JEE Main Mathematics — Algebra previous year questions with solutions.
Let ${A}_{1},{A}_{2},{A}_{3},\ldots ..$ be squares such that for each $n\geqslant 1,$ the length of the side of ${A}_{n}$ equals the length of diagonal of ${A}_{n+1}$. If the length of ${A}_{1}$ is $12\mathrm{cm}$, then the smallest value of $n$ for which area of ${A}_{n}$ is less than one, is
A man is walking on a straight line. The arithmetic mean of the reciprocals of the intercepts of this line on the coordinate axes is $\frac{1}{4}$. Three stones $A,B$ and $C$ are placed at the points $(1,1),(2,2)$ and $(4,4)$ respectively. Then which of these stones is$/$are on the path of the man?
If the sum of the coefficients in the expansion of $(x+y{)}^{n}$ is $4096,$ then the greatest coefficient in the expansion is _____.
If $z$ and $\omega$ are two complex numbers such that $|z\omega |=1$ and $\mathrm{arg}(z)-\mathrm{arg}(\omega )=\frac{3\pi }{2}$, then $\mathrm{arg}(\frac{1-2\bar{z}\omega }{1+3\bar{z}\omega })$ is: (Here $\mathrm{arg}(z)$ denotes the principal argument of complex number $z$)
Let $A=[\begin{matrix}i & -i \\ -i & i\end{matrix}],i=\sqrt{-1}$. Then, the system of linear equations ${A}^{8}[\begin{matrix}x \\ y\end{matrix}]=[\begin{matrix}8 \\ 64\end{matrix}]$ has :
Consider a rectangle $ABCD$ having $5,6,7,9$ points in the interior of the line segments $AB,BC,CD,DA$ respectively. Let $\alpha$ be the number of triangles having these points from different sides as vertices and $\beta$ be the number of quadrilaterals having these points from different sides as vertices. Then $(\beta -\alpha )$ is equal to
$3\times {7}^{22}+2\times {10}^{22}-44$ when divided by $18$ leaves the remainder
Consider the following system of equations: $x+2y-3z=a$ $2x+6y-11z=b$ $x-2y+7z=c$ where $a,b$ and $c$ are real constants. Then the system of equations :
The domain of the function ${cosec}^{-1}(\frac{1+x}{x})$ is :
For the natural numbers $m,n,$ if $(1-y{)}^{m}(1+y{)}^{n}=1+{a}_{1}y+{a}_{2}{y}^{2}+\ldots .+{a}_{m+n}{y}^{m+n}$ and ${a}_{1}={a}_{2}=10,$ then the value of $m+n,$ is equal to:
Let $A=[\begin{matrix}{a}_{1} \\ {a}_{2}\end{matrix}]$ and $B=[\begin{matrix}{b}_{1} \\ {b}_{2}\end{matrix}]$ be two $2\times 1$matrices with real entries such that $A=XB,$ where $X=\frac{1}{\sqrt{3}}[\begin{matrix}1 & -1 \\ 1 & k\end{matrix}],$ and $k\in R.$ If ${a}_{1}^{2}+{a}_{2}^{2}=\frac{2}{3}({b}_{1}^{2}+{b}_{2}^{2})$ and $({k}^{2}+1){b}_{2}^{2}\neq -2{b}_{1}{b}_{2}$, then the value of $k$ is __________.
The values of $\lambda$ and $\mu$ such that the system of equations $x+y+z=6,3x+5y+5z=26$ and $x+2y+\lambda z=\mu$ has no solution, are:
The values of $a$ and $b$, for which the system of equations $2x+3y+6z=8$ $x+2y+az=5$ $3x+5y+9z=b$ has no solution, are :
The probability of selecting integers $a\in [-5,30]$ such that ${x}^{2}+2(a+4)x-5a+64>0$, for all $x\in R$, is:
Let $p$ and $q$ be two positive numbers such that $p+q=2$ and ${p}^{4}+{q}^{4}=272$. Then $p$ and $q$ are roots of the equation:
If the system of equations$kx+y+2z=1$ $3x-y-2z=2$ $-2x-2y-4z=3$ has infinitely many solutions, then $k$ is equal to ______ .
The equation $arg(\frac{z-1}{z+1})=\frac{\pi }{4}$ represents a circle with:
The value of $4+\frac{1}{5+\frac{1}{4+\frac{1}{5+\frac{1}{4+\ldots \ldots \infty }}}}$ is:
A scientific committee is to be formed from $6$ Indians and $8$ foreigners, which includes at least $2$ Indians and double the number of foreigners as Indians. Then the number of ways, the committee can be formed, is:
The sum of all the elements in the set ${n\in {1,2,\ldots \ldots ,100}\mid$ H.C.F. of $n$ and $2040$ is $1$} is equal to __________.
The number of elements in the set ${n\in {1,2,3,\ldots ,100}\mid (11{)}^{n}>(10{)}^{n}+(9{)}^{n}}$ is ___________.
Consider function $f:A\rightarrow B$ and $g:B\rightarrow C(A,B,C\subseteq R)$ such that ${(gof)}^{-1}$ exists, then:
If $[x]$ be the greatest integer less than or equal to $x,$ then $\sum _{n=8}^{100}[\frac{{(-1)}^{n}n}{2}]$ is equal to:
Let ${S}_{n}$ denote the sum of first $n$-terms of an arithmetic progression. If ${S}_{10}=530,{S}_{5}=140,$ then ${S}_{20}-{S}_{6}$ is equal to: