General Solution of cos theta = cos alpha

Q.

cos 10 cos 30 cos 50 cos 70 =

Q. If $4{\sin }^2\theta +2(\sqrt{3}+1)\cos \theta =4+\sqrt{3}$, then the general solution is

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

Q. If $\sqrt{3}\left({\cos }^{2}x\right)=(\sqrt{3}-1)\cos x+1,$ the number of solutions of the given equation when $x\in [0,\frac{\pi }{2}]$ is

Q. Considering the principal values of the inverse trigonometric functions, the sum of all the solutions of the equation ${\cos }^{–1}\left(x\right)–2{\sin }^{–1}\left(x\right)={\cos }^{–1}\left(2x\right)$ is equal to

Q. $Let\overrightarrow{\alpha }=3\hat{i}+\hat{j}and\overrightarrow{\beta }=2\hat{i}-\hat{j}+3\hat{k}.If\overrightarrow{\beta }=\overrightarrow{{\beta }_1}-\overrightarrow{{\beta }_2},where\overrightarrow{{\beta }_1}$
is parallel to $\overrightarrow{\alpha }$ and ${\overrightarrow{\beta }}_2$ is perpendicular to $\overrightarrow{\alpha }$, then $\overrightarrow{{\beta }_1}\times \overrightarrow{{\beta }_2}$ is equal to:

1. $\frac{1}{2}(-3\hat{i}+9\hat{j}+5\hat{k})$
2. $\frac{1}{2}(3\hat{i}-9\hat{j}+5\hat{k})$
3. $-3\hat{i}+9\hat{j}+5\hat{k}$
4. $3\hat{i}-9\hat{j}-5\hat{k}$

Q. The number of solution(s) for $\left\{x\right\}+\left\{\sin x\right\}=2,$ if $x\in [0,2\pi ],$ where $\{.\}$ is the fractional part function, is

1. $0$
2. $1$
3. $2$
4. infinitely many

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. 4
2. 2
3. 1
4. 0

Q. The equation of the bisector of the obtuse angle between the planes 3x + 4y – 5z + 1 = 0, 5x + 12y – 13z = 0 is :

1. 11x + 4y – 3z = 0
2. 14x – 8y + 13 = 0
3. x + y + z = 9
4. 13x – 7z + 18 = 0

Q.

The value of${\int }_0^{2\pi }\frac{1}{e^{\sin x}+1}dx$ is

1. $\pi$

2. $0$

3. $2\pi$

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

Q.

The number of solutions of the equation x/100=sinx

A) 63

B)32

C) 33

D) 0

Q. The solution of the equation $8\sin x=\frac{\sqrt{3}}{\cos x}+\frac{1}{\sin x}$ is/are given by

1. 
   $x=\mathrm{n\pi }+\frac{\pi }{6},n\in Z.$
2. 
   $x=n\pi -\frac{\pi }{6},n\in Z.$
3. 
   $x=\frac{\mathrm{n\pi }}{2}+\frac{\pi }{12},n\in Z.$
4. 
   $x=\frac{\mathrm{n\pi }}{2}-\frac{\pi }{12},n\in Z.$

Q.

If $2{\sin }^2\theta +\sqrt{3}\cos \theta +1=0$, then the value of $\theta$ is

1. $\frac{\mathrm{\pi }}{6}$

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

3. $\frac{5\mathrm{\pi }}{6}$

4. $\mathrm{\pi }$

Q. The solution(s) of the equation $9{\cos }^{12}x+{\cos }^{2}2x+1=6{\cos }^{6}x\cos 2x+6{\cos }^{6}x-2\cos 2x$ is/are $(n\in I)$.

1. $n=n\pi +\frac{\pi }{2}$
2. $n=n\pi +{\cos }^{-1}\left(\sqrt[4]{4\frac{2}{3}}\right)$
3. $n=n\pi -{\cos }^{-1}\left(\sqrt[4]{4\frac{2}{3}}\right)$
4. $n=n\pi +{\cos }^{-1}\left(\sqrt{\frac{2}{3}}\right)$

Q. The number of solution(s) of $\sin 3x-\sin x=4{\cos }^{2}x-2$ for $x\in [0,2\pi ]$ is

1. $2$
2. More than $2$ but finite
3. $1$
4. infinitely many

Q. hiw many solution can we find in principal solution and general solution ????

Q.

$\text{If cos x=k has exactly one solution in}[0,2\pi ],thenwritethevalue\left(s\right)ofk.$

Q. $\mathrm{If}\mathrm{cosp\theta }+\mathrm{cosq\theta }=0,\mathrm{then}\mathrm{the}\mathrm{different}\mathrm{values}\mathrm{of}\theta \mathrm{are}\mathrm{in}A.P.\mathrm{with}a\mathrm{common}\mathrm{difference}$
.

1. $2\pi /\left(p\pm q\right)$
2. $\pi /\left(p\pm q\right)$
3. $3\pi /\left(p\pm q\right)$

Q. General solutions of $\text{cosec}\theta =\text{cosec}\alpha ,\alpha \ne n\pi$ are
$\theta =m\pi +(-1{)}^m\alpha ,$
where $n,m\in Z$

1. True
2. False

Q.

How do you solve $\sin \left(x\right)=0.5$?

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

Q.

The number of solutions of the equation $\left(1-\cos \left(2x\right)\right)\left(\cos \left(2x\right)+\cot \left(2x\right)\right)=0,0\le x\le 2\pi$ is

1. $3$

2. $2$

3. $1$

4. $0$

Q. Solve the equation
$\left(ix\right)\sin 2x-\sin 4x+\sin 6x=0$

Q. The general solution of $\sin 2x+\sin 4x+\sin 6x=0$ is/are

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

Q. If $3\sin x+4\cos ax=7$ has at least one solution and $a=\frac{pm}{qn+1};$ where $m,n\in Z,$ then the value of $p+q$ is

Q. What is the polar form of -1-i? Argument mostly given in solutions is -3pi/4.... but can we write 5pi/4?both mean same only ryt with a phase difference of 2pi?

Q. If $\frac{k\pi }{10}$ is the least positive value of $\theta$ which satisfy the equation $\sin 3\theta +\cos 2\theta =0$, then $k$ is

Q.

If $\alpha$ be a real root of the equation $x^3+p{x}^2+qx+r=0$ where p, q and r are real. If $p^2-4q-2p\alpha -3{\alpha }^2\ge 0$ then other roots are ________.

1. Real numbers

2. Imaginary conjugate

3. Irrational conjugate

4. None of these

Q. In a triangle $ABC$ if $\cos A+\cos B+\cos C=\frac{3}{2},$ then the triangle is

1. Right angled
2. Right angled isoceles
3. Equilateral
4. None of these

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

1. 
2. 
3. 
4. 

Q. The number of values of $x$ in $[0,2\pi ]$ that satisfy the equation ${\sin }^{2}x-\cos x=\frac{1}{4}$

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