Mathematics Algebra questions from JEE Main 2010.
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 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
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
If $\alpha$ and $\beta$ are the roots of the equation $x^2-x+1=0$, then $\alpha^{2009}+\beta^{2009}=$
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: $|\mathrm{A}|=1$
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 $P$ ?
The number of complex numbers $z$ such that $|z-1|=|z+1|=|z-i|$ equals
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