Domain of a Function

Q. The range of ${\sin }^{-1}\left(2^x\right)$ is

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

Q. The value of ${\sin }^{-1}\left(2\right)$ can be obtained.

1. True
2. False

Q. Inverse of All trigonometric functions exist over their respective domains.

1. False
2. True

Q.

Q. $(\sum _{n=1}^{10}{\int }_{-2n-1}^{-2n}{\sin }^{27}xdx)+(\sum _{n=1}^{10}{\int }_{2n}^{2n+1}si{n}^{27}$ $xdx)=$

1. ${27}^2$
2. -54
3. 54
4. 0

Q. ${\int }_0^1{\sin }^{-1}x$ $dx$

Q. If $\sum _{i-1}^{2n}{\sin }^{-1}{x}_i=n\pi$, then show that $\sum _{i=1}^{2n}{x}_i=2n$.

Q. The range of ${\sin }^{-1}\left(2^x\right)$ is

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

Q. The range of ${\sin }^{-1}\left(2^x\right)$ is

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

Q. The value of ${\sin }^{-1}\left(2\right)$ can be obtained.

1. True
2. False

Q. If $y=e^x\sin x$ and $z=e^x\cos x$ then

1. $y^{\prime \prime }=2z$
2. $y^{\prime }=z$
3. $z^{\prime \prime }=-2y$
4. $z^{\prime }=-y$

Q. Inverse of All trigonometric functions exist over their respective domains.

1. False
2. True

Q. The range of the function ${\sin }^{-1}\sqrt{x}$ is

1. $(0,\frac{\pi }{2})$
2. $[0,\frac{\pi }{2}]$
3. $[0,\frac{\pi }{2})$
4. $(0,\frac{\pi }{2}]$

Q. Inverse of All trigonometric functions exist over their respective domains.

1. False
2. True

Q. The value of the $\sin 1^0+\sin 2^0+.....+\sin {359}^0$ is equal to

1. $0$
2. $1$
3. $-1$
4. $180$

Q. The value of ${\sin }^{-1}\left(2\right)$ can be obtained.

1. True
2. False

Q. ${sin}^2{5}^{\circ }+{sin}^2{10}^{\circ }+.....+{sin}^2{180}^{\circ }=$

Q. Evaluate
$\frac{{\sin }^2{63}^{\circ }+{\sin }^2{27}^{\circ }}{{\cos }^2{17}^{\circ }+{\cos }^2{73}^{\circ }}$

1. 1
2. 0
   
3. $\sqrt{2}$
4. $\sqrt{3}$

Q. What is $\int {\sin }^{2}xdx+\int {\cos }^{2}xdx$ equal to?

1. $x+c$
2. $\frac{x^2}{2}+c$
3. None of the above
   where $c$ is an arbitrary constant
4. $x^2+c$

Q. If$\int \frac{sin2x}{a^{2}co{s}^{2}x+b^2{\sin }^{2}x}dx=$$k.\log |a^2{\cos }^{2}x+b^2{\sin }^{2}x|+c$,

1. $\frac{1}{b^2-a^2}$
2. $\frac{1}{(b^2-a^2{)}^2}$
3. $\frac{1}{a^2-b^2}$
4. $\frac{1}{a^2+b^2}$

Q. Value of $\frac{{\cos }^2{20}^o+{\cos }^2{70}^o}{{\sin }^2{59}^o+{\sin }^2{31}^o}$

1. $0$
2. $1$
3. $\frac{1}{2}$
4. $-1$

Q. I.F of $(1+x^3)\frac{dy}{dx}+3x^{2}y=si{n}^{2}x$:

1. $1+x^3$
2. $3x^2$
3. $\frac{1}{1+x^3}$
4. $\frac{1}{x^3}$

Q. Prove that: ${\sin }^2{5}^o+{\sin }^2{10}^o+{\sin }^2{15}^o....+{\sin }^2{85}^o+{\sin }^2{90}^o=9\frac{1}{2}$

Q. The range of $\Psi \left(x\right)$ is given by

1. $[e^{-1/2},e^{1/2}]$
2. $[2^{-1/2},2^{1/2}]$
3. $[-1,1]$
4. $[-2,2]$

Q. Evaluate: ${\int }_0^{\pi }{\sin }^{6}xdx$

1. $\frac{5\pi }{16}$
2. $\frac{35\pi }{128}$
3. $\frac{5\pi }{8}$
4. $\frac{5\pi }{18}$