1
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
$\operatorname{Arg}\left(z-z_1\right)=\operatorname{Arg}\left(z_2-z_1\right)$
2
IIT-JEE 2010 Paper 1 Offline
MCQ (Single Correct Answer)
+3
-1

Let $p$ be an odd prime number and $T_p$ be the following set of $2 \times 2$ matrices :

$$ \mathrm{T}_{\mathrm{p}}=\left\{\mathrm{A}=\left[\begin{array}{ll} a & b \\ c & a \end{array}\right]: a, b, c \in\{0,1,2, \ldots, p-1\}\right\} $$

The number of $A$ in $T_p$ such that $A$ is either symmetric or skew-symmetric or both, and $\operatorname{det}(\mathrm{A}) \operatorname{divisible}$ by $p$ is :
A
$(p-1)^2$
B
$2(p-1)$
C
$(p-1)^2+1$
D
$2 p-1$
3
IIT-JEE 2010 Paper 1 Offline
MCQ (Single Correct Answer)
+3
-1

Let $p$ be an odd prime number and $T_p$ be the following set of $2 \times 2$ matrices :

$$ \mathrm{T}_{\mathrm{p}}=\left\{\mathrm{A}=\left[\begin{array}{ll} a & b \\ c & a \end{array}\right]: a, b, c \in\{0,1,2, \ldots, p-1\}\right\} $$

The number of A in $\mathrm{T}_p$ such that the trace of A is not divisible by $p$ but $\operatorname{det}(\mathrm{A})$ is divisible by $p$ is

[Note : The trace of a matrix is the sum of its diagonal entries.]

A
$(p-1)\left(p^2-p+1\right)$
B
$p^3-(p-1)^2$
C
$(p-1)^2$
D
$(p-1)\left(p^2-2\right)$
4
IIT-JEE 2010 Paper 1 Offline
MCQ (Single Correct Answer)
+3
-1

Let $p$ be an odd prime number and $T_p$ be the following set of $2 \times 2$ matrices :

$$ \mathrm{T}_{\mathrm{p}}=\left\{\mathrm{A}=\left[\begin{array}{ll} a & b \\ c & a \end{array}\right]: a, b, c \in\{0,1,2, \ldots, p-1\}\right\} $$

The number of A in $\mathrm{T}_p$ such that $\operatorname{det}(\mathrm{A})$ is not divisible by $p$ is :
A
$2 p^2$
B
$p^3-5 p$
C
$p^3-3 p$
D
$p^3-p^2$
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