General Solution of cos theta = cos alpha

Q. Find the value of θ and p, if the equation x cos θ + y sin θ = p is the normal form of the line $\sqrt{3}x+y+2=0$.

Q. If $\cos \mathrm{\theta }=-\frac{1}{2}\mathrm{and}0<\mathrm{\theta }<360°$, then the solutions are
(a) $\theta =60°,240°$
(b) $\theta =120°,240°$
(c) $\theta =120°,210°$
(d) $\theta =120°,300°$

Q. $cos(\alpha -\beta )=1$ and $cos(\alpha +\beta )=\frac{1}{e}$, where $\alpha ,\beta \in [-\pi ,\pi ]$. Number of pairs of $\alpha ,\beta$ which satisfy both the equation is

1. \N
2. 1
3. 2
4. 4

Q.

Cos 4x cos 8x - cos 5x cos 9x = 0 if

1. Cos 12x = cos 14x

2. Sin 13x = 0

3. Sin x = 0

4. Cos x = 0

Q.

Find the values of θand p, if the equation is the normal form of the line.

Q. If $3\cos x\ne 2\sin x$, then the general solution of ${\sin }^{2}x-\cos 2x=2-\sin 2x$ is

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

Q. 33. solve the equation, 2(sinx +siny)-2cos(x-y)=3 for smallest positive value of x and y

Q. Find the values of θ and p, if the equation x cos θ + y sin θ = p is the normal form of the line $\sqrt{3}x+y+2=0$.

Q. If the equation $2\cos x+\cos 2\lambda x=3$ has only one solution, then $\lambda$ is

1. 1
2. a rational number
3. an irrational number
4. None of these

Q. The general value of θ in the equation $\cos \theta =\frac{1}{\sqrt{2}},\tan \theta =-1$ is

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

Q. If $\sin 3\theta =\sin \theta ,$ how many solutions exist such that $-2\pi <\theta <2\pi$ ?

1. 8
2. 9
3. 11
4. 7

Q. The sum of all the solutions of the equation $\cos \theta \cos (\frac{\pi }{3}+\theta )\cos (\frac{\pi }{3}-\theta )=\frac{1}{4},\theta \in [0,6\pi ]$ is

1. $15\pi$
2. $30\pi$
3. $\frac{100\pi }{3}$
4. None of these