General Solutions of (sin theta)^2 = (sin alpha)^2 , (cos theta)^2 = (cos alpha)^2 , (tan theta)^2 = (tan alpha)^2

Q.

The number of solutions of the pair of equations $2{\sin }^2\theta -\cos 2\theta =0$and $2{\cos }^2\theta -3\sin \theta =0$ in the interval $[0,2\pi ]$ is

1. $0$

2. $1$

3. $2$

4. $4$

Q.

If ${\sin }^{-1}\left(a\right)+{\sin }^{-1}\left(b\right)+{\sin }^{-1}\left(c\right)=\pi$ then the value of $a\sqrt{\left(1-a^2\right)}+b\sqrt{\left(1-b^2\right)}+c\sqrt{\left(1-c^2\right)}$ will be

1. $2abc$

2. $abc$

3. $\frac{1}{2}abc$

4. $\frac{1}{3}abc$

Q. The number of solutions of ${\sin }^{5}x-{\cos }^{5}x=\frac{1}{\cos x}-\frac{1}{\sin x}$ (where $\sin x\ne \cos x)$ is

1. $0;x\in (0,\pi )-\left\{\frac{\pi }{2}\right\}$
2. $1;x\in (0,\pi )-\left\{\frac{\pi }{2}\right\}$
3. $2;x\in (0,2\pi )-\{\frac{\pi }{2},\frac{3\pi }{2},\pi \}$
4. $0;x\in (0,2\pi )-\{\frac{\pi }{2},\frac{3\pi }{2},\pi \}$

Q. Consider an equation with $x$ as a variable $7\sin 3x-2\sin 9x={\sec }^2\theta +4{\text{cosec}}^2\theta .$ Then the value of $\frac{15}{\pi }(\text{(minimum positive root)}-\text{(maximum negative root)})$ is

Q. If the inequality ${\sin }^{2}x+a\cos x+a^2>1+\cos x$ holds for any $x\in R$, then the largest negative integral value of $a$ is

Q. If the function $f\left(x\right)=\lambda |\sin x|+{\lambda }^2|\cos x|+g\left(\lambda \right),\lambda \in R,$ where $g$ is a function of $\lambda ,$ is periodic with fundamental period $\frac{\pi }{2},$ then

1. $\lambda =0,1$
2. $\lambda =1$
3. $\lambda =-1$
4. $\lambda =0$

Q. In a $△ABC$ with usual notations, if $\tan A,\tan B$ satisfy the equation $\sqrt{3}{x}^2-4x+\sqrt{3}<0$ and $(a-b{)}^2>c^2-ab$, then number of such triangles possible is

Q. $\mathrm{If}{\cos }^{-1}x+{\cos }^{-1}y+{\cos }^{-1}z=3\pi ,$then the value of $xy+yz+zx$ =

Q. In a $△ABC$ with usual notation $CF$ is the internal angle bisector of $\angle C$ and $F$ lies on side $AB$, then the length of $CF$ is equal to

1. $\frac{a\sin B}{\sin (B+\frac{C}{2})}$
2. $\frac{a+b\cos (\frac{C}{2})}{2ab}$
3. $\frac{2ab\cos (\frac{C}{2})}{a+b}$
4. $\frac{b\sin B}{\sin (C+\frac{B}{2})}$

Q.

If ${\sin }^{2}x$ + ${\cos }^{2}y$ = 2 ${\sec }^{2}z$, find the value of ${\cos }^{2}x$ + $si{n}^{2}y+2si{n}^{2}z$.

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Q. The inequality $2\cos x\le |\sqrt{1+\sin 2x}-\sqrt{1-\sin 2x}|\le \sqrt{2}$ where $x\in [0,2\pi ]$ holds true in the interval $[\frac{a\pi }{2},\frac{b\pi }{2}]$. Then the value of $a+b$ is

Q. There exists a value of $\theta$ between 0 and $2\pi$ that satisfies the equation $si{n}^4\theta -2si{n}^2\theta -1=0.$

1. True
2. False

Q. $\{x\in R:\cos 2x+2{\cos }^{2}x=2\}$ is equal to

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

Q.

If ${\sin }^{2}x$ + ${\cos }^{2}y$ = 2 ${\sec }^{2}z$, find the value of ${\cos }^{2}x$ + $si{n}^{2}y+2si{n}^{2}z$.

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Q.

How many values of $x$ in the interval $[0,2\pi ]$ satisfies the equation ${\sin }^{6}x=1+{\cos }^{4}3x$

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Q. The number of solutions of ${\sin }^{5}x-{\cos }^{5}x=\frac{1}{\cos x}-\frac{1}{\sin x}$ (where $\sin x\ne \cos x)$ is

1. $0;x\in (0,\pi )-\left\{\frac{\pi }{2}\right\}$
2. $1;x\in (0,\pi )-\left\{\frac{\pi }{2}\right\}$
3. $2;x\in (0,2\pi )-\{\frac{\pi }{2},\frac{3\pi }{2},\pi \}$
4. $0;x\in (0,2\pi )-\{\frac{\pi }{2},\frac{3\pi }{2},\pi \}$

Q.

If $\alpha$ is the value of $xϵ[0,2\pi ]$ which is a solution of the equation

$2co{s}^2$ $\frac{x^2+x}{6}=2^x$ + $2^{-x}$, then find the value of $\frac{2\alpha }{3\pi }$ .

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