JEE Main 2010 — Algebra

A person is to count 4500 currency notes. Let $a_n$ denote the number of notes he counts in the $\mathrm{n}^{\text {th }}$ minute. If $\mathrm{a}_1=\mathrm{a}_2=\ldots \ldots=\mathrm{a}_{10}=150$ and $\mathrm{a}_{10}, \mathrm{a}_{11}, \ldots \ldots$ are in A.P. with common difference $-2$, then the time taken by him to count all notes is

Consider the system of linear equations: $$ \begin{aligned} & x_1+2 x_2+x_3=3 \\ & 2 x_1+3 x_2+x_3=3 \\ & 3 x_1+5 x_2+2 x_3=1 \end{aligned} $$ The system has

Let $S$ be a non-empty subset of R. Consider the following statement: $\mathrm{P}$ : There is a rational number $\mathrm{x} \in \mathrm{S}$ such that $\mathrm{x}>0$. Which of the following statements is the negation of the statement $P$ ?

Let $A$ be a $2 \times 2$ matrix with non-zero entries and let $A^2=1$, where 1 is $2 \times 2$ identity matrix. Define $\operatorname{Tr}(\mathrm{A})=$ sum of diagonal elements of $A$ and $|A|=$ determinant of matrix $A$. Statement-1: $\operatorname{Tr}(\mathrm{A})=0$ Statement-2: $|\mathrm{A}|=1$

Consider the following relations: $R=\{(x, y) \mid x, y$ are real numbers and $x=$ wy for some rational number w $\}$; $S=\left\{\left(\frac{m}{n}, \frac{p}{q}\right) \mid m, n, p\right.$ and $q$ are integers such that $n, q \neq 0$ and $\left.q m=p n\right\}$. Then

The number of $3 \times 3$ non-singular matrices, with four entries as 1 and all other entries as 0 , is

There are two urns. Urn A has 3 distinct red balls and urn B has 9 distinct blue balls. From each urn two balls are taken out at random and then transferred to the other. The number of ways in which this can be done is

The number of complex numbers $z$ such that $|z-1|=|z+1|=|z-i|$ equals

If $\alpha$ and $\beta$ are the roots of the equation $x^2-x+1=0$, then $\alpha^{2009}+\beta^{2009}=$