16. The minimum value of $\left| a + b w + c w ^ { 2 } \right|$, where $a , b$ and $c$ are all not equal integers and $w \left( \begin{array} { l l } 1 & 1 \end{array} \right)$ is a cube root of unity, is: (a) $\sqrt { } 3$ (b) $1 / 3$ (c) 1 (d) 0
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If $P = \left[ \begin{array} { c c } \sqrt { 3 } / 2 & 1 / 2 \\ - 1 / 2 & \sqrt { 3 } / 2 \end{array} \right] , A = \left[ \begin{array} { l l } 1 & 1 \\ 0 & 1 \end{array} \right]$ and $\mathrm { Q } = \mathrm { PAP } ^ { \top }$, then $\mathrm { P } ^ { \top } \mathrm { Q } ^ { 2005 } \mathrm { P }$ is: (a) $\left[ \begin{array} { c c } 1 & 2005 \\ 0 & 1 \end{array} \right]$ (b) $\left[ \begin{array} { c c } 1 & 2005 \\ 2005 & 1 \end{array} \right]$ (c) $\left[ \begin{array} { c c } 1 & 0 \\ 2005 & 1 \end{array} \right] ^ { 1 }$ (d) $\left[ \begin{array} { l l } 1 & 0 \\ 0 & 1 \end{array} \right]$
The shaded region, where $P \equiv ( - 1,0 ) , Q \equiv ( - 1 + \sqrt { } 2 , \sqrt { } 2 )$ $R \equiv ( - 1 + \sqrt { } 2 , - \sqrt { } 2 ) , S \equiv ( 1,0 )$ is represented by: (a) $| z + 1 | > 2 , | \arg ( z + 1 ) | < n / 4$ (b) $| z + 1 | < 2 , | \arg ( z + 1 ) | < n / 2$ (c) $| z - 1 | > 2 , | \arg ( z + 1 ) | > \pi / 4$ (d) $| z - 1 | < 2 , | \arg ( z + 1 ) | > n / 2$
The number of ordered pairs ( $a , \beta$ ), where $a , \beta \hat { I } ( - \Pi , \Pi )$ satisfying $\cos ( a - \beta ) = 1$ and $\cos ( a + \beta ) = 1 / e$ is : (a) 0 (b) 1 (c) 2 (d) 4
Let $f ( x ) = | x | - 1$, then points where $f ( x )$ is not differentiable is/(are): (a) $0 , + 1$ (b) + 1 (c) 0 (d) 1
The second degree polynomial $f ( x )$, satisfying $f ( 0 ) = 0 , f ( 1 ) = 1 , f ^ { \prime } ( x ) > 0$ for all $x \hat { I }$ ( 0 , 1) : (a) $f ( x ) = f$ (b) $f ( x ) = a x + ( 1 - a ) x ^ { 2 } ; \forall a \hat { I } ( 0 , ¥ )$ (c) $f ( x ) = a x + ( 1 - a ) x ^ { 2 } ; \forall a \hat { I } ( 0,2 )$ (d) no such polynomial
If $f$ is a differentiable function satisfying $f ( 1 / n ) = 0$ for all $n > 1 , n \hat { I } I$, then : (a) $f ( x ) = 0 , x \hat { I } ( 0,1 ]$ (b) $f ^ { \prime } ( 0 ) = 0 = f ( 0 )$
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... Powered By IITians (c) $f ( 0 ) = 0$ but $f ^ { \prime } ( 0 )$ not necessarily zero (d) $| f ( x ) | < 1 , x \hat { I } ( 0,1 ]$
A tangent at a point on the ellipse $\frac { x ^ { 2 } } { a ^ { 2 } } + \frac { y ^ { 2 } } { b ^ { 2 } } = 1$ cuts the coordinate axes at $P$ and $Q$. The minimum area of the triangle $O P Q$ is
16. The minimum value of $\left| a + b w + c w ^ { 2 } \right|$, where $a , b$ and $c$ are all not equal integers and $w \left( \begin{array} { l l } 1 & 1 \end{array} \right)$ is a cube root of unity, is:\\
(a) $\sqrt { } 3$\\
(b) $1 / 3$\\
(c) 1\\
(d) 0
\section*{III askllTians ||}
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\begin{enumerate}
\setcounter{enumi}{16}
\item If $P = \left[ \begin{array} { c c } \sqrt { 3 } / 2 & 1 / 2 \\ - 1 / 2 & \sqrt { 3 } / 2 \end{array} \right] , A = \left[ \begin{array} { l l } 1 & 1 \\ 0 & 1 \end{array} \right]$ and $\mathrm { Q } = \mathrm { PAP } ^ { \top }$, then $\mathrm { P } ^ { \top } \mathrm { Q } ^ { 2005 } \mathrm { P }$ is:\\
(a) $\left[ \begin{array} { c c } 1 & 2005 \\ 0 & 1 \end{array} \right]$\\
(b) $\left[ \begin{array} { c c } 1 & 2005 \\ 2005 & 1 \end{array} \right]$\\
(c) $\left[ \begin{array} { c c } 1 & 0 \\ 2005 & 1 \end{array} \right] ^ { 1 }$\\
(d) $\left[ \begin{array} { l l } 1 & 0 \\ 0 & 1 \end{array} \right]$
\item The shaded region, where\\
$P \equiv ( - 1,0 ) , Q \equiv ( - 1 + \sqrt { } 2 , \sqrt { } 2 )$\\
$R \equiv ( - 1 + \sqrt { } 2 , - \sqrt { } 2 ) , S \equiv ( 1,0 )$ is represented by:\\
(a) $| z + 1 | > 2 , | \arg ( z + 1 ) | < n / 4$\\
(b) $| z + 1 | < 2 , | \arg ( z + 1 ) | < n / 2$\\
(c) $| z - 1 | > 2 , | \arg ( z + 1 ) | > \pi / 4$\\
(d) $| z - 1 | < 2 , | \arg ( z + 1 ) | > n / 2$
\item The number of ordered pairs ( $a , \beta$ ), where $a , \beta \hat { I } ( - \Pi , \Pi )$ satisfying $\cos ( a - \beta ) = 1$ and $\cos ( a + \beta ) = 1 / e$ is :\\
(a) 0\\
(b) 1\\
(c) 2\\
(d) 4
\item Let $f ( x ) = | x | - 1$, then points where $f ( x )$ is not differentiable is/(are):\\
(a) $0 , + 1$\\
(b) + 1\\
(c) 0\\
(d) 1
\item The second degree polynomial $f ( x )$, satisfying $f ( 0 ) = 0 , f ( 1 ) = 1 , f ^ { \prime } ( x ) > 0$ for all $x \hat { I }$ ( 0 , 1) :\\
(a) $f ( x ) = f$\\
(b) $f ( x ) = a x + ( 1 - a ) x ^ { 2 } ; \forall a \hat { I } ( 0 , ¥ )$\\
(c) $f ( x ) = a x + ( 1 - a ) x ^ { 2 } ; \forall a \hat { I } ( 0,2 )$\\
(d) no such polynomial
\item If $f$ is a differentiable function satisfying $f ( 1 / n ) = 0$ for all $n > 1 , n \hat { I } I$, then :\\
(a) $f ( x ) = 0 , x \hat { I } ( 0,1 ]$\\
(b) $f ^ { \prime } ( 0 ) = 0 = f ( 0 )$
\end{enumerate}
\section*{III askllTians ||}
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(c) $f ( 0 ) = 0$ but $f ^ { \prime } ( 0 )$ not necessarily zero\\
(d) $| f ( x ) | < 1 , x \hat { I } ( 0,1 ]$\\