Trigonometric Ratios of Multiple of an Angle

Q. The number of solutions of the equation ${32}^{{\tan }^{2}x}+{32}^{{\sec }^{2}x}=81,$ $0\le x\le \frac{\pi }{4}$ is

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

Q. The set of value(s) of $a$ for which $x^2+ax+{\sin }^{-1}(x^2-4x+5)+{\cos }^{-1}(x^2-4x+5)=0$ has at least one solution is

Q. Let $S$ be the set of all $\alpha \in R$ such that the equation, $\cos 2x+\alpha \sin x=2\alpha -7$ has a solution. Then $S$ is equal to:

1. $R$
2. $[2,6]$
3. $[1,4]$
4. $[3,7]$

Q. The solution set of the equation $2{\sin }^{2}x+\sqrt{3}\cos x+1=0$ is $\{2n\pi \pm \frac{a\pi }{b},n\in Z;\frac{a}{b}\in [0,1\left]\right\}$ where $a,b$ are co-prime. Then the value of $a+b$ is

1. $10$
2. $11$
3. $9$
4. $15$

Q. Let $0\le x,y,z\le \frac{\pi }{2}$ such that $\sin x\cdot \sin y\cdot \cos z=\frac{1}{4\sqrt{2}},\frac{{\sin }^{2}x\cdot {\sin }^{3}y}{\cos z}=\frac{1}{4\sqrt{2}}\text{and}\frac{\sin x\cdot {\sin }^{4}y}{{\cos }^{2}z}=\frac{1}{8}$, then which of the following is/are correct?

1. $\sin x+\sin y=\frac{3}{2}$
2. $\cos (x-y)=\frac{\sqrt{3}}{2}$
3. $\tan (x+y+z)=\sqrt{3}-2$
4. $\sec (y-z)=\sqrt{6}-\sqrt{2}$

Q. If the sum of the solutions of the equation $\cos (\frac{\pi }{3}-\theta )\cos (\frac{\pi }{3}+\theta )-\frac{\sec \theta }{4}=0$ in $[0,10\pi ]$ is $k\pi ,$ then the value of $k$ is

Q.

If y(x) is a solution of $\left(\frac{2+sinx}{1+y}\right)\frac{dy}{dx}=-cosxandy\left(0\right)=1,$then find the value of $y\left(\frac{\pi }{2}\right).$

Q. For $x\in (0,\pi ),$ the equation $\sin x+2\sin 2x-\sin 3x=3$ has

1. no solution
2. one solution
3. two solutions
4. three solutions

Q. Total number of solutions for the equation ${\sin }^{4}x+{\cos }^{4}x=\sin x\cos x,x\in [0,2\pi ]$ is

1. $4$
2. $2$
3. $6$
4. $3$

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

Q. The number of solutions of the equation ${\sin }^{-1}[x^2+\frac{1}{3}]+{\cos }^{-1}[x^2-\frac{2}{3}]=x^2$, for $x\in [-1,1]$ and $\left[x\right]$ denotes the greatest integer less than or equal to $x$, is

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

Q.

The complex number sin x+i cos 2x and cos x - i sin 2x are conjugate to each other for :

1. 

2. 

3. 

4. 

Q. The number of solutions of the equation ${\cos }^{-1}x+{\cos }^{-1}\sqrt{1-x^2}=\pi$ is

1. $1$
2. $2$
3. none of these
4. $0$

Q. Solution of the equation $a\sin x+b\cos x=c$ exits if and only if $|c|\ge \sqrt{a^2+b^2}$

1. False
2. True

Q. Is ${\sec }^2\theta =\frac{4xy}{(x+y{)}^2}$ possible for real value of $x$ and $y$?

Q.

If $\frac{\pi }{2}<\theta <\pi$, then write the value of $\sqrt{2+\sqrt{2+2cos2\theta }}$ in the simplest form.

Q.

If $\frac{\pi }{2}<\theta <\pi$, then write the value of $\sqrt{\frac{1-cos2\theta }{1+cos2\theta }}$

Q.

If $\pi <\theta <\frac{3\pi }{2}$, then write the value of

$\sqrt{\frac{1-cos2\theta }{1+cos2\theta }}$

Q. The number of values of $x$ between $0$ and $2\pi$ that satisfies the equation $\sin x+\sin 2x+\sin 3x=\cos x+\cos 2x+\cos 3x$ must be-

Q. The number of real solutions of; ${\cos }^{-1}x+{\cos }^{-1}2x=-\pi$ is :

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

Q. For $x\in (0,\pi ),$ the equation $\sin x+2\sin 2x-\sin 3x=3$ has

1. no solution
2. one solution
3. two solutions
4. three solutions

Q. $(2\sin x-\cos x)(1+\cos x)={\sin }^{2}x$
If true then enter $1$ and if false then enter $0$

Q. Evaluate $\int \frac{5\cos x+6}{2\cos x+\sin x+3}dx$

Q. If $\cos (x-y)+\cos (y-z)+\cos (z-x)=-\frac{3}{2}$, if and only if

1. $\sin x+\sin y+\sin z=0$
2. $\cos x+\cos y-\cos z=0$
3. $\sin x+\sin y+\sin z=\cos x+\cos y+\cos z$
4. $\sin x+\sin y+\sin z=\cos x+\cos y+\cos z=0$

Q. Let $S$ be the set of all $\alpha \in R$ such that the equation, $\cos 2x+\alpha \sin x=2\alpha -7$ has a solution. Then $S$ is equal to:

1. $R$
2. $[2,6]$
3. $[1,4]$
4. $[3,7]$

Q. If the solution set of inequality $\left({\cot }^{-1}x\right)\left({\tan }^{-1}x\right)+(2-\frac{\pi }{2}){\cot }^{-1}x-3{\tan }^{-1}x-3(2-\frac{\pi }{2})>0$ is $(a,b)$, then the value of ${\cot }^{-1}a+{\cot }^{-1}b$ is