General Solution of sin theta = sin alpha

Q. The number of solutions of the equation ${\sin }^{4}x+{\cos }^{4}x=$ $\sin x\cos x$ in $[\pi ,5\pi ]$ is

Q.

Given $0\le x\le \frac{1}{2}$, then the value of $tan\left[{\sin }^{-1}\left(\frac{x}{\surd 2}+\frac{\surd 1-x^2}{\surd 2}\right)-{\sin }^{-1}x\right]$ is

1. $1$

2. $\sqrt{3}$

3. $-1$

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

Q.

The period of ${\sin }^2\theta$ is

1. ${\pi }^2$

2. $\pi$

3. $2\pi$

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

Q. The general solution for $\sin x=\frac{\sqrt{3}}{2}$ is given as

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

Q. General solution of the trigonometric equation
$\sin x+\sin 5x=\sin 2x+\sin 4x$ is

1. $\frac{2n\pi }{3}$
2. $\frac{n\pi }{3}$
3. $2n\pi$
4. $n\pi$

Q. If $\alpha$ and $\beta$ $(\alpha >\beta )$ are the roots of the equation $x^2-\sqrt{2}x+\sqrt{3-2\sqrt{2}}=0,$ then the value of $({\cos }^{-1}\alpha +{\tan }^{-1}\alpha +{\tan }^{-1}\beta )$ is equal to

1. $\frac{3\pi }{8}$
2. $\frac{5\pi }{8}$
3. $\frac{7\pi }{8}$
4. $\frac{\pi }{3}$

Q.

$\text{The smallest value of}\left(\theta \right)\text{satisfying the equation}\sqrt{3}(cot\theta +tan\theta )=4is$

1. $2\pi /3$

2. $\pi /3$

3. $\pi /6$

4. $\pi /12$

Q.

The solution of trigonometric equation ${\cos }^4\left(x\right)+{\sin }^4\left(x\right)=2\cos (2x+\mathrm{\pi })\cos (2\mathrm{x}-\mathrm{\pi })$ is

1. $x=\frac{n\mathrm{\pi }}{2}\pm {\sin }^{-1}\left(\frac{1}{5}\right)$

2. $x=\frac{n\mathrm{\pi }}{4}+\frac{{\left(-1\right)}^n}{4}{\sin }^{-1}\left(\frac{\pm 2\sqrt{2}}{3}\right)$

3. $x=\frac{n\mathrm{\pi }}{2}\pm {\cos }^{-1}\left(\frac{1}{5}\right)$

4. $x=\frac{n\mathrm{\pi }}{2}+\frac{{\left(-1\right)}^n}{4}{\cos }^{-1}\left(\frac{1}{5}\right)$

Q. The ratio of the lengths of the sides $BC$ and $AC$ of $\Delta ABC$ to the radius of circumscribed circle are equal to $2$ and $1.5$ respectively. The ratio of length of the bisectors of the interior angles $B$ and $C$ is

1. $\frac{7(\sqrt{7}-1)}{9\sqrt{2}}$
2. $\frac{7(\sqrt{7}+1)}{9\sqrt{2}}$
3. $\frac{7(\sqrt{7}-1)}{9\sqrt{2}}$
4. $\text{None of these}$

Q.

(1+cosx) (1+sinx) = 5/4 Find (1-cosx) and (1-sinx)

Q. $\mathrm{The}\mathrm{general}\mathrm{solution}\mathrm{of}\mathrm{the}\mathrm{equation}\sqrt{5-2\sin x}=6\sin x-1\mathrm{is}\mathrm{given}\mathrm{by}$.

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

Q. If $\sin \theta ,1,\cos 2\theta$ are in G.P., then

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

Q.

If $\sin \left(\mathrm{A}\right)+\sin \left(\mathrm{B}\right)+\sin \left(\mathrm{C}\right)+s\mathrm{in}\left(\mathrm{D}\right)=4$ then find the value of $\sin \left(\mathrm{A}\right)\times \sin \left(\mathrm{B}\right)\times \sin \left(\mathrm{C}\right)\times \sin \left(\mathrm{D}\right)$.

Q.

Give the general solution of the equation tan 5x = cot 2x

Q. Equation sin 7x + cos 2x = -2 Solution is A) x=(2kpi)/7 + 3pi/14, kI B) x=npi + pi/4 , n I C)x= 2npi + pi/2 , n I D)none of these

Q. $\cos (2x+7)=a(2-\sin x)$ can have a real solution for

1. $a\in [2,6]$
2. $a\in (0,\infty )$
3. all real values of $a$
4. $a\in (-\infty ,2)\setminus \left\{0\right\}$

Q. The general solution of the equation ${\sin }^3\theta \cos \theta -{\cos }^3\theta \sin \theta =\frac{1}{4}$is

1. 
   $\frac{n\pi }{4}+{(-1)}^{n+1}\left(\frac{\pi }{4}\right),n\in Z$
2. 
   $\frac{n\pi }{4}+{(-1)}^{n+1}\left(\frac{\pi }{8}\right),n\in Z$
3. 
   $\frac{n\pi }{8}+{(-1)}^{n+1}\left(\frac{\pi }{4}\right),n\in Z$
4. 
   $\frac{n\pi }{8}+{(-1)}^{n+1}\left(\frac{\pi }{8}\right),n\in Z$

Q. If $x^{3x}-2x^x\cot y-1=0,$ then the value of $y\text{'}\left(1\right)$ is

1. $-\ln 2$
2. $-\frac{3}{2}$
3. $1$
4. $\ln 2$

Q.

If p and q are the lengths of perpendiculars from the origin to the lines x cos θ – y sin θ = k cos 2θ and x sec θ+ y cosec θ = k, respectively, prove that p² + 4q² = k²

Q. The general solution of the equation ${\sin }^3\frac{x}{2}-{\cos }^3\frac{x}{2}=\frac{\cos x\times (2+\mathrm{sinx})}{3}$ is

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

Q. Answer Q.15 and Q.16 by appropriately matching the lists based on the information given in the paragraph.

Let $f\left(x\right)=\sin \left(\pi \cos x\right)$ and $g\left(x\right)=\cos \left(2\pi \sin x\right)$ be two functions defined for $x>0.$ Define the following sets whose elements are written in the increasing order:
$X=\{x:f(x)=0\},Y=\{x:f^{\prime }(x)=0\},$
$Z=\{x:g(x)=0\},W=\{x:g^{\prime }(x)=0\},$.
$List-I$ contains the sets $X,Y,Z$ and $W$. $List-II$ contains some information regarding these sets.

$\begin{array}{cccc}& ListI& & ListII\\ \left(I\right)& X& \left(P\right)& \supseteq \{\frac{\pi }{2},\frac{3\pi }{2},4\pi ,7\pi \}\\ \left(II\right)& Y& \left(Q\right)& \text{an arithmetic progression}\\ \left(III\right)& Z& \left(R\right)& \text{NOT an arithmetic progression}\\ \left(IV\right)& Z& \left(S\right)& \supseteq \{\frac{\pi }{6},\frac{7\pi }{6},\frac{13\pi }{6}\}\\ & & \left(T\right)& \supseteq \{\frac{\pi }{3},\frac{2\pi }{3},\pi \}\\ & & \left(U\right)& \supseteq \{\frac{\pi }{6},\frac{3\pi }{4}\}\end{array}$
Q.15 which of the following is the only CORRECT combination?

1. $\left(I\right),\left(P\right),\left(R\right)$
2. $\left(II\right),\left(Q\right),\left(T\right)$
3. $\left(I\right),\left(Q\right),\left(U\right)$
4. $\left(II\right),\left(R\right),\left(S\right)$

Q. The smallest positive value of $\theta$' satisfying the equation $\sqrt{3}(cot\theta +tan\theta )=4$ is

1. 
2. 
3. 
4. 

Q. Let A be the set containing value(s) of $x$ which satisfy the equation $3{\sin }^{2}x-\cos x=\sin x(3\cos x-1)$, where $x\in [0,9\pi ]$, then the correct statement(s) is/are -

1. Number of elements in set A is $17$
2. Number of elements in set A is $18$
3. Sum of elements in set A is $\frac{297}{4}\pi$
4. Sum of elements in set A is $\frac{295}{4}\pi$

Q. Number of solutions of the equation $tanx+secx=2cosx$ lying in the interval $[0,2\pi ]$ is

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

Q. If $\sin 2\theta \cos 2\theta =-\frac{1}{4},$ then the values of $\theta$ which satisfy the equation is

1. $\theta =\frac{n\pi }{8}+(-1{)}^n(-\frac{\pi }{48}),n\in Z$
2. $\theta =\frac{n\pi }{2}+(-1{)}^n(-\frac{\pi }{24}),n\in Z$
3. $\theta =\frac{n\pi }{4}+(-1{)}^n(-\frac{\pi }{12}),n\in Z$
4. $\theta =\frac{n\pi }{4}+(-1{)}^n(-\frac{\pi }{24}),n\in Z$

Q. Find the no. of solutions of the equation $2^x+3^x+4^x-5^x=0$

Q. Solve: $\frac{\sec 8A-1}{\sec 4A-1}-\frac{\tan 8A}{\tan 4A}$

Q. Complete the set of values of $x$ in $(0,\pi )$ satisfying the inequation $1+{\log }_2\sin x+{\log }_2\sin 3x\ge 0$ is

1. $[0,\frac{\pi }{6}]$
2. $[\frac{\pi }{2},\frac{3\pi }{4}]$
3. $[\frac{\pi }{6},\frac{\pi }{4}]\cup [\frac{3\pi }{4},\frac{5\pi }{6}]$
4. $[\frac{5\pi }{6},\pi ]$

Q. For $0\le x\le 2\pi$, then $2^{{\text{cosec}}^{2}x}\sqrt{\frac{1}{2}{y}^2-y+1}\le \sqrt{2}$

1. is satisfied by exactly one value of $y$
2. is satisfied by exactly two values of $x$
3. is satisfied by $x$ for which $\cos x=\frac{1}{2}$
4. is satisfied by $x$ for which $\sin x=0$

Q. $\mathrm{The}\mathrm{general}\mathrm{solution}\mathrm{of}\mathrm{the}\mathrm{equation}2{\cos }^2\theta +3\sin \theta =0\mathrm{is}$
.

1. $n\pi +{(-1)}^n\frac{\pi }{6},n\in Z$
2. $n\pi +{(-1)}^{n+1}\frac{\pi }{6},n\in Z$
3. $n\pi +{(-1)}^n\frac{5\pi }{6},n\in Z$