JEE Main Mathematics — Algebra previous year questions with solutions.
Let $\alpha, \beta$ be real and $z$ be a complex number. If $z^2+\alpha z+\beta=0$ has two distinct roots on the line $\operatorname{Re} z=1$, then it is necessary that
This question has Statement $-1$ and Statement $-2$. Of the four choices given after the statements, choose the one that best describes the two statements. Statement - 1 : The number of ways of distributing 10 identical balls in 4 distinct boxes such that no box is empty is ${ }^9 \mathrm{C}_3$ Statement-2: The number of ways of choosing any 3 places from 9 different places is ${ }^9 \mathrm{C}_3$.
If $\omega(\neq 1)$ is a cube root of unity, and $(1+\omega)^7=A+B \omega$. Then $(A, B)$ equals
The coefficient of $x^7$ in the expansion of $\left(1-x-x^2+x^3\right)^6$ is
If $\alpha$ and $\beta$ are the roots of the equation $x^2-x+1=0$, then $\alpha^{2009}+\beta^{2009}=$
The number of complex numbers $z$ such that $|z-1|=|z+1|=|z-i|$ equals
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
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
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
Let $f(x)=(x+1)^2-1, x \geq-1$ Statement-1: The set $\left\{x: f(x)=f^{-1}(x)\right\}=\{0,-1\}$ Statement-2 : $\mathrm{f}$ is a bijection.
If $\left|z-\frac{4}{z}\right|=2$, then the maximum value of $|z|$ is equal to
Let A be a $2 \times 2$ matrix Statement-1 $: \operatorname{adj}(\operatorname{adj} A)=A$ Statement-2 : $|\operatorname{adj} \mathrm{A}|=|\mathrm{A}|$
If the roots of the equation $b x^2+c x+a=0$ be imaginary, then for all real values of $x$, the expression $3 b^2 x^2+6 b c x+2 c^2$ is
Let $a, b, c$ be such that $b(a+c) \neq 0$. If $\left|\begin{array}{ccc}a & a+1 & a-1 \\ -b & b+1 & b-1 \\ c & c-1 & c+1\end{array}\right|+\left|\begin{array}{ccc}a+1 & b+1 & c-1 \\ a-1 & b-1 & c+1 \\ (-1)^{n+2} a & (-1)^{n+1} b & (-1)^n c\end{array}\right|=0$, then the value of ' $n$ ' is
Let $A$ and $B$ denote the statements A: $\cos \alpha+\cos \beta+\cos \gamma=0$ B: $\sin \alpha+\sin \beta+\sin \gamma=0$ If $\cos (\beta-\gamma)+\cos (\gamma-\alpha)+\cos (\alpha-\beta)=-\frac{3}{2}$, then
For real $x$, let $f(x)=x^3+5 x+1$, then
If $A, B$ and $C$ are three sets such that $A \cap B=A \cap C$ and $A \cup B=A \cup C$, then
The remainder left out when $8^{2 n}-(62)^{2 n+1}$ is divided by 9 is
From 6 different novels and 3 different dictionaries, 4 novels and 1 dictionary are to be selected and arranged in a row on the shelf so that the dictionary is always in the middle. Then the number of such arrangements is
Let $\mathrm{A}$ be a $2 \times 2$ matrix with real entries. Let I be the $2 \times 2$ identity matrix. Denote by $\operatorname{tr}(\mathrm{A})$, the sum of diagonal entries of $A$. Assume that $A^2=1$. Statement -1: If $A \neq 1$ and $A \neq-1$, then $\operatorname{det} A=-1$. Statement $-2$ : If $A \neq 1$ and $A \neq-1$, then $\operatorname{tr}(A) \neq 0$.