k+8$ to have real solutions $(x,y)$, the possible value of k is

1. 4
2. -4
3. -5
4. none of these

Q. Solve the equation
$\left(i\right)\sin x+\cos x=\sqrt{2}$

Q. If $x$ and $y$ satisfy the equation $12\sin x+5\cos x=2y^2-8y+21$, then the value of $12\cot \left(\frac{xy}{2}\right)$ is

Q. If $0<x,y<\pi$ and $\cos x+\cos y-\cos (x+y)=\frac{3}{2},$ then $\sin x+\cos y$ is equal to :

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

Q. The number of solution of equation $2\sin 2x+\sin x(1-2\cos 2x)-4=0$ in the interval $(0,\pi$) is -

Q. Let $x,y\in [0,2\pi ]$ and satisfy the equation ${\sin }^{3}x+{\cos }^{3}y+6\sin x\cos y=8.$ If $a=x+y$ and $b=x-y,$ then

1. minimum value of $a$ is $\frac{\pi }{2}$
2. maximum value of $a$ is $\frac{5\pi }{2}$
3. minimum value of $b$ is $-\frac{3\pi }{2}$
4. maximum value of $b$ is $\pi$

Q. The equation
$2{\cos }^2\frac{x}{2}{\sin }^{2}x=x^2+x^{-2};0<x\le \frac{\pi }{2}$
has

1. No real solution
2. One real solution
3. more than one solution
4. none of these

Q. The general solution of $8\cos x\cdot \cos 2x\cdot \cos 4x=\frac{\sin 6x}{\sin x}$ is

1. $\frac{(2n+1)\pi }{14},n\in Z$
2. $\left\{n\pi \right\}\cup \left\{\frac{(2n+1)\pi }{14}\right\},n\in Z$
3. $\frac{n\pi }{7},n\in Z$
4. $\frac{(2n+1)\pi }{7},n\in Z$

Q. The sum of all the values of $\theta (0\le \theta \le 2\pi )$ which satisfies both the equations $r\sin \theta =3$ and $r=4(1+\sin \theta )$ is

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

Q.

If $cos\theta =\frac{cos\alpha +cos\beta }{1+cos\alpha cos\beta }$, prove that $tan\frac{\theta }{2}=\pm tan\frac{\alpha }{2}tan\frac{\beta }{2}$

Q. The general solution of the equation $2{\cos }^{2}x=-3\sin x$ is

1. $x=n\pi +\frac{7\pi }{6},n\in Z$
2. $x=n\pi +(-1{)}^n\frac{7\pi }{6},n\in Z$
3. $x=n\pi +(-1{)}^n\frac{\pi }{6},n\in Z$
4. $x=n\pi +\frac{\pi }{6},n\in Z$

Q. General solution of the equation $4\cos x-3\sec x=\tan x,(\cos x\ne 0)$ can be

1. $x=n\pi +(-1{)}^n\alpha ,\alpha ={\sin }^{-1}\left(\frac{-1+\sqrt{17}}{8}\right),n\in Z$
2. $x=n\pi +(-1{)}^n\alpha ,\alpha ={\sin }^{-1}\left(\frac{1+\sqrt{17}}{8}\right),n\in Z$
3. $x=n\pi +(-1{)}^n\beta ,\beta ={\sin }^{-1}\left(\frac{1-\sqrt{17}}{8}\right),n\in Z$
4. $x=n\pi +(-1{)}^n\beta ,\beta ={\sin }^{-1}\left(\frac{-1-\sqrt{17}}{8}\right),n\in Z$

Q. Number of solutions of the equation $2({\sin }^3\theta +{\sin }^2\theta )+2({\cos }^3\theta +{\cos }^2\theta )=3\sin 2\theta$
in the interval $[0,4\pi ]$ is