JEE Main Mathematics — Algebra previous year questions with solutions.
The number of points, having both co-ordinates as integers, that lie in the interior of the triangle with vertices $(0,0),(0,41)$ and $(41,0)$ is
In a geometric progression, if the ratio of the sum of first 5 terms to the sum of their reciprocals is 49 , and the sum of the first and the third term is 35. Then the first term of this geometric progression is:
If $\frac{1}{\sqrt{\alpha }},\frac{1}{\sqrt{\beta }}$ are the roots of the equation $a{x}^{2}+bx+1=0,(a\neq 0,a,b\in R)$, then the equation $x(x+{b}^{3})+({a}^{3}-3abx)=0$ has roots:
If $g$ is the inverse of a function $f$ and ${f}^{'}(x)=\frac{1}{1+{x}^{5}},$ then ${g}^{'}(x)$ is equal to
An eight digit number divisible by 9 is to be formed using digits from 0 to 9 without repeating the digits. The number of ways in which this can be done is:
The coefficient of ${ x }^{ 1 0 1 2 }$ in the expansion of ${(1+{x}^{n}+{x}^{253})}^{10},$ (where $n\leq 22$ is any positive integer), is
Let $w(Imw\neq 0)$ be a complex number. Then, the set of all complex numbers $z$ satisfying the equation $w-\bar{w}z=k(1-z)$, for some real number $k$, is
The equation $\sqrt{3{x}^{2}+x+5}=x-3$, where $x$ is real, has
If equations $a{x}^{2}+bx+c=0,(a,b,c\in R,a\neq 0)$ and $2{x}^{2}+3x+4=0$ have a common root, then $a:b:c$ equals :
The coefficient of $x^{50}$ in the binomial expansion of $(1+x)^{1000}+x(1+x)^{999}+x^2(1+x)^{998}+\ldots$ $+x^{1000}$ is:
If $X={{4}^{n}-3n-1:n\in N}$ and $Y={9(n-1):n\in N}$, where $N$ is the set of natural numbers, then $X\cup Y$ is equal to
The least positive integer $\mathrm{n}$ such that $1-\frac{2}{3}-\frac{2}{3^2}-\ldots .-\frac{2}{3^{n-1}} < \frac{1}{100}$, is:
The sum of the first 20 terms common between the series $3+7+11+15+$ and $1+6+11+$ $16+\ldots .$. is
If $\alpha ,\beta \neq 0$, $f(n)={\alpha }^{n}+{\beta }^{n}$ and $|\begin{matrix}3 & 1+f(1) & 1+f(2) \\ 1+f(1) & 1+f(2) & 1+f(3) \\ 1+f(2) & 1+f(3) & 1+f(4)\end{matrix}|=K{(1-\alpha )}^{2}{(1-\beta )}^{2}{(\alpha -\beta )}^{2}$, then $K$ is equal to
If $$ \begin{aligned} &\left.\mid \begin{array}{ccc} a^2 & b^2 & c^2 \\ (a+\lambda)^2 & (b+\lambda)^2 & \left(c+\lambda^2\right) \\ (a-\lambda)^2 & \left(b-\lambda^2\right) & \left(-\lambda^2\right. \end{array}\right) \\ &=k \lambda\left|\begin{array}{ccc} a^2 & b^2 & c^2 \\ a & b & c \\ 1 & 1 & 1 \end{array}\right|, \lambda \neq 0 \end{aligned} $$ then $\mathrm{k}$ is equal to:
The number of terms in the expansion of ${(1+x)}^{101}{(1-x+{x}^{2})}^{100}$ in powers of $x$ is
If $\mathrm{f}(\theta)=\left|\begin{array}{ccc}1 & \cos \theta & 1 \\ -\sin \theta & 1 & -\cos \theta \\ -1 & \sin \theta & 1\end{array}\right|$ and $\mathrm{A}$ and $\mathrm{B}$ are respectively the maximum and the minimum values of $f(\theta)$, then $(A, B)$ is equal to:
If $\left(2+\frac{x}{3}\right)^{55}$ is expanded in the ascending powers of $x$ and the coefficients of powers of $x$ in two consecutive terms of the expansion are equal, then these terms are:
If the sum $\frac{3}{{1}^{2}}+\frac{5}{{1}^{2}+{2}^{2}}+\frac{7}{{1}^{2}+{2}^{2}+{3}^{2}}+.....+$ up to $20$ terms is equal to $\frac{k}{21},$ then $k$ is equal to
If $A=\left[\begin{array}{ccc}1 & 2 & x \\ 3 & -1 & 2\end{array}\right]$ and $B=\left[\begin{array}{c}y \\ x \\ 1\end{array}\right]$ be such that $\mathrm{AB}=\left[\begin{array}{l}6 \\ 8\end{array}\right]$, then:
The function $f(x)=|sin4x|+|cos2x|,$ is a periodic function with a fundamental period
If $f(x)=x^2-x+5, x>\frac{1}{2}$, and $\mathrm{g}(\mathrm{x})$ is its inverse function, then $\mathrm{g}^{\prime}(7)$ equals:
If ${\Delta }_{r}=|\begin{matrix}r & 2r-1 & 3r-2 \\ \frac{n}{2} & n-1 & a \\ \frac{1}{2}n(n-1) & {(n-1)}^{2} & \frac{1}{2}(n-1)(3n+4)\end{matrix}|$, then the value of $\sum _{r=1}^{n-1}{\Delta }_{r}$
The number of terms in an $A.P.$ is even, the sum of the odd terms in it is $24$ and that the even terms is $30.$ If the last term exceeds the first term by $10\frac{1}{2},$ then the number of terms in the $A.P.$ is