JEE Main 2020 — Algebra

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:

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

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

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 $A={a,b,c}$ 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 $\ldots$

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 :

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 :

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 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

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 $A$

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 :

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

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 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 $f(x)$ 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

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

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:

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

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:

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

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

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

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?

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_____________.

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 :

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___.

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

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 :

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

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?

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.

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

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 ..... .

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

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

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:

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

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 :

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.....

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 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

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

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

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

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 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

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 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 __________

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

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

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:

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 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 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.

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

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 :

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

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 $\lambda \neq 0$ 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 $\beta$ 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 :

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 :

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 $\alpha$ 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

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 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 :

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

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

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 ___________

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

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

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

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

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

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?

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

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

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

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

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

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:

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 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

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

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

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

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

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

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

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 ___________.

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 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

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 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 :

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:

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

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

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

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 ${R}_{1}$ 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

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

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

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 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

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

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?

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: