Mathematics Mock Test - 11

Let $y=f\left(\frac{x+e^x}{e^x}\right)$ satisfy $f^{\prime }(1)=2$. Then the value of $\frac{dy}{dx}$ at $x=0$ equals:

Match List I with List II:List I (Integrals)List II (Values)A. ${\int }_0^{\pi /2}\frac{\sin 2x}{{\sin }^{4}x+{\cos }^{4}x}dx$ I. $\pi /4$ B. ${\int }_0^{\pi /2}\frac{\sin x}{\sin x+\cos x}dx$ II. $\pi$ C. ${\int }_0^{2\pi }\frac{1}{1+e^{\sin x}}dx$ III. $\pi /12$ D. ${\int }_0^{\pi /2}\frac{1}{9{\sin }^{2}x+4{\cos }^{2}x}dx$ IV. $\pi /2$ Choose the correct answer from the options given below:

If $x=a(\cos \theta +\theta \sin \theta )$ and $y=a(\sin \theta -\theta \cos \theta )$, then at $\theta =\frac{\pi }{4}$, which of the following is true?

If $f(x+y)=f(x)\cdot f(y)$ for all $x,y\in \mathbb{R}$ such that $f(5)=2$ and $f^{\prime }(0)=3$, then the value of $f^{\prime }(5)$ is:

If $A=\{1,2,3\}$ and $R$ is a relation defined on $A$ such that $R=\{(1,1),(1,2),(2,3),(2,2),(3,3),(3,1)\}$. Then the relation $R$ is: