JEE Main Mathematics — Algebra previous year questions with solutions.
If the system of equations $2x+3y-z=0, x+ky-2z=0$ and $2x-y+z=0$ has a non-trivial solution $(x,y,z),$ then $\frac{x}{y}+\frac{y}{z}+\frac{z}{x}+k$ is equal to
The number of values of $\theta \in (0, \pi )$ for which the system of linear equations $x+3y+7z=0$ $-x+4y+7z=0$ $(\mathrm{sin}3\theta )x+(\mathrm{cos}2\theta )y+2z=0$ has a non-trivial solution, is:
If the system of equations $x+y+z=5,$ $x+2y+3z=9,$ $x+3y+\alpha z=\beta$ has inifinitely many solutions, then $\beta -\alpha$ equals
The sum of the solutions of the equation $|\sqrt{x}-2|+\sqrt{x}(\sqrt{x}-4)+2=0,(x>0)$ is equal to
Let $A=$ { $x\in R:x$ is not a positive integer} $.$ Define a function $f:A\rightarrow R$ as $f(x)=\frac{2x}{x-1}$ , then $f$ is:
If the system of linear equations $2 x+2 y+3 z=a$ $3 x-y+5 z=b$ $x-3 y+2 z=c$ where, $a, b,$ care non-zero real numbers, has more than onc solution, then
If the fourth term in the Binomial expansion of ${(\frac{2}{x}+{x}^{{log}_{8}x})}^{6},(x>0)$ is $20\times {8}^{7},$ then a value of $x$ is
If the coefficients of ${x}^{2}$ and ${x}^{3}$, are both zero, in the expansion of the expression $(1+ax+b{x}^{2}){(1-3x)}^{15}$, in powers of $x$ , then the ordered pair $(a,b)$ is equal to
Let $f(x)={a}^{x} (a>0)$ be written as $f(x)={f}_{1}(x)+{f}_{2}(x),$ where ${f}_{1}(x)$ is an even function and ${f}_{2}(x)$ is an odd function. Then ${f}_{1}(x+y)+{f}_{1}(x-y)$ equals:
If $[x]$ denotes the greatest integer $\leq x,$ then the system of linear equations $[sin\theta ]x+[-cos\theta ]y=0$, $[cot\theta ]x+y=0$
If the system of linear equations $x+y+z=5$, $x+2y+2z=6$, $x+3y+\lambda z=\mu , (\lambda , \mu \in R)$ , has infinitely many solutions, then the value of $\lambda +\mu$ is:
Let $a,b$ and $c$ be in $G.P.$ with common ratio $r,$ where $a\neq 0$ and $0<r\leq \frac{1}{2}.$ If $3a,7b$ and $15c$ are the first three terms of an $A.P.,$ then the ${4}^{th}$ term of this $A.P.$ is :
If some three consecutive coefficients in the binomial expansion of ${(x+1)}^{n}$ in powers of $x$ are in the ratio $2:15:70,$ then the average of these three coefficients is:
If $\lambda$ be the ratio of the roots of the quadratic equation in $x, 3{m}^{2}{x}^{2}+m(m-4)x+2=0$, then the least value of $m$ for which $\lambda +\frac{1}{\lambda }=1$, is :
Let $N$ be the set of natural numbers and two functions $f$ and $g$ be defined as $f,g:N\rightarrow N$ such that $f(n)={\begin{matrix}\frac{n+1}{2}, if n is odd \\ \frac{n}{2}, if n is even\end{matrix}$ and $g(n)=n-{(-1)}^{n}.$ Then $fog$ is:
Let $f(x)={\mathrm{log}}_{e}(sinx), (0<x<\pi )$ and $g(x)={\mathrm{sin}}^{-1}({e}^{-x}), (x\geq 0).$ If $\alpha$ is a positive real number such that $a={(fog)}^{'}(\alpha )$ and $b=(fog)(\alpha ),$ then
The number of functions $f$ from $\{1,2,3, \ldots, 20\}$ onto $\{1,2,3, \ldots, 20\}$ such that $f(k)$ is a multiple of $3,$ whenever $k$ is a multiple of 4 is:
Let ${z}_{1}$ and ${z}_{2}$ be two complex numbers satisfying $|{z}_{1}|=9$ and $|{z}_{2}-3-4i|=4$. Then the minimum value of $|{z}_{1}-{z}_{2}|$ is :
If the third term in the binomial expansion of ${(1+{x}^{{\mathrm{log}}_{2}x})}^{5}$ equals $2560,$ then a possible value of $x$ is
In a class of $140$ students numbered $1$ to$140$ , all even numbered students opted Mathematics course, those whose number is divisible by $3$ opted Physics course and those whose number is divisible by $5$ opted Chemistry course. Then the number of students who did not opt for any of the three courses is:
Let $a_{1}, a_{2}, \ldots, a_{10}$ be a G.P. If $\frac{a_{3}}{a_{1}}=25,$ then $\frac{a_{9}}{a_{5}}$ equals :
If 19 th term of a non-zero A.P. is zero, then its (49th term): (29th term) is:
For $x\in R-{0, 1},$ let ${f}_{1}(x)=\frac{1}{x},{f}_{2}(x)=1-x$ and ${f}_{3}(x)=\frac{1}{1-x}$ be three given functions. If a function, $J(x)$ satisfies $({f}_{2}oJo{f}_{1})(x)={f}_{3}(x)$ then $J(x)$ is equal to:
If $f(x)={\mathrm{log}}_{e}(\frac{1-x}{1+x}), |x|< 1,$ then $f(\frac{2x}{1+{x}^{2}})$ is equal to