1
IIT-JEE 2010 Paper 1 Offline
MCQ (Single Correct Answer)
+4
-1
Let $$ABC$$ be a triangle such that $$\angle ACB = {\pi \over 6}$$ and let $$a, b$$ and $$c$$ denote the lengths of the sides opposite to $$A$$, $$B$$ and $$C$$ respectively. The value(s) of $$x$$ for which $$a = {x^2} + x + 1,\,\,\,b = {x^2} - 1\,\,\,$$ and $$c = 2x + 1$$ is (are)
A
$$ - \left( {2 + \sqrt 3 } \right)$$
B
$${1 + \sqrt 3 }$$
C
$${2 + \sqrt 3 }$$
D
$${4 \sqrt 3 }$$
2
IIT-JEE 2010 Paper 1 Offline
MCQ (Single Correct Answer)
+3
-1

The number of $3 \times 3$ matrices A whose entries are either 0 or 1 and for which the system

$\mathrm{A}\left[\begin{array}{l}x \\ y \\ z\end{array}\right]=\left[\begin{array}{l}1 \\ 0 \\ 0\end{array}\right]$ has exactly two distinct solutions, is

A
0
B
$2^9-1$
C
168
D
2
3
IIT-JEE 2010 Paper 1 Offline
MCQ (Single Correct Answer)
+3
-1

Let $f, g$ and $h$ be real valued functions defined on the interval $[0,1]$ by

$f(x)=e^{x^2}+e^{-x^2}$,

$g(x)=x e^{x^2}+e^{-x^2}$

and $h(x)=x^2 e^{x^2}+e^{-x^2}$.

If $a, b$ and $c$ denote, respectively, the absolute maximum of $f, g$ and $h$ on $[0,1]$, then :

A
$a=b$ and $c \neq b$
B
$a=c$ and $a \neq b$
C
$a \neq b$ and $c \neq b$
D
$a=b=c$
4
IIT-JEE 2010 Paper 1 Offline
MCQ (More than One Correct Answer)
+3
-0

Let $z_1$ and $z_2$ be two distinct complex numbers let $z=(1-t) z_1+t z_2$ for some real number t with $0 < t < 1$.

If $\operatorname{Arg}(w)$ denotes the principal argument of a nonzero complex number $w$, then :

A
$\left|z-z_1\right|+\left|z-z_2\right|=\left|z_1-z_2\right|$
B
$\operatorname{Arg}\left(z-z_1\right)=\operatorname{Arg}\left(z-z_2\right)$
C
$\left|\begin{array}{cc}z-z_1 & \bar{z}-\bar{z}_1 \\ z_2-z_1 & \bar{z}_2-\bar{z}_1\end{array}\right|=0$
D
$\operatorname{Arg}\left(z-z_1\right)=\operatorname{Arg}\left(z_2-z_1\right)$
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