JEE Main Mathematics — Algebra previous year questions with solutions.
The function $f:N-{1}\rightarrow N$; defined by $f(n)=$ the highest prime factor of $n$, is :
Consider the system of linear equation $x+y+z=$ $4\mu ,x+2y+2\lambda z=10\mu ,x+3y+4{\lambda }^{2}z={\mu }^{2}+15$, where $\lambda ,\mu \in R$. Which one of the following statements is NOT correct?
There are 4 men and 5 women in Group A, and 5 men and 4 women in Group B. If 4 persons are selected from each group, then the number of ways of selecting 4 men and 4 women is _____
Let $\alpha$ and $\beta$ be the roots of the equation $p{x}^{2}+qx-r=0$, where $p\neq 0$. If $p,q$ and $r$ be the consecutive terms of a non-constant G.P and $\frac{1}{\alpha }+\frac{1}{\beta }=\frac{3}{4}$, then the value of ${(\alpha -\beta )}^{2}$ is:
Let $A=\{(x, y): 2 x+3 y=23, x, y \in \mathbb{N}\}$ and $B=\{x:(x, y) \in A\}$. Then the number of one-one functions from $A$ to $B$ is equal to _______
Let $A$ be a non-singular matrix of order 3 . If $\operatorname{det}(3 \operatorname{adj}(2 \operatorname{adj}((\operatorname{det} A) A)))=3^{-13} \cdot 2^{-10}$ and $\operatorname{det}(3 \operatorname{adj}(2 \mathrm{~A}))=2^{\mathrm{m}} \cdot 3^{\mathrm{n}}$, then $|3 \mathrm{~m}+2 \mathrm{n}|$ is equal to $\qquad$
If the system of equations $x+4 y-z=\lambda, 7 x+9 y+\mu z=-3,5 x+y+2 z=-1$ has infinitely many solutions, then $(2 \mu+3 \lambda)$ is equal to :
Let $S_1=\{z \in C:|z| \leq 5\}, S_2=\left\{z \in C: \operatorname{Im}\left(\frac{z+1-\sqrt{3} i}{1-\sqrt{3} i}\right) \geq 0\right\}$ and $S_3=\{z \in C: \operatorname{Re}(z) \geq 0\}$. Then the area of the region $S_1 \cap S_2 \cap S_3$ is :
Let $A$ be a $2\times 2$ real matrix and $I$ be the identity matrix of order $2.$ If the roots of the equation $|A-\mathrm{xI}|=0$ be $-1$ and $3,$ then the sum of the diagonal elements of the matrix ${A}^{2}$ is _____.
Let for any three distinct consecutive terms $a,b,c$ of an A.P, the lines $ax+by+c=0$ be concurrent at the point $P$ and $Q(\alpha ,\beta )$ be a point such that the system of equations $x+y+z=6$, $2x+5y+\alpha z=\beta$ and $x+2y+3z=4$, has infinitely many solutions. Then $(PQ{)}^{2}$ is equal to _______.
In an A.P., the sixth term ${a}_{6}=2$. If the ${a}_{1}{a}_{4}{a}_{5}$ is the greatest, then the common difference of the A.P., is equal to
The number of common terms in the progressions $4,9,14,19,\ldots \ldots$, up to ${25}^{\text{th }}$ term and $3,6,9,12$,.... up to ${37}^{\text{th }}$ term is :
If the system of equations $\begin{aligned} & x+(\sqrt{2} \sin \alpha) y+(\sqrt{2} \cos \alpha) z=0 \\ & x+(\cos \alpha) y+(\sin \alpha) z=0 \\ & x+(\sin \alpha) y-(\cos \alpha) z=0 \end{aligned}$ has a non-trivial solution, then $\alpha \in\left(0, \frac{\pi}{2}\right)$ is equal to :
If the system of equations $\begin{aligned} & 2 x+7 y+\lambda z=3 \\ & 3 x+2 y+5 z=4 \\ & x+\mu y+32 z=-1 \end{aligned}$ has infinitely many solutions, then $(\lambda-\mu)$ is equal to________
The ${20}^{\text{th }}$ term from the end of the progression $20,19\frac{1}{4},18\frac{1}{2},17\frac{3}{4},\ldots ,-129\frac{1}{4}$ is :-
The values of $m, n$, for which the system of equations $\begin{aligned} & x+y+z=4, \\ & 2 x+5 y+5 z=17, \\ & x+2 y+\mathrm{m} z=\mathrm{n} \end{aligned}$ has infinitely many solutions, satisfy the equation:
Let $R$ be a relation on $Z\times Z$ defined by $(a,b)R(c,d)$ if and only if $ad-bc$ is divisible by $5$ . Then $R$ is
Let $\alpha, \beta ; \alpha>\beta$, be the roots of the equation $x^2-\sqrt{2} x-\sqrt{3}=0$. Let $\mathrm{P}_n=\alpha^n-\beta^n, n \in \mathrm{N}$. Then $(11 \sqrt{3}-10 \sqrt{2}) \mathrm{P}_{10}+(11 \sqrt{2}+10) \mathrm{P}_{11}-11 \mathrm{P}_{12}$ is equal to
The coefficient of ${x}^{2012}$ in the expansion of ${(1-x)}^{2008}{(1+x+{x}^{2})}^{2007}$ is equal to _____.
Consider the function $f: \mathbb{R} \rightarrow \mathbb{R}$ defined by $f(x)=\frac{2 x}{\sqrt{1+9 x^2}}$. If the composition of $f, \underbrace{(f \circ f \circ f \circ \cdots \circ f)}_{10 \text { times }}(x)=\frac{2^{10} x}{\sqrt{1+9 \alpha x^2}}$, then the value of $\sqrt{3 \alpha+1}$ is equal to ______
Let $\alpha \beta \gamma=45 ; \alpha, \beta, \gamma \in \mathbb{R}$. If $x(\alpha, 1,2)+y(1, \beta, 2)+z(2,3, \gamma)=(0,0,0)$ for some $x, y, z \in \mathbb{R}, x y z \neq 0$, then $6 \alpha+4 \beta+\gamma$ is equal to _______
Let $f:R\rightarrow R$ and $g:R\rightarrow R$ be defined as $f(x)={\begin{matrix}{\mathrm{log}}_{e}x, & x>0 \\ {e}^{-x}, & x\leq 0\end{matrix}$ and $g(x)={\begin{matrix}x, & x\geq 0 \\ {e}^{x}, & x<0\end{matrix}$. Then, $gof:R\rightarrow R$ is:
In a survey of 220 students of a higher secondary school, it was found that at least 125 and at most 130 students studied Mathematics; at least 85 and at most 95 studied Physics; at least 75 and at most 90 studied Chemistry; 30 studied both Physics and Chemistry; 50 studied both Chemistry and Mathematics; 40 studied both Mathematics and Physics and 10 studied none of these subjects. Let $\mathrm{m}$ and $\mathrm{n}$ respectively be the least and the most number of students who studied all the three subjects. Then $\mathrm{m}+\mathrm{n}$ is equal to ______
If 2 and 6 are the roots of the equation $a x^2+b x+1=0$, then the quadratic equation, whose roots are $\frac{1}{2 a+b}$ and $\frac{1}{6 a+b}$, is :