JEE Main 2018 — Algebra

$n$ - digit numbers are formed using only three digits 2,5 and 7 . The smallest value of $n$ for which 900 such distinct numbers can be formed, is

$n$-digit numbers are formed using only three digits $2,5$ and $7$. The smallest value of $n$ for which $900$ such distinct numbers can be formed is :

Consider the following two binary relations on the set $A={a,b,c}:{R}_{1}={(c,a),(b,b),(a,c),(c,c),(b,c),(a,a)}$ and ${R}_{2}={(a,b),(b,a),(c,c),(c,a),(a,a),(b,b),(a,c)}$, then :

Consider the following two binary relations on the set $A=\{a, b, c\}: R_1=\{(\mathrm{c}, a)(b, b),(\mathrm{a}, c),(c$, $c),(b, c),(a, a)\}$ and $\mathrm{R}_2=\{(\mathrm{a}, \mathrm{b}),(\mathrm{b}, \mathrm{a}),(\mathrm{c}, \mathrm{c})$, (c, a), (a, a), (b, b), (a, c). Then

From $6$ different novels and $3$ different dictionaries, $4$ novels and $1$ dictionary are to be selected and arranged in a row on a shelf so that the dictionary is always in the middle. The number of such arrangements is:

If an angle $A$ of a $\Delta ABC$ satisfies $5\mathrm{cos}A+3=0$, then the roots of the quadratic equation $9{x}^{2}+27x+20=0$ are

If $\mathrm{tan}⁡A$ and $\mathrm{tan}⁡B$ are the roots of the quadratic equation $3{x}^{2}-10x-25=0$, then the value of $3{\mathrm{sin}}^{2}⁡(A+B)-10\mathrm{sin}⁡(A+B)\mathrm{cos}⁡(A+B)-25{\mathrm{cos}}^{2}⁡(A+B)$ is :

If $x_1, x_2, \ldots ., x_n$ and $\frac{1}{h_1}, \frac{1}{h^2}, \ldots . . \frac{1}{h_n}$ are two A.P's such that $x_3=h_2=8$ and $x_8=h_7=20$, then $x_5 . h_{10}$ equals.

If ${x}_{1},{x}_{2},\ldots ..,{x}_{n}$ and $\frac{1}{{h}_{1}},\frac{1}{{h}_{2}},\ldots ..,\frac{1}{{h}_{n}}$ are two A.P.s such that ${x}_{3}={h}_{2}=8&{x}_{8}={h}_{7}=20$, then ${x}_{5}\cdot {h}_{10}$ is equal to

If $a, b, c$ are in A.P. and $a^2, b^2, c^2$ are in G.P. such that $a < b < c$ and $a+b+c=\frac{3}{4}$, then the value of $a$ is

If $\alpha , \beta \in C$ are the distinct roots of the equation ${x}^{2}-x+1=0,$ then ${\alpha }^{101}+{\beta }^{107}$ is equal to

If $\lambda \in \mathrm{R}$ is such that the sum of the cubes of the roots of the equation, $x^2+(2-\lambda) x+(10-\lambda)=0$ is minimum, then the magnitude of the difference of the roots of this equation is

If $\lambda \in R$ is such that the sum of the cubes of the roots of the equation ${x}^{2}+(2-\lambda )x+(10-\lambda )=0$ is minimum, then the magnitude of the difference of the roots of this equation is :

If $n$ is the degree of the polynomial, $$ \left[\frac{1}{\sqrt{5 x^3+1}-\sqrt{5 x^3-1}}\right]^8+\left[\frac{1}{\sqrt{5 x^3+1}+\sqrt{5 x^3-1}}\right]^8 $$ and $\mathrm{m}$ is the coefficient of $\mathrm{x}^{\mathrm{n}}$ in it, then the ordered pair $(\mathrm{n}, \mathrm{m})$ is equal to

If $n$ is the degree of the polynomial, ${[\frac{2}{\sqrt{5{x}^{3}+1}-\sqrt{5{x}^{3}-1}}]}^{8}+ {[\frac{2}{\sqrt{5{x}^{3}+1}+\sqrt{5{x}^{3}-1}}]}^{8}$ and $m$ is the coefficient of ${x}^{n}$ in it, then the ordered pair $(n,m)$ is equal to

If $b$ is the first term of an infinite geometric progression whose sum is five, then $b$ lies in the interval

If $b$ is the first term of an infinite G. P whose sum is five, then $b$ lies in the interval.

If the system of linear equations $x+ky+3z=0$ $3x+ky-2z=0$ $2x+4y-3z=0$ has a non-zero solution $(x, y, z),$ then $\frac{xz}{{y}^{2}}$ is equal to:

If the system of linear equations $$ \begin{aligned} &x+a y+z=3 \\ &x+2 y+2 z=6 \\ &x+5 y+3 z=b \end{aligned} $$ has no solution, then

If $f(x)=|\begin{matrix}\mathrm{cos}x & x & 1 \\ 2\mathrm{sin}x & {x}^{2} & 2x \\ \mathrm{tan}x & x & 1\end{matrix}|$, then $\underset{x\rightarrow 0}{\mathrm{lim}}⁡\frac{{f}^{'}(x)}{x}$

If $f(x)=\left|\begin{array}{ccc}\cos x & x & 1 \\ 2 \sin x & x^2 & 2 x \\ \tan x & x & 1\end{array}\right|$, then $\lim _{x \rightarrow 0} \frac{f^{\prime}(x)}{x}$

If $|z-3+2 i| \leq 4$ then the difference between the greatest value and the least value of $|z|$ is

If $|\begin{matrix}x-4 & 2x & 2x \\ 2x & x-4 & 2x \\ 2x & 2x & x-4\end{matrix}|=(A+Bx) {(x-A)}^{2},$ then the ordered pair $(A,B)$ is equal to

Let $p,q$ and $r$ be real numbers $(p\neq q,r\neq 0)$, such that the roots of the equation $\frac{1}{x+p}+\frac{1}{x+q}=\frac{1}{r}$ are equal in magnitude but opposite in sign, then the sum of squares of these roots is equal to

Let $A_n=\left(\frac{3}{4}\right)-\left(\frac{3}{4}\right)^2+\left(\frac{3}{4}\right)^3-\ldots+(-1)^{n-1}\left(\frac{3}{4}\right)^n$ and $B_n=1-A_n$. Then, the least odd natural number $p$, so that $B_n>A_n$, for all $n \geq p$ is

Let $A=[\begin{matrix}1 & 0 & 0 \\ 1 & 1 & 0 \\ 1 & 1 & 1\end{matrix}]$ and $B={A}^{20}$. Then the sum of the elements of the first column of $B$ is

Let $f: \mathrm{A} \rightarrow$ B be a function defined as $f(x)=$ $\frac{x-1}{x-2}$, where $A=R-\{2\}$ and $B=R-\{1\}$. Then $f$ is

Let $A$ be a matrix such that $A$. $\left[\begin{array}{ll}1 & 2 \\ 0 & 3\end{array}\right]$ is a scalar matrix and $|3 A|=108$. Then $A^2$ equals

Let $A$ be a matrix such that $A\cdot [\begin{matrix}1 & 2 \\ 0 & 3\end{matrix}]$ is a scalar matrix and $|3A|=108$. Then, ${A}^{2}$ equals :

Let ${a}_{1}, {a}_{2}, {a}_{3},\ldots \ldots ,{a}_{49}$ be in $A.P.$ such that $\Sigma _{ k⁡ = 0 }^{ 1 2 } {a}_{4k+1}=416$ and ${a}_{9}+{a}_{43}=66.$ If ${a}_{1}^{2}+{a}_{2}^{2}+\ldots +{a}_{17}^{2}=140m,$ then $m$ is equal to:

Let $S$ be the set of all real values of $k$ for which the system of linear equations $x+y+z=2$ $2x+y-z=3$ $3x+2y+kz=4$ has a unique solution. Then, $S$is :

Let $S$ be the set of all real values of $k$ for which the system of linear equations $$ \begin{aligned} &x+y+z=2 \\ &2 x+y-z=3 \\ &3 x+2 y+k z=4 \end{aligned} $$ has a unique solution. Then $S$ is

Let $A$ be the sum of the first $20$ terms and $B$ be the sum of the first $40$ terms of the series ${1}^{2}+{2\cdot 2}^{2}+{3}^{2}+{2\cdot 4}^{2}+{5}^{2}+{2\cdot 6}^{2}+\ldots$ If $B-2A=100\lambda ,$ then $\lambda$ is equal to :

Let $N$ denote the set of all natural numbers. Define two binary relations on $N$ as ${R}_{1}={(x,y)\in N\times N:2x+y=10}$ and ${R}_{2}={(x,y)\in N\times N:x+2y=10}$. Then

Let $\frac{1}{{x}_{1}},\frac{1}{{x}_{2}},\ldots ,\frac{1}{{x}_{n}}({x}_{i}\neq 0$ for $i=1,2,\ldots .,n)$ be in A.P. such that ${x}_{1}=4$ and ${x}_{21}=20$$.$ If $n$ is the least positive integer for which ${x}_{n}>50$, then $\sum _{i=1}^{n}(\frac{1}{{x}_{i}})$ is equal to

Let $S={x\in R :x\geq 0 & 2|\sqrt{x}-3|+\sqrt{x} (\sqrt{x}-6)+6=0}$ . Then $S$:

Suppose $A$ is any $3 \times 3$ non-singular matrix and $(A-3 I)(A-5 I)=O$, where $I=I_3$ and $O=O_3$. If $\alpha A+$ $\beta A^{-1}=4 I$, then $\alpha+\beta$ is equal to

The coefficient of $x^{10}$ in the expansion of $(1+x)^2$ $\left(1+x^2\right)^3\left(1+x^3\right)^4$ is equal to

The coefficient of ${x}^{2}$ in the expansion of the product $(2-{x}^{2}){{(1+2x+3{x}^{2})}^{6}+{(1-4{x}^{2})}^{6}}$ is

The least positive integer $n$ for which ${(\frac{1+i\sqrt{3}}{1-i\sqrt{3}})}^{n}=1$ is

The number of four letter words that can be formed using the letters of the word BARRACK is

The number of numbers between $2,000$ and $5,000$ that can be formed with the digits $0,1,2,3,4$ (repetition of digits is not allowed) and are multiple of $3$ is

The number of values of $k$ for which the system of linear equations $(k+2)x+10y=k&$ $kx+(k+3)y=k-1$ has no solution is

The set of all $\alpha \in R$, for which $w=\frac{1+(1-8 \alpha) z}{1-z}$ is a purely imaginary number, for all $z \in C$ satisfying $|z|=1$ and $\operatorname{Re} z \neq 1$, is

The set of all $\alpha \in R$, for which $w=\frac{1+(1-8\alpha )z}{1-z}$ is a purely imaginary number, for all $z\in C$ satisfying $|z|=1$ and $Re(z)\neq 1$, is :

The sum of the first $20$ terms of the series $1+\frac{3}{2}+\frac{7}{4}+\frac{15}{8}+\frac{31}{16}+\ldots$ is