Chapter 5

If $\text{Missing left or extra right}$ then (1) $A={\tan }^{-1}\frac{2}{3}$ (2) $\mathrm{A}={\tan }^{-1}\frac{2}{\sqrt{13}}$ (3) $A={\mathrm{cosec}}^{-1}\frac{3}{2}$ (4) $\mathrm{A}={\cot }^{-1}\frac{2}{3}$

If $\underset{h\to 0}{lim}\left[\frac{f(a+h)}{f(a)}\right]=c$ where $f(x)$ is a continuous function such that $f(x)>0$ for all $x\in R$ and $[.]$ denotes greatest integer function then which of the following statements is always true? (1) If $x=a$ is a point of local minima then $c\in N$ (2) If $x=a$ is a point of local maxima then $c\in N$ (3) If $x=a$ is a point of local minima then $c\in I^{-}$ (4) If $x=a$ is a point of local maxima then $c\in I^{-}$

If $\underset{\theta \to 0}{lim}(\frac{1+a\cos \theta }{{\theta }^2}-\frac{b\sin \theta }{{\theta }^3})=1$, then the value of $\mathrm{a}+\mathrm{b}$ is $-\lambda$, find the value of $\lambda$. (1) 4 (2) 1 (3) 2 (4) 0

The maximum values of $\frac{{\log }_{e}x}{x}$ is (1) 1 (2) $\frac{2}{e}$ (3) $\mathrm{e}$ (4) $\frac{1}{e}$

A curve passes through the point $(2,0)$ and the slope of the tangent at any point $(x,y)$ is $x^2-2x$ for all values of $x$. The point of local maximum on the curve is (1) $(0,\frac{4}{3})$ (2) $(0,\frac{2}{3})$ (3) $(1,\frac{2}{3})$ (4) $(2,\frac{4}{3})$

If in a $\mathrm{△}\mathrm{ABC},{\mathrm{acos}}^2\left(\frac{\mathrm{C}}{2}\right)+\mathrm{c}{\cos }^2\left(\frac{\mathrm{A}}{2}\right)=\frac{3\mathrm{b}}{2}$, then the sides $a,c,b$ (1) are in A.P. (2) are in G.P. (3) are in H.P. (4) satisfy $a+c=2b$

If $f(\theta )=(\sin \theta +\mathrm{cosec}\theta {)}^2+(\cos \theta +\sec \theta {)}^2$, then minimum value of $f(\theta )$ is (1) 7 (2) 8 (3) 9 (4) 4

The set $(A\cup B\cup C)\cap {(A\cap B^{\prime }\cap C^{\prime })}^{\prime }\cap C^{\prime }$ is equal to (1) $\mathrm{B}\cap {\mathrm{C}}^{\prime }$ (2) $\mathrm{A}\cap \mathrm{C}$ (3) ${\mathrm{B}}^{\prime }\cap {\mathrm{C}}^{\prime }$ (4) $\mathrm{A}\cap {(\mathrm{B}\cup {\mathrm{C}}^{\prime })}^{\prime }$

If $x^{3/4{({\log }_{3}x)}^2+{\log }_{3}x-5/4}=\sqrt{3}$ then $x$ can not assume (1) one positive integral value (2) one irrational value (3) two positive rational values (4) two distinct positive integral values

$\underset{x\to \mathrm{\infty }}{lim}\frac{2\cdot x^{1/2}+3\cdot x^{1/3}+4\cdot x^{1/4}+\dots ..+n\cdot x^{1/n}}{(3x-4{)}^{1/2}+(3x-4{)}^{1/3}+\dots +(3x-4{)}^{1/n}}($ here $n\in N,n\ge 2)$ is equal to (1) $\frac{2}{\sqrt{3}}$ (2) $\frac{\sqrt{3}}{2}$ (3) $\frac{1}{2}$ (4) $\frac{1}{\sqrt{3}}$