Function Defintion

Q.

What is $\sin \left(x\right)+\cos \left(x\right)$in terms of $\mathrm{sine}$?

Q.

The value of $\int {\mathrm{e}}^{2\mathrm{x}}(2\sin 3\mathrm{x}+3\cos 3\mathrm{x})\mathrm{dx}$ is

Q. If $0<a<b<1,$ then which of the following is true

1. $\frac{1}{\sqrt{1-a^2}}<\frac{{\sin }^{-1}b-{\sin }^{-1}a}{b-a}<\frac{1}{\sqrt{1-b^2}}$
2. $\frac{1}{\sqrt{1-b^2}}<\frac{{\sin }^{-1}b-{\sin }^{-1}a}{b-a}<\frac{1}{\sqrt{1-a^2}}$
3. $\frac{{\sin }^{-1}b-{\sin }^{-1}a}{b-a}=\frac{1}{\sqrt{1-a^2}}$
4. $\frac{{\sin }^{-1}b-{\sin }^{-1}a}{b-a}=\frac{1}{\sqrt{1-b^2}}$

Q. The least integral value of $k$ for $f\left(x\right)=\frac{e^{-x}}{\sqrt{k+8x^2}}$ is decreasing for all $x\in R$ is

Q. If $f\left(x\right)=\frac{x^2-8}{x-3}$ is an increasing function, then

1. $x\in (-\infty ,2]\cup [4,\infty )$
2. $x\in (2,4)$
3. $x\in (-\infty ,3)$
4. $x\in (2,\infty )$

Q. The number of distinct real roots of $x^4-4x+1=0$ is:

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

Q. The value of $\underset{t\to 0}{lim}(\frac{-1}{t^2}\cdot \underset{2t}{\overset{t^2}{\int }}\sin tdt)$ equals to

Q. If $f\text{'}\text{'}\left(x\right)>0,\forall x\in R,f\text{'}\left(3\right)=0$ and $g\left(x\right)=f({\tan }^{2}x-2\tan x+4),0<x<\frac{\pi }{2}$, then $g\left(x\right)$ is increasing in

1. $(0,\frac{\pi }{4})$
2. $(\frac{\pi }{6},\frac{\pi }{3})$
3. $(0,\frac{\pi }{3})$
4. $(\frac{\pi }{4},\frac{\pi }{2})$

Q. If function $f\left(x\right)=\lambda \sin x+\cos x$ has two extremum points in $[0,2\pi ],$ then the value of $\lambda$ can be

1. $-\frac{3}{13}$
2. $0$
3. $\frac{2}{7}$
4. $1$

Q. Which of the following is (are) correct about the function $y=x^4-\frac{4x^3}{3}$ ?

1. f is decreasing in (– ∞, 1]
2. f is strictly increasing in (– ∞, 1]
3. f is increasing in [1, ∞).
4. f is decreasing in (1, ∞).