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

Q. The general solution(s) of the equation $4{\cos }^{2}x+6{\sin }^{2}x=5$ is/are

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

Q. The number of solutions of the equation $\sqrt{1-\sin x}=\cos x$ in the interval $[0,3\pi ]$ is

1. $2$
2. $4$
3. $5$
4. $6$

Q.

Prove that the $\frac{d}{dx}\left(secx\right)=secx\tan x$

Q. The general solution of the equation ${\sin }^2\theta ={\sin }^2\alpha$ is/are

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

Q. The number of real solutions of the equation $(\sin x-x)(\cos x-x^2)=0$ is

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

Q. The sum of all the solutions of the equation $\cos 2x+{\sin }^{2}2x=1$ which lie in the interval$[0,2\pi ]$ is equal to

1. $4\pi$
2. $7\pi$
3. $8\pi$
4. $5\pi$

Q. The set of value(s) of $x$ for which $7{\cos }^{2}x+3{\sin }^{2}x=4$ is :

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

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. $\text{If}{\cos }^{5}x+{\cos }^5(x+\frac{2\pi }{3})+{\cos }^5(x+\frac{4\pi }{3})=0$,
then the number of solution(s) in $[0,2\pi ]$ is

Q.

Principal solution and general solution

Q. Let ${\sec }^2\left(\frac{\pi }{9}\right)+{\sec }^2\left(\frac{2\pi }{9}\right)+{\sec }^2\left(\frac{4\pi }{9}\right)=S$ and ${\sum }_{k=1}^{89}{\cos }^6\left(k^{\circ }\right)=\frac{a}{b}$, where $a,b$ are coprime, then which of the following is/are correct?

1. Number of positive divisors of $a+b+S$ is $4$.
2. Number of positive divisors of $a+b+S$ is $8$
3. $S$ is a perfect square
4. $a+b$ is a prime number.

Q. The number of values of $x$ in the interval $[0,5\pi ]$ satisfying the equation $3{\sin }^{2}x-7\sin x+2=0$ is

1. $10$
2. $5$
3. $6$
4. $0$

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. Find the value of x which satisfy equation $\cos \left(2si{n}^{-1}x\right)=\frac{1}{3}$.

1. $x=\frac{1}{\sqrt{5}}$ and $x=\frac{-1}{\sqrt{5}}$
2. $x=\frac{1}{\sqrt{3}}$ and $x=\frac{-1}{\sqrt{3}}$
3. $x=\frac{1}{\sqrt{2}}$ and $x=\frac{-1}{\sqrt{2}}$
4. None of these

Q. For the given graph of $y=\tan 2x;\forall x\in R$
Select the correct statements.

1. Period: $\pi /2$
2. Period: $\pi /4$
3. Domain: $R-(2n+1)\pi /4;n\in Z$
4. Domain: $R$

Q.

Prove the following:-

Cos4x=1-8sin^2xcos^2x

Q. General solution of $\tan 5x=\cot 2x$ is

1. $x=\frac{n\pi }{7}+\frac{\pi }{2},n\in Z$
2. $x=\frac{n\pi }{7}-\frac{\pi }{14},n\in Z$
3. $x=\frac{n\pi }{7}+\frac{\pi }{14},n\in Z$
4. $x=\frac{n\pi }{7}+\frac{\pi }{3},n\in Z$

Q. If $x$ satisfies the inequalities $x+7<2x+3$ and $2x+4<5x+3$, then $x$ lies in the interval.

1. $(-\infty ,-1)$
2. $(-\infty ,3)$
3. $(1,3)$
4. $(4,\infty )$
5. $(3,4)$

Q.

Statement (1): Maximum value of $si{n}^2$x + $co{s}^2$y is 2

Statement (2): Maximum value of 2 $se{c}^2$z is 2

1. Both the statements are wrong.

2. Only (1) is correct

3. Both the statements are correct

4. Only (2) is correct

Q.

The solutions of the equation $4co{s}^{2}x+6si{n}^{2}x=5$ are

1. x = nπ ± π /3, nϵZ

2. x = nπ ± π /2, nϵZ

3. x = nπ ± π / 4, nϵZ

4. x = nπ ± 2π /3, nϵZ

Q.

The number of solutions of the equation tanx + secx= 2cosx lying in the

Internal [0, 2$\pi$] is

1. 2

2. 0

3. 1

4. 3

Q.

$cos\frac{\pi }{7}$. $cos\frac{3\pi }{7}$. $cos\frac{5\pi }{7}$ are the roots of the equation $8x^3-4x^2-4x+1=0$. Then the value of $sec\frac{\pi }{7}+sec\frac{3\pi }{7}+sec\frac{5\pi }{7}$ is

1. None of these

2. 2

3. 4

4. 8

Q. Let $z,w$ be complex numbers such that $\overline{z}+i\overline{w}=0$ and arg $zw=\pi$. Then arg $z$ equals

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

Q. The most general solutions $2^{\sin x}+2^{\cos x}=2^{1+\frac{1}{\sqrt{2}}}$ are

1. 
2. 
3. 
4. 

Q. If $A+B+C=\pi$ and $\cos A+\cos B+\cos C$$=0=\sin A+\sin B+\sin C$ then $\cos 3A+\cos 3B+\cos 3C=1$

1. True
2. False

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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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.

Statement (1): Maximum value of $si{n}^2$x + $co{s}^2$y is 2

Statement (2): Maximum value of 2 $se{c}^2$z is 2

1. Both the statements are correct

2. Only (1) is correct

3. Only (2) is correct

4. Both the statements are wrong.

Q. The number of real solutions of the equation
$\sqrt{1+\cos 2x}=\sqrt{2}{\sin }^{-1}(\sin x),-\mathrm{\pi }\le x\mathit{\le }\mathrm{\pi }$ is
(a) 0
(b) 1
(c) 2
(d) infinite

Q. The number of solutions of the equation ${\int }_{-2}^x\left|\cos x\right|dx=0$ is?

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