JEE Main 2020 — Algebra

If the sum of the second, third and fourth terms of a positive term G.P. is $3$ and the sum of its sixth, seventh and eighth terms is $243$, then the sum of the first $50$ terms of this G.P. is :

If ${3}^{2\mathrm{sin}2\alpha -1},14$ and ${3}^{4-2\mathrm{sin}2\alpha }$ are the first three terms of an A.P. for some $\alpha$ , then the sixth term of this A.P. is

If the sum of the first $20$ terms of the series ${\mathrm{log}}_{({7}^{1/2})}x+{\mathrm{log}}_{({7}^{1/3})}x+{\mathrm{log}}_{({7}^{1/4})}x+\ldots$is $460$, then $x$ is equal to:

The value of log₂8 is:

If ${2}^{10}+{2}^{9}\cdot {3}^{1}+{2}^{8}\cdot {3}^{2}+\ldots \ldots +2\cdot {3}^{9}+{3}^{10}=S-{2}^{11}$, then $S$ is equal to

If the letters of the word ${}^{'}{\mathrm{MOTHER}}^{'}$ be permuted and all the words so formed (with or without meaning) be listed as in a dictionary, then the position of the word ${}^{'}{\mathrm{MOTHER}}^{'}$ is.....

Let $\alpha =\frac{-1+i\sqrt{3}}{2}$. If $a=(1+\alpha )\sum _{k=0}^{100}{\alpha }^{2k}$ and $b=\sum _{k=0}^{100}{\alpha }^{3k}$, then $a$ and $b$, are the roots of the quadratic equation.

Let $z=x+\mathrm{iy}$ be a non-zero complex number such that ${z}^{2}=i{|z|}^{2}$, where $i=\sqrt{-1}$, then $z$ lies on the :

If for some positive integer $n$, the coefficients of three consecutive terms in the binomial expansion of ${(1+x)}^{n+5}$ are in the ratio $5:10:14$, then the largest coefficient in the expansion is :

Let $\cup _{i=1}^{50}{X}_{i}=\cup _{i=1}^{n}{Y}_{i}=T$, where each ${X}_{i}$ contains $10$ elements and each ${Y}_{i}$ contains $5$ elements. If each element of the set $T$ is an element of exactly $20$ of sets ${X}_{i}$'s and exactly $6$ of sets ${Y}_{i}$'s then $n$ is equal to :

Let A be a $3\times 3$ matrix such that $adjA=[\begin{matrix}2 & -1 & 1 \\ -1 & 0 & 2 \\ 1 & -2 & -1\end{matrix}]$ and $B=adj(adjA)$. If $|A|=\lambda$ and $|{({B}^{-1})}^{⊤}|=\mu$, then the ordered pair $(|\lambda |,\mu )$ is equal to

If $m$ arithmetic means (A.Ms) and three geometric means (G.Ms) are inserted between $3$ and $243$ such that ${4}^{th}$ A.M. is equal to ${2}^{nd}$ G.M., then $m$ is equal to:

Let $S$ be the set of all $\lambda \in R$ for which the system of linear equations $2x-y+2z=2$ $x-2y+\lambda z=-4$ $x+\lambda y+z=4$ has no solution. Then the set $S$

If the constant term in the binomial expansion of ${(\sqrt{x}-\frac{k}{{x}^{2}})}^{10}$ is $405$, then $|k|$ equals :

The coefficient of ${x}^{4}$ in the expansion of ${(1+x+{x}^{2}+{x}^{3})}^{6}$ in powers of $x,$ is $\ldots ..$

If ${p}$ denotes the fractional part of the number $p,$ then ${\frac{{3}^{200}}{8}}$ is equal to

If the number of integral terms in the expansion of ${({3}^{\frac{1}{2}}+{5}^{\frac{1}{8}})}^{n}$ is exactly $33$, then the least value of $n$ is

If the term independent of $x$ in the expansion of ${(\frac{3}{2}{x}^{2}-\frac{1}{3x})}^{9}$ is $k$, then $18k$ is equal to:

If the sum of the coefficients of all even powers of $x$ in the product $(1+x+{x}^{2}+\ldots +{x}^{2n})(1-x+{x}^{2}-{x}^{3}+\ldots +{x}^{2n})$ is $61,$ then n is equal to

If $A=[\begin{matrix}1 & 1 & 2 \\ 1 & 3 & 4 \\ 1 & -1 & 3\end{matrix}],B=adjA$ and $C=3A,$ then $\frac{|adjB|}{|C|}$ is equal to

The number of words, with or without meaning, that can be formed by taking 4 letters at a time from the letters of the word 'SYLLABUS' such that two letters are distinct and two letters are alike, is

Let $a,b,c\in R$ be all non-zero and satisfies ${a}^{3}+{b}^{3}+{c}^{3}=2$. If the matrix $A=[\begin{matrix}a & b & c \\ b & c & a \\ c & a & b\end{matrix}]\begin{matrix} \\ \\ \end{matrix}$ satisfies ${A}^{T}A=I,$ then a value of $abc$ can be

If the sum of the series $20+19\frac{3}{5}+19\frac{1}{5}+18\frac{4}{5}+..........$ up to ${n}^{\text{th }}$ term is $488$and the ${n}^{\text{th }}$ term is negative, then :

A survey shows that $63%$ of the people in a city read newspaper $A$ whereas $76%$ read news paper $B$. If $x%$ of the people read both the newspapers, then a possible value of $x$ can be:

A survey shows that $73%$ of the persons working in an office like coffee, whereas $65%$ like tea. If $x$ denotes the percentage of them, who like both coffee and tea, then $x$ cannot be:

Set $A$ has $m$elements and set $B$ has $n$elements. If the total number of subsets of $A$ is $112$ more than the total number of subsets of $B$, then the value of $m\cdot n$ is___.

Let ${R}_{1}$ and ${R}_{2}$ be two relations defined as follows :${R}_{1}={(a,b)\in {R}^{2}:{a}^{2}+{b}^{2}\in Q}$ and ${R}_{2}={(a,b)\in {R}^{2}:{a}^{2}+{b}^{2}\notin Q}$, where $Q$ is the set of all rational numbers, then

If the sum of first $11$ terms of an A.P. ,${a}_{1},{a}_{2},{a}_{3}\ldots \ldots$ is $0({a}_{1}\neq 0)$ then the sum of the A.P ${a}_{1},{a}_{3},{a}_{5},\ldots ..{a}_{23}$ is $k{a}_{1}$ where $k$ is equal to

If $a,b$ and $c$ are the greatest values of ${C}_{p}19,{C}_{q}20$ and ${C}_{r}21$ respectively, then:

If the minimum and the maximum values of the function $f:[\frac{\pi }{4},\frac{\pi }{2}]\rightarrow R,$ defined by $f(\theta )=|\begin{matrix}-{\mathrm{sin}}^{2}\theta & -1-{\mathrm{sin}}^{2}\theta & 1 \\ -{\mathrm{cos}}^{2}\theta & -1-{\mathrm{cos}}^{2}\theta & 1 \\ 12 & 10 & -2\end{matrix}|$ are $m$ and $M$respectively, then the ordered pair $(m,M)$ is equal to :

Let $\theta =\frac{\pi }{5}$ and $A=[\begin{matrix}cos\theta & sin\theta \\ -sin\theta & cos\theta \end{matrix}]$. If $B=A+{A}^{4}$, then det $(B)$ :

If$A=[\begin{matrix}\mathrm{cos}\theta & \text{ isin}\theta \\ \text{isin}\theta & \mathrm{cos}\theta \end{matrix}],(\theta =\frac{\pi }{24})$ and ${A}^{5}=[\begin{matrix}a & b \\ c & d\end{matrix}],$ where $i=\sqrt{-1}$, then which one of the following is not true?

Let $A={X={(x,y,z)}^{T}:PX=0\mathrm{and}{x}^{2}+{y}^{2}+{z}^{2}=1}$ where $P=[\begin{matrix}1 & 2 & 1 \\ -2 & 3 & -4 \\ 1 & 9 & -1\end{matrix}]$ then the set $A$

Let $A=[\begin{matrix}x & 1 \\ 1 & 0\end{matrix}],x\epsilon R$ and ${A}^{4}=[{a}_{ij}].$ If ${a}_{11}=109,$ then ${a}_{22}$ is equal to_____________.

The number of all $3\times 3$ matrices $A,$ with entries from the set ${-1,0,1}$ such that the sum of the diagonal elements of $A{A}^{T}$ is $3,$ is ___________.

Let $A$ be a $2\times 2$ real matrix with entries from ${0,1}$ and $|A|\neq 0$. Consider the following two statements; $(P)$ If $A\neq {l}_{2}$, then $|A|=-1$ $(Q)$ If $|A|=1$, then $tr(A)=2$ Where ${l}_{2}$ denotes $2\times 2$ identity matrix and $tr(A)$ denotes the sum of the diagonal entries of $A$. Then

If $A=(\begin{matrix}2 & 2 \\ 9 & 4\end{matrix})$ and $I=(\begin{matrix}1 & 0 \\ 0 & 1\end{matrix}),$ then $10 {A}^{-1}$, is equal to.

The sum of distinct values of $\lambda$ for which the system of equations : $(\lambda -1)x+(3\lambda +1)y+2\lambda z=0$ $(\lambda -1)x+(4\lambda -2)y+(\lambda +3)z=0$ $2x+(3\lambda +1)y+3(\lambda -1)z=0$, Has non-zero solutions, is ....... .

If the system of linear equations $x+y+3z=0$ $x+3y+{k}^{2}z=0$ $3x+y+3z=0$ has a non-zero solution $(x,y,z)$ for some $k\in R,$ then $x+(\frac{y}{z})$ is equal to :

Let m and M be respectively the minimum and maximum value values of $|\begin{matrix}{\mathrm{cos}}^{2}x & 1+{\mathrm{sin}}^{2}x & \mathrm{sin}2x \\ 1+{\mathrm{cos}}^{2}x & {\mathrm{sin}}^{2}x & \mathrm{sin}2x \\ {\mathrm{cos}}^{2}x & {\mathrm{sin}}^{2}x & 1+\mathrm{sin}2x\end{matrix}|$ Then the ordered pair (m, M) is equal to:

Let $\lambda \in R$. The system of linear equations $2{x}_{1}-4{x}_{2}+\lambda {x}_{3}=1$ ${x}_{1}-6{x}_{2}+{x}_{3}=2$ $\lambda {x}_{1}-10{x}_{2}+4{x}_{3}=3$ is inconsistent for :

If the system of equations $x+y+z=2$ $2x+4y-z=6$ $3x+2y+\lambda z=\mu$ has infinitely many solutions, then :

Let $S$ be the set of all integer solutions $(x,y,z)$ of the system of equations $x-2y+5z=0$ $-2x+4y+z=0$ $-7x+14y+9z=0$ such that $15\leq {x}^{2}+{y}^{2}+{z}^{2}\leq 150$. Then, the number of elements in the set $S$ is equal to ..........

The following system of linear equations $7x+6y-2z=0$ $3x+4y+2z=0$ $x-2y-6z=0,$ has

The region represented by ${z=x+iy\in C:|z|-Re(z)\leq 1}$ is also given by the inequality

If the system of linear equations, $x+y+z=6$ $x+2y+3z=10$ $3x+2y+\lambda z=\mu$ has more than two solutions, then $\mu -{\lambda }^{2}$, is equal to.

If the system of linear equations $2x+2ay+az=0$ $2x+3by+bz=0$ $2x+4cy+cz=0,$ where $a,b,c\in R$ are non-zero and distinct; has a non-zero solution, then

The value of $\sum _{r=0}^{20}C650-r$ is equal to:

The system of linear equations $\lambda x+2y+2z=5$ $2\lambda x+3y+5z=8$ $4x+\lambda y+6z=10$ has

If $|x|<1,|y|<1$ and $x\neq 1$, then the sum to infinity of the following series $(x+y)+({x}^{2}+xy+{y}^{2})+({x}^{3}+{x}^{2}y+x{y}^{2}+{y}^{3})+.....$ is

Let $A={a,b,c}$ and $B={1,2,3,4}.$ Then the number of elements in the set$C={f:A\rightarrow B\mid 2\in f(A)$ and $f$ is not one-one$}$ is $\ldots$

For a suitably chosen real constant $a$, let a function, $f:R-{-a}\rightarrow R$ be defined by $f(x)=\frac{a-x}{a+x}$. Further supposed that for any real number $x\neq -a,$and $f(x)\neq -a,(\mathrm{fof})(x)=x$. Then $f(-\frac{1}{2})$ is equal to :

Suppose that a function $f:R\rightarrow R$ satisfies $f(x+y)=f(x)f(y)$ for all $x,y\epsilon R$ and $f(1)=3$. If $\sum _{i=1}^{n}f(i)=363$, then $n$ is equal to ..... .

Suppose a differentiable function $f(x)$ satisfies the identity $f(x+y)=f(x)+f(y)+x{y}^{2}+{x}^{2}y,$ for all real $x$ and $y$. If $\underset{x\rightarrow 0}{\mathrm{lim}}\frac{f(x)}{x}=1,$ then ${f}^{'}(3)$ is equal to :

The minimum value of ${2}^{\mathrm{sin}x}+{2}^{\mathrm{cos}x}$ is :

Let $f:R\rightarrow R$ be a function which satisfies $f(x+y)=f(x)+f(y),\forall x,y\in R$ . If $f(1)=2$ and $g(n)=\sum _{k=1}^{(n-1)}f(k),$$n\in N$ then the value of $n$, for which $g(n)=20$, is

Let $f:(1, 3)\rightarrow R$, be a function defined by $f(x)=\frac{x[x]}{1+{x}^{2}},$ where $[x]$, denotes the greatest integer $\leq x.$ Then the range of $f$, is

If $g(x)={x}^{2}+x-1$ and $(gof)(x)=4{x}^{2}-10x+5,$ then $f(\frac{5}{4})$ is equal to

The inverse function of $f(x)=\frac{{8}^{2x}-{8}^{-2x}}{{8}^{2x}+{8}^{-2x}},x\in (-1,1),$ is __________.

Let $\alpha$ be a root of the equation ${x}^{2}+x+1=0$ and the matrix $A=\frac{1}{\sqrt{3}}[\begin{matrix}1 & 1 & 1 \\ 1 & \alpha & {\alpha }^{2} \\ 1 & {\alpha }^{2} & {\alpha }^{4}\end{matrix}],$ then the matrix ${A}^{31}$ is equal to

If $x=\sum _{n=0}^{\infty }{(-1)}^{n}{\mathrm{tan}}^{2}⁡\theta$ and $y=\sum _{n=0}^{\infty }{\mathrm{cos}}^{2n}⁡\theta ,$ for $0<\theta <\frac{\pi }{4},$ then:

The product ${2}^{\frac{1}{4}}\cdot {4}^{\frac{1}{16}}\cdot {8}^{\frac{1}{48}}\cdot {16}^{\frac{1}{128}}\cdot ....$ to $\infty$ is equal to:

The sum, $\sum _{n=1}^{7}\frac{n(n+1)(2n+1)}{4}$, is equal to

Let $f:R\rightarrow R$ be such that for all $x\in R({2}^{1+x}+{2}^{1-x}),f(x)$ and $({3}^{x}+{3}^{-x})$ are in A.P., then the minimum value of $f(x)$ is

The sum $\sum _{k=1}^{20}(1+2+3+\ldots +k)$ is ___________.

Let ${a}_{1},{a}_{2},{a}_{3},\ldots$, be a $G.P.$ such that ${a}_{1}<0,{a}_{1}+{a}_{2}=4$ and ${a}_{3}+{a}_{4}=16$. If $\sum _{i=1}^{9}{a}_{i}=4\lambda$, then $\lambda$, is equal to.

The sum of the first three terms of $G.P$ is $S$and their products is $27$. Then all such $S$ lie in

If $R={(x,y):x,y\in Z,{x}^{2}+3{y}^{2}\leq 8}$ is a relation on the set of integers $Z$, then the domain of ${R}^{-1}$ is

The number of words (with or without meaning) that can be formed from all the letters of the word$"LETTER"$ in which vowels never come together is.....

Five numbers are in $A.P.,$ whose sum is $25$ and product is $2520.$ If one of these five numbers is $-\frac{1}{2},$ then the greatest number amongst them is

If the determinant |1 2 3; 4 5 6; 7 8 k| = 0, then k equals:

If the ${10}^{th}$, term of an A.P. is $\frac{1}{20}$, and its ${20}^{th}$, term is $\frac{1}{10}$, then the sum of its first $200$, terms is.

Let $S$, be the set of all real roots of the equation, ${3}^{x}({3}^{x}-1)+2=|{3}^{x}-1|+|{3}^{x}-2|$, then

Let $u=\frac{2z+i}{z-ki},z=x+iy$ and $k>0$. If the curve represented by$Re(u)+Im(u)=1$ intersects the $y$-axis at points $P$ and $Q$ where $\mathrm{PQ}=5$ then the value of $k$ is

Let $\alpha >0,\beta >0$ be such that ${\alpha }^{3}+{\beta }^{2}=4$. If the maximum value of the term independent of $x$ in the binomial expansion of ${(\alpha {x}^{\frac{1}{9}}+\beta {x}^{-\frac{1}{6}})}^{10}$ is $10k$, then $k$ is equal to

If $f(x+y)=f(x)f(y)$ and $\Sigma _{x=1}^{\infty }f(x)=2,x,y\in N$, where $N$ is the set of all natural numbers, then the value of $\frac{f(4)}{f(2)}$ is

Suppose the vectors ${x}_{1},{x}_{2}$ and ${x}_{3}$ are the solutions of the system of linear equations, $Ax=b$ when the vector $b$ on the right side is equal to ${b}_{1},{b}_{2}$ and ${b}_{3}$ respectively. If ${x}_{1}=[\begin{matrix}1 \\ 1 \\ 1\end{matrix}],{x}_{2}=[\begin{matrix}0 \\ 2 \\ 1\end{matrix}],{x}_{3}=[\begin{matrix}0 \\ 0 \\ 1\end{matrix}]$; ${b}_{1}=[\begin{matrix}1 \\ 0 \\ 0\end{matrix}],{b}_{2}=[\begin{matrix}0 \\ 2 \\ 0\end{matrix}],{b}_{3}=[\begin{matrix}0 \\ 0 \\ 2\end{matrix}]$, then the determinant of $A$ is equal to

Let $S$ be the sum of the first $9$ term of the series : ${x+ka}+{{x}^{2}+(k+2)a}+{{x}^{3}+(k+4)a}+{{x}^{4}+(k+6)a}+\ldots$ where $a\neq 0$ and $x\neq 1$. If $S=\frac{{x}^{10}-x+45a(x-1)}{x-1}$ , then $k$ is equal to

There are $3$ sections in a question paper and each section contains $5$ questions. A candidate has to answer a total of $5$ questions, choosing at least one question from each section. Then the number of ways, in which the candidate can choose the questions, is:

The product of the roots of the equation $9{x}^{2}-18|x|+5=0$ is :

The domain of the function $f(x)={\mathrm{sin}}^{-1}(\frac{|x|+5}{{x}^{2}+1})$ is $(-\infty ,-a]\cup [a,\infty )$, then $a$ is equal to

Let $a,1{a}_{2},\ldots ,{a}_{n}$ be a given A.P. whose common difference is an integer and ${S}_{n}={a}_{1}+{a}_{2}+\ldots +{a}_{n}$. If ${a}_{1}=1,{a}_{n}=300$ and $15\leq n\leq 50,$ then the ordered pair $({S}_{n-4},{a}_{n-4})$ is equal to:

If $a+x=b+y=c+z+1,$ where $a,b,c,x,y,z$ are non-zero distinct real numbers, then$|\begin{matrix}x & a+y & x+a \\ y & b+y & y+b \\ z & c+y & z+c\end{matrix}|$ is equal to :

If $1+(1-{2}^{2}\cdot 1)+(1-{4}^{2}\cdot 3)+(1-{6}^{2}\cdot 5)+\ldots \ldots +(1-{20}^{2}\cdot 19)=\alpha -220\beta$, then an ordered pair $(\alpha ,\beta )$ is equal to:

If $\Delta =|\begin{matrix}x-2 & 2x-3 & 3x-4 \\ 2x-3 & 3x-4 & 4x-5 \\ 3x-5 & 5x-8 & 10x-17\end{matrix}|=A{x}^{3}+B{x}^{2}+Cx+D$, then $B+C$ is equal to :

If the system of equations $x-2y+3z=9$ $2x+y+z=b$ $x-7y+az=24,$ has infinitely many solutions, then $a-b$ is equal to ______

Let $A=[{a}_{ij}]$ and $B=[{b}_{ij}]$ be two $3\times 3$ real matrices such that ${b}_{ij}={(3)}^{(i+j-2)}{a}_{ij}$ , where $i,j=1,2,3$ . If the determinant of $B$ is $81$ , then determinant of $A$ i s

Consider the two sets: $A={m\in R:$ both the roots of ${x}^{2}-(m+1)x+m+4=0$ are real $}$ and $B=[-3,5)$ Which of the following is not true?

In the expansion of ${(\frac{x}{\mathrm{cos}⁡\theta }+\frac{1}{x\mathrm{sin}⁡\theta })}^{16},$ if ${l}_{1}$ is the least value of the term independent of $x$ when $\frac{\pi }{8}\leq \theta \leq \frac{\pi }{4}$ and ${l}_{2}$ is the least value of the term independent of $x$ when $\frac{\pi }{16}\leq \theta \leq \frac{\pi }{8},$ then the ratio ${l}_{2}:{l}_{1}$ is equal to:

Let ${(2{x}^{2}+3x+4)}^{10}=\sum _{r=0}^{20}{a}_{r}{x}^{r}.$ Then $\frac{{a}_{7}}{{a}_{13}}$ is equal to ______

The natural number $m$, for which the coefficient of $x$ in the binomial expansion of ${({x}^{m}+\frac{1}{{x}^{2}})}^{22}$ is 1540, is

Let ${a}_{n}$ be the ${n}^{th}$ term of a G.P. of positive terms. If $\sum _{n=1}^{100}{a}_{2n+1}=200$ and $\sum _{n=1}^{100}{a}_{2n}=100,$ then $\sum _{n=1}^{200}{a}_{n}$ is equal to:

The values of $\lambda$ and $\mu$ for which the system of linear equations $x+y+z=2$, $x+2y+3z=5$, $x+3y+\lambda z=\mu$ has infinitely many solutions, are respectively

Let $X={n\in N:1\leq n\leq 50}$. If $A={n\in X:n is a multiple of2}$ and $B={n\in X:n is a multiple of 7}$, then the number of elements in the smallest subset of $X$, containing both $A$ and $B$, is.

The value of $0.{16}^{{\mathrm{log}}_{2.5}(\frac{1}{3}+\frac{1}{{3}^{2}}+\frac{1}{{3}^{3}}+\ldots .\infty )}$ is __________

The number of real roots of the equation, ${e}^{4x}+{e}^{3x}-4{e}^{2x}+{e}^{x}+1=0$ is:

If the first term of an $A.P.$ is $3$ and the sum of its first $25$ terms is equal to the sum of its next $15$ terms, then the common difference of this $A.P.$is

The number of ordered pairs $(r,k)$ for which $6.{C}_{r}35=({k}^{2}-3).{C}_{r+1}36,$ where $k$ is an integer is

Let $a-2b+c=1.$ If $f(x)=|\begin{matrix}x+a & x+2 & x+1 \\ x+b & x+3 & x+2 \\ x+c & x+4 & x+3\end{matrix}|,$ then:

Let $f(x)$ be a quadratic polynomial such that $f(–1)+f(2)=0$. If one of the roots of $f(x)=0$ is $3$, then its other root lies in

The number of terms common to the two A.P.’s $3,7,11,\ldots ,407$ and $2,9,16,\ldots ,709$ is ____________.

For which of the following ordered pairs $(\mu ,\delta ),$ the system of linear equations $x+2y+3z=1$ $3x+4y+5z=\mu$ $4x+4y+4z=\delta$ is inconsistent?

If for some $\alpha$ and $\beta$ in $R$ , the intersection of the following three planes $x+4y-2z=1$ $x+7y-5z=\beta$ $x+5y+\alpha z=5$ is a line in ${R}^{3}$ , then $\alpha +\beta$ is equal to:

The greatest positive integer $k,$ for which ${49}^{k}+1$ is a factor of the sum ${49}^{125}+{49}^{124}+\ldots +{49}^{2}+49+1,$ is

The number of distinct solutions of the equation, ${\mathrm{log}}_{\frac{1}{2}}⁡|\mathrm{sin}⁡x|=2-{\mathrm{log}}_{\frac{1}{2}}⁡|\mathrm{cos}⁡x|$ in the interval $[0,2\pi ],$ is ________

If $A={x\in R:|x|<2}$ and $B={x\in R:|x-2|\geq 3};$ then

If $\alpha$ and $\beta$ are the roots of the equation $2x(2x+1)=1$, then $\beta$ is equal to :

Let $\alpha$ and $\beta$ be the roots of ${x}^{2}-3x+p=0$ and $\gamma$ and $\delta$ be the roots of ${x}^{2}-6x+q=0.$ If $\alpha ,\beta ,\gamma ,\delta$ from a geometric progression. Then ratio $(2q+p):(2q-p)$ is

Let $\lambda \neq 0$ be in $R$. If $\alpha$ and $\beta$ are the roots of the equation, ${x}^{2}-x+2\lambda =0$ and $\alpha$ and $\gamma$ are the roots of the equation, $3{x}^{2}-10x+27\lambda =0,$ then $\frac{\beta \gamma }{\lambda }$ is equal to:

If $\alpha$ and $\beta$ are the roots of the equation, $7{x}^{2}-3x-2=0,$ then the value of$\frac{\alpha }{1-{\alpha }^{2}}+\frac{\beta }{1-{\beta }^{2}}$ is equal to:

If $\alpha$ and $\beta$ be two roots of the equation ${x}^{2}-64x+256=0.$ Then the value of ${(\frac{{\alpha }^{3}}{{\beta }^{5}})}^{\frac{1}{8}}+{(\frac{{\beta }^{3}}{{\alpha }^{5}})}^{\frac{1}{8}}$ is :

If $\alpha$ and $\beta$ are the roots of the equation ${x}^{2}+px+2=0$ and $\frac{1}{\alpha }$ and $\frac{1}{\beta }$ are the roots of the equation $2{x}^{2}+2qx+1=0,$ then $(\alpha -\frac{1}{\alpha })(\beta -\frac{1}{\beta })(\alpha +\frac{1}{\beta })(\beta +\frac{1}{\alpha })$ is equal to :

Let $[t]$ denote the greatest integer$\leq t$. Then the equation in $x,{[x]}^{2}+2[x+2]-7=0$ has :

The set of all real values of $\lambda$ for which the quadratic equation $({\lambda }^{2}+1){x}^{2}-4\lambda x+2=0$ always have exactly one root in the interval $(0,1)$ is :

Let $\alpha$ and $\beta$ be the roots of the equation, $5{x}^{2}+6x-2=0$. If ${S}_{n}={\alpha }^{n}+{\beta }^{n},n=1,2,3,....,$ then

Let $a,b\in R,a\neq 0$ be such that the equation, $a{x}^{2}-2bx+5=0$ has a repeated root $\alpha ,$ which is also a root of the equation, ${x}^{2}-2bx-10=0.$ If $\beta$ is the other root of this equation, then ${\alpha }^{2}+{\beta }^{2}$ is equal to:

Let $\alpha$ and $\beta$ be two real roots of the equation $(k+1){tan}^{2}x-\sqrt{2}\cdot \lambda \mathrm{tan}⁡x=(1-k),$ where $k(\neq -1)$ and $\lambda$ are real numbers. If ${tan}^{2}(\alpha +\beta )=50,$ then a value of $\lambda$ is

Let $\alpha$ and $\beta$ be the roots of the equation ${x}^{2}-x-1=0$ . If ${p}_{k}={(\alpha )}^{k}+{(\beta )}^{k},k\geq 1,$ then which one of the following statements is not true?

The least positive value of ‘ $a$ ’ for which the equation, $2{x}^{2}+(a-10)x+\frac{33}{2}=2a$ has real roots is ___________.

If $a$ and $b$ are real numbers such that ${(2+\alpha )}^{4}=a+b\alpha$, where $\alpha =\frac{-1+i\sqrt{3}}{2}$, then $a+b$ is equal to:

The value of${(\frac{-1+i\sqrt{3}}{1-i})}^{30}$ is :

If the four complex numbers $z,\bar{z},\bar{z}-2Re(\bar{z})$ and $z-2Re(z)$ represent the vertices of a square of side $4$ units in the Argand plane, then $|z|$ is equal to :

The value of ${(\frac{1+\mathrm{sin}\frac{2\pi }{9}+i\mathrm{cos}\frac{2\pi }{9}}{1+\mathrm{sin}\frac{2\pi }{9}-i\mathrm{cos}\frac{2\pi }{9}})}^{3}$ is

If ${(\frac{1+i}{1-i})}^{\frac{m}{2}}={(\frac{1+i}{i-1})}^{\frac{n}{3}}=1,(m,n\in N)$ then the greatest common divisor of the least values of $m$ and $n$ is

If ${z}_{1},{z}_{2}$ are complex numbers such that $Re({z}_{1})=|{z}_{1}-1|$ and $Re({z}_{2})=|{z}_{2}-1|$ and $\mathrm{arg}({z}_{1}-{z}_{2})=\frac{\pi }{6}$, then $Im({z}_{1}+{z}_{2})$ is equal to :

Let $z$ be a complex number such that $|\frac{z-i}{z+2i}|=1$ and $|z|=\frac{5}{2}$ . Then, the value of $|z+3i|$ is

If the equation ${x}^{2}+bx+45=0,b\in R$ has conjugate complex roots and they satisfy $|z+1|=2\sqrt{10},$ then

If $z$ is a complex number satisfying $|Re(z)|+|Im(z)|=4,$ then $|z|$ cannot be

If $\frac{3+isin\theta }{4-icos\theta },\theta \in [0,2\pi ],$ is a real number, then an argument of $sin\theta +icos\theta$ is

If $Re(\frac{z-1}{2z+i})=1,$ where $z=x+iy,$ then the point $(x,y)$ lies on a

A test consists of $6$ multiple choice questions, each having $4$ alternative answers of which only one is correct. The number of ways, in which a candidate answers all six questions such that exactly four of the answers are correct, is ___________

The total number of $3-$digit numbers whose sum of digits is $10$, is ..........

Two families with three members each and one family with four members are to be seated in a row. In how many ways can they be seated so that the same family members are not separated ?

The number of $4$ letter words (with or without meaning) that can be formed from the eleven letters of the word $\mathrm{EXAMINATION}$ is

An urn contains $5$ red marbles, $4$ black marbles and $3$ white marbles. Then, the number of ways in which $4$ marbles can be drawn so that at the most three of them are red is ___________.

Let$n>2$ be an integer. Suppose that there are $n$ Metro stations in a city located around a circular path. Each pair of the nearest stations is connected by a straight track only. Further, each pair of the nearest station is connected by blue line, whereas all remaining pairs of stations are connected by red line. If number of red lines is $99$ times the number of blue lines, then the value of $n$ is

If the number of five digit numbers with distinct digits and $2$ at the ${10}^{th}$ place is $336k$ , then $k$ is equal to:

Total number of $6-$ digit numbers in which only and all the five digits $1,3,5,7$ and $9$ appears, is

The common difference of the $A.P.{b}_{1},{b}_{2},....,{b}_{m}$ is $2$ more than common difference of $A.P.{a}_{1},{a}_{2},.....,{a}_{n}$. If ${a}_{40}=-159,{a}_{100}=-399$ and ${b}_{100}={a}_{70}$, then ${b}_{1}$ is equal to :

Let $a,b,c,d\text{and}p$ be non-zero distinct real numbers such that $({a}^{2}+{b}^{2}+{c}^{2}){p}^{2}-2(ab+bc+cd)p+({b}^{2}+{c}^{2}+{d}^{2})=0$. Then