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

Q. The product of the perpendiculars from the foci on any tangent to the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ is

1. 
2. 
3. 
4. 

Q.

The general solution to $si{n}^{10}x+co{s}^{10}x=\frac{29}{16}co{s}^{4}2x$ is

1. 

2. 

3. 

4. 

Q.

If sin 3x + cos 2x = -2, then find the value of x

1. (2m+1)

2. (4m-1)

3. (4m+1)

4. (2m+1)

Q.

Evaluate the following limit:
$\underset{x\to 0}{\text{lim}}\frac{\text{sin ax + bx}}{\text{ax+sin bx}},a,b,a+b\ne 0$

Q. 45. Find number of solutions of equation in[0, 2] tan(5cosα )=cot(5sinα )

Q. If log${}_{0.5}$ sinx = 1 - log${}_{0.5}$ cosx then number of values of $x\in [-2\pi ,2\pi ]$ is

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

Q.

Let $\alpha ,\beta$ be any two positive value of x for which 2cosx, |cosx| and 1 - 3$co{s}^{2}x$ are in G.P. The minimum value of $|\alpha -\beta |$ is

1. 

2. 

3. 

4. 

Q.

If $z_r$ = cos $\left(\frac{\pi }{2^r}\right)$ + i sin $\left(\frac{\pi }{2^r}\right)$ where i = $\sqrt{-1}$ . Find the value of $z_1$.$z_2$.$z_3$.$z_4\dots \dots \infty$

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Q. If $2{\sin }^{2}x=3\cos x$, where $0\le x\le 2\mathrm{\pi }$, then find the value of x.

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

1. 
2. 
3. 
4. 

Q.

Find the value of $xϵ(0,\pi )$ which satisfies the equation sin x +$\sqrt{3}$ cos x = $\sqrt{2}$

1. 

2. 

3. 

4. 

Q. Tan^-1(tan(-6))

Q. If α, β are two different values of x lying between 0 and 2π, which satisfy the equation 6 cos x + 8 sin x = 9, find the value of sin (α + β).

Q. The general solution of the equation $7{\cos }^{2}x+3{\sin }^{2}x=4$ is
(a) $x=2n\pi \pm \frac{\pi }{6},n\in Z$

(b) $x=2n\pi \pm \frac{2\pi }{3},n\in Z$

(c) ​$x=n\pi \pm \frac{\pi }{3},n\in Z$

(d) none of these

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

Q.

How many values of x in the interval [0, 2π] satisfies the equation $si{n}^6$ x = 1 + $co{s}^4$ 3x

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Q.

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

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

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Q.

Evaluate the following limit:
$\underset{x\to 0}{\text{lim}}\frac{\text{ax+x cos x}}{\text{b sin x}}$

Q. Find the general solution:
$sinx+sin3x+sin5x=0$

Q. In (0, π), the number of solutions of the equation ​$\tan x+\tan 2x+\tan 3x=\tan x\tan 2x\tan 3x$ is
(a) 7
(b) 5
(c) 4
(d) 2.

Q.

If $si{n}^2$z = 1 + $co{s}^2$ y, find the value of $co{s}^2$ z + $si{n}^2$ y

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Q.

If $si{n}^2$ x + $co{s}^2$y = 2 $se{c}^2$ z, find the value of $co{s}^2$ x + $si{n}^2$ y + 2 $si{n}^2$ z.

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Q.

Evaluate the following limit:
$\underset{x\to \pi }{\text{lim}}\frac{\text{sin}(\pi -x)}{\pi (\pi -x)}$

Q.

If $x\ne \frac{n\pi }{2}and(cosx{)}^{si{n}^{2}x-3sinx+2}=1$ then all solutions of x are given by

1. 

2. 

3. 

4. None of these

Q. Write the general solutions of tan² 2x = 1.

Q.

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

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 set of all x in the interval $[0,\pi ]$ for which ${\sin }^{2}x-3\sin x+1\ge 0$ is

1. $\pi /2$
2. $\varphi$
3. $[0,\pi /4]$
4. none of these

Q.

Evaluate the following limit:
$\underset{x\to 0}{\text{lim}}\frac{\text{cos 2x-1}}{\text{cos x-1}}$

Q. Find no. Of solutions of sinx=x/10