Chapter 23

Let $$\begin{gathered} S= \\ 1,2,3,\dots ,9 \end{gathered}$$. For $k=1,2,\dots ,5$, let $N_k$ be the number of subsets of $S$, each containing five elements out of which exactly $k$ are odd. Then $N_1+N_2+N_3+N_4+N_5=$ [A] 210 [B] 252 [C] 125 [D] 126

Three randomly chosen nonnegative integers $x,y$ and $z$ are found to satisfy the equation $x+y+z=10$. Then the probability that $z$ is even, is [A] $\frac{36}{55}$ [B] $\frac{6}{11}$ $[\mathrm{C}]\frac{1}{2}$ [D] $\frac{5}{11}$

If $g(x)={\int }_{\sin x}^{\sin (2x)}{\sin }^{-1}(t)dt$, then [A] $g^{\prime }\left(\frac{\pi }{2}\right)=-2\pi$ [B] $g^{\prime }(-\frac{\pi }{2})=2\pi$ [C] $g^{\prime }\left(\frac{\pi }{2}\right)=2\pi$ [D] $g^{\prime }(-\frac{\pi }{2})=-2\pi$

If $f:\mathbb{R}\to \mathbb{R}$ is a differentiable function such that $f^{\prime }(x)>2f(x)$ for all $x\in \mathbb{R}$, and $f(0)=1$, then [A] $f(x)$ is increasing in $(0,\mathrm{\infty })$ [B] $f(x)$ is decreasing in $(0,\mathrm{\infty })$ [C] $f(x)>e^{2x}$ in $(0,\mathrm{\infty })$ [D] \(f^{\prime}(x)

Let $f(x)=\frac{1-x(1+|1-x|)}{|1-x|}\cos \left(\frac{1}{1-x}\right)$ for $x\ne 1$. Then [A] $\underset{x\to 1^{-}}{lim}f(x)=0$ [B] $\underset{x\to 1^{-}}{lim}f(x)$ does not exist [C] $\underset{x\to 1^{+}}{lim}f(x)=0$ [D] $\underset{x\to 1^{+}}{lim}f(x)$ does not exist

If $I=\sum _{k=1}^{98}{\int }_k^{k+1}\frac{k+1}{x(x+1)}dx$, then [A] $I>{\log }_{e}99$ [B] $I<{\log }_{e}99$ [C] $I<\frac{49}{50}$ [D] $I>\frac{49}{50}$