JEE Main Mathematics — Algebra previous year questions with solutions.
If the system of linear equations $2x+y-z=7$ $x-3y+2z=1$ $x+4y+\delta z=k$, where $\delta ,k\in R$ has infinitely many solutions, then $\delta +k$ is equal to
If the system of linear equations $2x-3y=\gamma +5$ $\alpha x+5y=\beta +1$, where $\alpha ,\beta ,\gamma \in R$ has infinitely many solutions, then the value of $|9\alpha +3\beta +5\gamma |$ is equal to
If the system of linear equations. $8x+y+4z=-2$ $x+y+z=0$ $\lambda x-3y=\mu$ has infinitely many solutions, then the distance of the point $(\lambda ,\mu ,-\frac{1}{2})$ from the plane $8x+y+4z+2=0$ is:
If the system of equations $x+y+z=6$ $2x+5y+\alpha z=\beta$ $x+2y+3z=14$ has infinitely many solutions, then $\alpha +\beta$ is equal to
If the system of equations $\alpha x+y+z=5,x+2y+3z=4,x+3y+5z=\beta$. Has infinitely many solutions, then the ordered pair $(\alpha ,\beta )$ is equal to
If the sum of the squares of the reciprocals of the roots $\alpha$ and $\beta$ of the equation $3{x}^{2}+\lambda x-1=0$ is $15$, then $6{({\alpha }^{3}+{\beta }^{3})}^{2}$ is equal to
If the sum of the coefficients of all the positive powers of $x$, in the binomial expansion of ${({x}^{n}+\frac{2}{{x}^{5}})}^{7}$ is $939$, then the sum of all the possible integral values of $n$ is
If the sum of the co-efficients of all the positive even powers of $x$ in the binomial expansion of ${(2{x}^{3}+\frac{3}{x})}^{10}$ is ${5}^{10}-\beta \cdot {3}^{9}$, then $\beta$ is equal to _____.
If the minimum value of $f(x)=\frac{5{x}^{2}}{2}+\frac{\alpha }{{x}^{5}},x>0$, is $14$, then the value of $\alpha$ is equal to
If the maximum value of the term independent of $t$ in the expansion of ${({t}^{2}{x}^{\frac{1}{5}}+\frac{{(1-x)}^{{}^{\frac{1}{10}}}}{t})}^{15},x\geq 0$, is $K$, then $8K$ is equal to _____ .
If the constant term in the expansion of ${(3{x}^{3}-2{x}^{2}+\frac{5}{{x}^{5}})}^{10}$ is ${2}^{k}.l$, where $l$ is an odd integer, then the value of $k$ is equal to
If the coefficients of $x$ and ${x}^{2}$ in the expansion of ${(1+x)}^{p}{(1-x)}^{q},p,q\leq 15$, are $-3$ and $-5$ respectively, then the coefficient of ${x}^{3}$ is equal to ______.
If the coefficient of ${x}^{10}$ in the binomial expansion of ${(\frac{\sqrt{x}}{{5}^{\frac{1}{4}}}+\frac{\sqrt{5}}{{x}^{\frac{1}{3}}})}^{60}$ is ${5}^{k}l$, where $l,k\in N$ and $l$ is coprime to $5$, then $k$ is equal to ______.
If $z=x+iy$ satisfies $|z|-2=0$ and $|z-i|-|z+5i|=0$, then
If $1+(2+C149+C249+\ldots .+C4949)(C250+C450+\ldots ..+C5050)$ is equal to ${2}^{n}.m$, where $m$ is odd, then $n+m$ is equal to _____ .
If for some $p,q,r\in R$, all have positive sign, one of the roots of the equation $({p}^{2}+{q}^{2}){x}^{2}-2q(p+r)x+{q}^{2}+{r}^{2}=0$ is also a root of the equation ${x}^{2}+2x-8=0$, then $\frac{{q}^{2}+{r}^{2}}{{p}^{2}}$ is equal to-
If $n$ arithmetic means are inserted between a and $100$ such that the ratio of the first mean to the last mean is $1:7$ and $a+n=33$, then the value of $n$ is
If $\alpha ,\beta$ are the roots of the equation ${x}^{2}-(5+{3}^{\sqrt{{\mathrm{log}}_{3}5}}-{5}^{\sqrt{{\mathrm{log}}_{5}3}})x+3({3}^{{({\mathrm{log}}_{3}5)}^{\frac{1}{3}}}-{5}^{{({\mathrm{log}}_{5}3)}^{\frac{2}{3}}}-1)=0$ then the equation, whose roots are $\alpha +\frac{1}{\beta }$ and $\beta +\frac{1}{\alpha }$,
If $\alpha ,\beta ,\gamma ,\delta$ are the roots of the equation ${x}^{4}+{x}^{3}+{x}^{2}+x+1=0$, then ${\alpha }^{2021}+{\beta }^{2021}+{\gamma }^{2021}+{\delta }^{2021}$ is equal to
If ${a}_{1}(>0),{a}_{2},{a}_{3},{a}_{4},{a}_{5}$ are in a G.P. , ${a}_{2}+{a}_{4}=2{a}_{3}+1$ and $3{a}_{2}+{a}_{3}=2{a}_{4}$, then ${a}_{2}+{a}_{4}+2{a}_{5}$ is equal to _____.
If $A=\sum _{n=1}^{\infty }\frac{1}{{(3+{(-1)}^{n})}^{n}}$ and $B=\sum _{n=1}^{\infty }\frac{{(-1)}^{n}}{{(3+{(-1)}^{n})}^{n}}$, then $\frac{A}{B}$ is equal to
If $p$ and $q$ are real number such that $p+q=3,{p}^{4}+{q}^{4}=369$, then the value of ${(\frac{1}{p}+\frac{1}{q})}^{-2}$ is equal to
If ${a}_{1},{a}_{2},{a}_{3}\ldots$ and ${b}_{1},{b}_{2},{b}_{3}\ldots .$ are A.P. and ${a}_{1}=2,{a}_{10}=3,{a}_{1}{b}_{1}=1={a}_{10}{b}_{10}$ then ${a}_{4}{b}_{4}$ is equal to
For $n\in N$, let ${S}_{n}={z\in C:|z-3+2i|=\frac{n}{4}}$ and ${T}_{n}={z\in C:|z-2+3i|=\frac{1}{n}}$. Then the number of elements in the set ${n\in N:{S}_{n}\cap {T}_{n}=\phi }$ is