JEE Main 2019 — Algebra

Let ${a}_{1}, {a}_{2}, \ldots , {a}_{30}$ be an A.P., $S= \sum _{i=1}^{30}{a}_{i}$ and $T= \sum _{i=1}^{15}{a}_{(2i-1)}.$ If ${a}_{5}=27$ and $S-2T=75,$ then ${a}_{10}$ is equal to:

If ${a}_{1},{a}_{2},{a}_{3},....$ are in A.P. such that ${a}_{1}+{a}_{7}+{a}_{16}=40,$ then the sum of the first $15$ terms of this A.P is:

The number of integral values of $m$ for which the quadratic expression $(1+2m) {x}^{2}-2(1+3m)x+4(1+m), x\in R$ is always positive, is

Let $p, q\in Q .$ If $2-\sqrt{3}$ is a root of the quadratic equation ${x}^{2}+px+q=0,$ then

The number of real roots of the equation $5+|{2}^{x}-1|={2}^{x}({2}^{x}-2)$ is :

If $[\begin{matrix}1 & 1 \\ 0 & 1\end{matrix}][\begin{matrix}1 & 2 \\ 0 & 1\end{matrix}][\begin{matrix}1 & 3 \\ 0 & 1\end{matrix}]\ldots .[\begin{matrix}1 & n-1 \\ 0 & 1\end{matrix}]=[\begin{matrix}1 & 78 \\ 0 & 1\end{matrix}],$ then the inverse of $[\begin{matrix}1 & n \\ 0 & 1\end{matrix}]$ is:

Let $\alpha$ and $\beta$ be the roots of the equation ${x}^{2}+x+1=0.$ Then for $y\neq 0$ in $R,|\begin{matrix}y+1 & \alpha & \beta \\ \alpha & y+\beta & 1 \\ \beta & 1 & y+\alpha \end{matrix}|$ is equal to

The total number of matrices $A=(\begin{matrix}0 & 2y & 1 \\ 2x & y & -1 \\ 2x & -y & 1\end{matrix}),(x,y\in R,x\neq y)$ for which ${A}^{T}A=3{I}_{3}$ is:

The sum of the real roots of the equation $|\begin{matrix}x & -6 & -1 \\ 2 & -3x & x-3 \\ -3 & 2x & x+2\end{matrix}|=0,$ is equal to:

If $A=[\begin{matrix}cos\theta & -sin\theta \\ sin\theta & cos\theta \end{matrix}]$ , then the matrix ${A}^{-50}$ when $\theta =\frac{\pi }{12},$ is equal to:

If $B=[\begin{matrix}5 & 2\alpha & 1 \\ 0 & 2 & 1 \\ \alpha & 3 & -1\end{matrix}]$ is the inverse of a $3\times 3$ matix $A,$ then the sum of all values of $\alpha$ for which $det(A)+1=0,$ is:

A value of $\theta \in (0,\frac{\pi }{3}),$ for which $|\begin{matrix}1+{cos}^{2}\theta & {sin}^{2}\theta & 4 cos6\theta \\ {cos}^{2}\theta & 1+{sin}^{2}\theta & 4 cos6\theta \\ {cos}^{2}\theta & {sin}^{2}\theta & 1+4 cos6\theta \end{matrix}|=0,$ is

Let $P=[\begin{matrix}1 & 0 & 0 \\ 3 & 1 & 0 \\ 9 & 3 & 1\end{matrix}]$ and $Q=[{q}_{ij}]$ be two $3\times 3$ matrices such that $Q-{P}^{5}={I}_{3}$. Then $\frac{{q}_{21}+{q}_{31}}{{q}_{32}}$ is equal to :

The coefficient of ${x}^{18}$ in the product $(1+x){(1-x)}^{10}{(1+x+{x}^{2})}^{9}$ is

The term independent of $x$ in the expansion of $(\frac{1}{60}-\frac{{x}^{8}}{81}).{(2{x}^{2}-\frac{3}{{x}^{2}})}^{6}$ is equal to

If the fourth term in the binomial expansion of ${(\sqrt{{x}^{\frac{1}{1+{log}_{10}x}}}+{x}^{\frac{1}{12}})}^{6}$ is equal to $200,$ and $x>1,$ then the value of $x$ is

The sum of the real values of $x$ for which the middle term in the binomial expansion of $\left(\frac{x^{3}}{3}+\frac{3}{x}\right)^{8}$ equals 5670 is :

Let $A=[\begin{matrix}\mathrm{cos}\alpha & -\mathrm{sin}\alpha \\ \mathrm{sin}\alpha & \mathrm{cos}\alpha \end{matrix}],(a\in R)$ such that ${A}^{32}=[\begin{matrix}0 & -1 \\ 1 & 0\end{matrix}].$ Then, a value of $\alpha$ is:

Let $A=[\begin{matrix}2 & b & 1 \\ b & {b}^{2}+1 & b \\ 1 & b & 2\end{matrix}],$ where $b>0$. Then the minimum value of $\frac{\mathrm{det}(A)}{b}$ is:

Let the numbers $2, b, c$ be in an A.P. and $A=[\begin{matrix}1 & 1 & 1 \\ 2 & b & c \\ 4 & {b}^{2} & {c}^{2}\end{matrix}]$ . If $det(A) \in [2,16],$ then $c$ lies in the interval:

Let ${z}_{1}$ and ${z}_{2}$ be any two non-zero complex numbers such that $3|{z}_{1}|=4|{z}_{2}|.$ If $z=\frac{3{z}_{1}}{2{z}_{2}}+\frac{2{z}_{2}}{3{z}_{1}}$ then maximum value of $|z|$ is Note: In actual paper value of $|z|$ was asked. Hence, none of the options given were correct. So we have modified the question as well as options.

Let ${a}_{1}, {a}_{2}, {a}_{3}\ldots ,{a}_{10}$ be in $G.P.$with ${a}_{i}>0$ for $i=1, 2, \ldots , 10$ and $S$ be the set of pairs $(r, k), r, k\in N$ (the set of natural numbers) for which $|\begin{matrix}{\mathrm{log}}_{e}⁡{a}_{1}^{r} {a}_{2}^{k} & {\mathrm{log}}_{e}⁡{a}_{2}^{r}{a}_{3}^{k} & {\mathrm{log}}_{e}⁡{a}_{3}^{r}{a}_{4}^{k} \\ {\mathrm{log}}_{e}⁡{a}_{4}^{r} {a}_{5}^{k} & {\mathrm{log}}_{e}⁡{a}_{5}^{r}{a}_{6}^{k} & {\mathrm{log}}_{e}⁡{a}_{6}^{r}{a}_{7}^{k} \\ {\mathrm{log}}_{e}⁡{a}_{7}^{r}{a}_{8}^{k} & {\mathrm{log}}_{e}⁡{a}_{8}^{r}{a}_{9}^{k} & {\mathrm{log}}_{e}⁡{a}_{9}^{r}{a}_{10}^{k}\end{matrix}|=0$ Then the number of elements in $S,$ is:

If $\alpha ,\beta$ and $\gamma$ are three consecutive terms of a non-constant G.P. Such that the equations $\alpha {x}^{2}+2\beta x+\gamma =0$ and ${x}^{2}+x-1=0$ have a common root, then $\alpha (\beta +\gamma )$ is equal to:

The sum of an infinite geometric series with positive terms is 3 and the sum of the cubes of its terms is $\frac{27}{19}$. Then the common ratio of this series is:

Let $Z$ be the set of integers. If $A={x\in Z :{2}^{(x+2)({x}^{2}-5x+6)}=1}$ and $B={x\in Z :- 3< 2x-1<9}$, then the number of subsets of the set $A\times B$, is :

If $\left|\begin{array}{ccc}a-b-c & 2 a & 2 a \\ 2 b & b-c-a & 2 b \\ 2 c & 2 c & c-a-b\end{array}\right|$ $=(a+b+c)(x+a+b+c)^{2}, x \neq 0$ and $a+b+c \neq 0,$ then $x$ is equal to

If $5, 5r, 5{r}^{2}$ are the lengths of the sides of a triangle, then $r$ can not be equal to:

If the system of linear equations $x-2y+kz=1$ $2x+y+z=2$ $3x-y-kz=3$ has a solution $(x,y,z),z\neq 0,$ then $(x,y)$ lies on the straight line whose equation is:

If $A=[\begin{matrix}{e}^{t} & {e}^{-t}cos t & {e}^{-t}\mathrm{sin}⁡ t \\ {e}^{t} & -{e}^{-t}\mathrm{cos}⁡t-{e}^{-t}\mathrm{sin}⁡t & -{e}^{-t}\mathrm{sin}⁡t+{e}^{-t}\mathrm{cos}⁡t \\ {e}^{t} & {2e}^{-t}\mathrm{sin}⁡t & -2{e}^{-t}\mathrm{cos}⁡t\end{matrix}],$ then $A$ is:

Let $z={(\frac{\sqrt{3}}{2}+\frac{i}{2})}^{5}+{(\frac{\sqrt{3}}{2}-\frac{i}{2})}^{5}.$ If $R(z)$ and $I(z)$ respectively denote the real and imaginary parts of $z,$ then

An ordered pair $(\alpha , \beta )$ for which the system of linear equations $(1+\alpha )x+\beta y+z=2$ $\alpha x+(1+\beta )y+z=3$ $\alpha x+\beta y+2z=2$ has a unique solution, is :

The system of linear equations $x+y+z=2$ $2x+3y+2z=5$ $2x+3y+({a}^{2}-1)z=a+1$

Let $A=\left(\begin{array}{ccc}0 & 2 q & r \\ p & q & -r \\ p & -q & r\end{array}\right)$. If $\mathrm{AA}^{\mathrm{T}}=\mathrm{I}_{3},$ then $|\mathrm{p}|$ is:

If the system of equations $2x+3y-z=0, x+ky-2z=0$ and $2x-y+z=0$ has a non-trivial solution $(x,y,z),$ then $\frac{x}{y}+\frac{y}{z}+\frac{z}{x}+k$ is equal to

The number of values of $\theta \in (0, \pi )$ for which the system of linear equations $x+3y+7z=0$ $-x+4y+7z=0$ $(\mathrm{sin}⁡3\theta )x+(\mathrm{cos}⁡2\theta )y+2z=0$ has a non-trivial solution, is:

If the system of equations $x+y+z=5,$ $x+2y+3z=9,$ $x+3y+\alpha z=\beta$ has inifinitely many solutions, then $\beta -\alpha$ equals

The sum of the solutions of the equation $|\sqrt{x}-2|+\sqrt{x}(\sqrt{x}-4)+2=0,(x>0)$ is equal to

Let $A=$ { $x\in R:x$ is not a positive integer} $.$ Define a function $f:A\rightarrow R$ as $f(x)=\frac{2x}{x-1}$ , then $f$ is:

If the system of linear equations $2 x+2 y+3 z=a$ $3 x-y+5 z=b$ $x-3 y+2 z=c$ where, $a, b,$ care non-zero real numbers, has more than onc solution, then

If the fourth term in the Binomial expansion of ${(\frac{2}{x}+{x}^{{log}_{8}x})}^{6},(x>0)$ is $20\times {8}^{7},$ then a value of $x$ is

If the coefficients of ${x}^{2}$ and ${x}^{3}$, are both zero, in the expansion of the expression $(1+ax+b{x}^{2}){(1-3x)}^{15}$, in powers of $x$ , then the ordered pair $(a,b)$ is equal to

Let $f(x)={a}^{x} (a>0)$ be written as $f(x)={f}_{1}(x)+{f}_{2}(x),$ where ${f}_{1}(x)$ is an even function and ${f}_{2}(x)$ is an odd function. Then ${f}_{1}(x+y)+{f}_{1}(x-y)$ equals:

If $[x]$ denotes the greatest integer $\leq x,$ then the system of linear equations $[sin\theta ]x+[-cos\theta ]y=0$, $[cot\theta ]x+y=0$

If the system of linear equations $x+y+z=5$, $x+2y+2z=6$, $x+3y+\lambda z=\mu , (\lambda , \mu \in R)$ , has infinitely many solutions, then the value of $\lambda +\mu$ is:

Let $a,b$ and $c$ be in $G.P.$ with common ratio $r,$ where $a\neq 0$ and $0<r\leq \frac{1}{2}.$ If $3a,7b$ and $15c$ are the first three terms of an $A.P.,$ then the ${4}^{th}$ term of this $A.P.$ is :

If some three consecutive coefficients in the binomial expansion of ${(x+1)}^{n}$ in powers of $x$ are in the ratio $2:15:70,$ then the average of these three coefficients is:

If $\lambda$ be the ratio of the roots of the quadratic equation in $x, 3{m}^{2}{x}^{2}+m(m-4)x+2=0$, then the least value of $m$ for which $\lambda +\frac{1}{\lambda }=1$, is :

Let $N$ be the set of natural numbers and two functions $f$ and $g$ be defined as $f,g:N\rightarrow N$ such that $f(n)={\begin{matrix}\frac{n+1}{2}, if n is odd \\ \frac{n}{2}, if n is even\end{matrix}$ and $g(n)=n-{(-1)}^{n}.$ Then $fog$ is:

Let $f(x)={\mathrm{log}}_{e}(sinx), (0<x<\pi )$ and $g(x)={\mathrm{sin}}^{-1}({e}^{-x}), (x\geq 0).$ If $\alpha$ is a positive real number such that $a={(fog)}^{'}(\alpha )$ and $b=(fog)(\alpha ),$ then

The number of functions $f$ from $\{1,2,3, \ldots, 20\}$ onto $\{1,2,3, \ldots, 20\}$ such that $f(k)$ is a multiple of $3,$ whenever $k$ is a multiple of 4 is:

Let ${z}_{1}$ and ${z}_{2}$ be two complex numbers satisfying $|{z}_{1}|=9$ and $|{z}_{2}-3-4i|=4$. Then the minimum value of $|{z}_{1}-{z}_{2}|$ is :

If the third term in the binomial expansion of ${(1+{x}^{{\mathrm{log}}_{2}x})}^{5}$ equals $2560,$ then a possible value of $x$ is

In a class of $140$ students numbered $1$ to$140$ , all even numbered students opted Mathematics course, those whose number is divisible by $3$ opted Physics course and those whose number is divisible by $5$ opted Chemistry course. Then the number of students who did not opt for any of the three courses is:

Let $a_{1}, a_{2}, \ldots, a_{10}$ be a G.P. If $\frac{a_{3}}{a_{1}}=25,$ then $\frac{a_{9}}{a_{5}}$ equals :

If 19 th term of a non-zero A.P. is zero, then its (49th term): (29th term) is:

For $x\in R-{0, 1},$ let ${f}_{1}(x)=\frac{1}{x},{f}_{2}(x)=1-x$ and ${f}_{3}(x)=\frac{1}{1-x}$ be three given functions. If a function, $J(x)$ satisfies $({f}_{2}oJo{f}_{1})(x)={f}_{3}(x)$ then $J(x)$ is equal to:

If $f(x)={\mathrm{log}}_{e}(\frac{1-x}{1+x}), |x|< 1,$ then $f(\frac{2x}{1+{x}^{2}})$ is equal to

If $\alpha$ and $\beta$ are the roots of the quadratic equation ${x}^{2}+xsin\theta -2sin\theta =0, \theta \in (0,\frac{\pi }{2})$ , then $\frac{{\alpha }^{12}+{\beta }^{12}}{({\alpha }^{-12}+{\beta }^{-12}){.(\alpha -\beta )}^{24}}$ is equal to :

If $a, b$ and $c$ be three distinct real numbers in G.P. and $a+b+c=xb,$ then $x$ cannot be:

Let a function $f:(0, \infty) \rightarrow(0, \infty)$ be defined by $f(x)=\left|1-\frac{1}{x}\right| .$ Then $f$ is :

If ${}^{n}{C}_{4},{ }^{n}{C}_{5}$ and ${}^{n}{C}_{6}$ are in A.P., then $n$ can be

The product of three consecutive terms of a $G.P.$ is $512$. If $4$ is added to each of the first and the second of these terms, the three terms now form an $A.P.$, then the sum of the original three terms of the given $G.P.$ is :

If $A$ is a symmetric matrix and $B$ is skew- symmetric matrix such that $A+B=[\begin{matrix}2 & 3 \\ 5 & -1\end{matrix}]$ , then $AB$ is equal to:

The coefficient of ${t}^{4}$ in the expansion of ${(\frac{1-{t}^{6}}{1-t})}^{3}$ is

Let $d\in R,$ and $A=[\begin{matrix}-2 & 4+d & (\mathrm{sin}\theta )-2 \\ 1 & (\mathrm{sin}\theta )+2 & d \\ 5 & (2\mathrm{sin}\theta )-d & (-\mathrm{sin}\theta )+2+2d\end{matrix}],$ $\theta \in [0, 2\pi ].$ If the minimum value of $det(A)$ is $8,$ then a value of $d$ is:

The number of integral values of $m$ for which the equation, $(1+{m}^{2}){x}^{2}-2(1+3m)x+(1+8m)=0$ has no real root, is

Let ${S}_{n}$ denote the sum of the first $n$ terms of an $A.P.$. If ${S}_{4}=16$ and ${S}_{6}=-48$ , then ${S}_{10}$ is equal to:

If $m$ is chosen in the quadratic equation $({m}^{2}+1){x}^{2}-3x+{({m}^{2}+1)}^{2}=0$ such that the sum of its roots is greatest, then the absolute difference of the cubes of its roots is:

If three distinct numbers $a, b, c$ are in G.P. and the equations $a{x}^{2}+2bx+c=0$ and $d{x}^{2}+2ex+f=0$ have a common root, then which one of the following statements is correct?

The value of $\lambda$ such that sum of the squares of the roots of the quadratic equation, ${x}^{2}+(3-\lambda ) x+2=\lambda$ has the least value is:

The number of all possible positive integral value of $\alpha$ for which the roots of the quadratic equation $6{x}^{2}-11x+\alpha =0$ are rational numbers is:

Let $\alpha$ and $\beta$ be the roots of the equation ${x}^{2}+2x+2=0,$ then ${\alpha }^{15}+{\beta }^{15}$ is equal to

All the points in the set $S={\frac{\alpha +i}{\alpha -i},\alpha \in R},i=\sqrt{-1}$ lie on a

If $z$ and $\omega$ are two complex numbers such that $|z\omega |=1$ and $arg(z)-arg(\omega )=\frac{\pi }{2}$, then:

Let $z$ be a complex number such that $|z|+z=3+i$ $($ where $i=\sqrt{-1})$ Then $|\mathrm{z}|$ is equal to :

If $\alpha$ and $\beta$ be the roots of the equation ${x}^{2}-2x+2=0,$ then the least value of $n$ for which ${(\frac{\alpha }{\beta })}^{n}=1$ is

Let $A={\theta \in (-\frac{\pi }{2},\pi ):\frac{3+2isin\theta }{1-2i sin\theta } is purely imaginary }.$ Then the sum of the elements in $A$ is:

If $z=\frac{\sqrt{3}}{2}+\frac{i}{2} (i=\sqrt{-1}),$ then ${(1+iz+{z}^{5}+i{z}^{8})}^{9}$ is equal to:

Suppose that $20$ pillars of the same height have been erected along the boundary of circular stadium. If the top of each pillar has been connected by beams with the top of all its non-adjacent pillars, then the total number of beams is:

The number of four-digit numbers strictly greater than $4321$ that can be formed using the digit $0,1,2,3,4,5$ (repetition of digits is allowed) is:

Consider three boxes, each containing $10$ balls labelled $1, 2, \ldots ., 10$. Suppose one ball is randomly drawn from each of the boxes. Denote by ${n}_{i}$, the label of the ball drawn from the ${i}^{th}$ box, $(i=1, 2, 3)$. Then, the number of ways in which the balls can be chosen such that ${n}_{1}<{n}_{2}<{n}_{3}$ is :

The number of natural numbers less than $7000$ which can be formed by using the digits $0, 1, 3, 7, 9$ (repetition of digits allowed) is equal to:

Consider a class of $5$ girls and $7$ boys. The number of different teams consisting of $2$ girls and $3$ boys that can be formed from this class, if there are two specific boys $A$ and $B,$ who refuse to be the members of the same team, is:

If the sum and product of the first three terms in an $A.P.$ are $33$ and $1155$, respectively, then a value of its ${11}^{th}$ term is:

Let ${S}_{k}=\frac{1 + 2 + 3+\ldots +k}{k}$. If ${S}_{1}^{2}+{S}_{2}^{2}+\ldots +{S}_{10}^{2}=\frac{5}{12}A$, then $A$ is equal to :

If $si{n}^{4}\alpha +4co{s}^{4}\beta +2=4\sqrt{2}sin\alpha cos\beta$, $\alpha ,\beta \in [0,\pi ]$, then $cos(\alpha +\beta )-cos(\alpha -\beta )$ is equal to

Let $a, b$ and $c$ be the ${7}^{th}, {11}^{th}$ and ${13}^{th}$ terms respectively of a non-constant A.P. . If these are also the three consecutive terms of a G.P. , then $\frac{a}{c}$ is equal to:

The sum of the following series $1+6+\frac{9({1}^{2}+{2}^{2}+{3}^{2})}{7}+\frac{12({1}^{2}+{2}^{2}+{3}^{2}+{4}^{2})}{9}+\frac{15({1}^{2}+{2}^{2}+\ldots +{5}^{2})}{11}+....$ up to $15$ terms, is:

For $x \epsilon (0,\frac{3}{2}),$ let $f(x)=\sqrt{x},g(x)=tanx$ and $h(x)=\frac{1-{x}^{2}}{1+{x}^{2}}$ . If $\phi (x)=(hof)og)(x),$ then $\phi (\frac{\pi }{3})$ is equal to:

The total number of irrational terms in the binomial expansion of ${({7}^{\frac{1}{5}}-{3}^{\frac{1}{10}})}^{60}$ is

The sum of the series $1+2\times 3+3\times 5+4\times 7+\ldots$ upto ${11}^{th}$ term is:

If the system of linear equations $x-4y+7z=g$; $3y-5z=h$; $-2x+5y-9z=k$ is consistent, then:

Let $\lambda$ be a real number for which the system of linear equations $x+y+z=6,$ $4x+\lambda y-\lambda z=\lambda -2$ and $3x+2y-4z=-5$ has infinitely many solutions. Then $\lambda$ is a root of the quadratic equation:

Let $\alpha$ and $\beta$ be the roots of the quadratic equation $x^{2} \sin \theta-x(\sin \theta \cos \theta+1)+\cos \theta=0\left(0 < \theta < 45^{\circ}\right),$ and $\alpha < \beta .$ Then $\sum_{n=0}^{\infty}\left(\alpha^{n}+\frac{(-1)^{n}}{\beta^{n}}\right)$ is equal to :

If $a>0$ and $z=\frac{{(1+i)}^{2}}{a-i}$ , has magnitude $\sqrt{\frac{2}{5}}$ , then $\overset{-}{z}$ is equal to:

If three of the six vertices of a regular hexagon are chosen at random, then the probability that the triangle formed with these chosen vertices is equilateral is:

The equation $|z-i|=|z-1|,i=\sqrt{-1},$ represents:

Let $f(x)={x}^{2}, x\in R$ . For any $A\subseteq R,$ define $g(A)={x\in R :f(x)\in A}$ . If $S=[0, 4]$ , then which one of the following statements is not true?

If $\alpha$and $\beta$ are the roots of the equation $375 {x}^{2}-25x-2=0,$ then $\underset{n\rightarrow \infty }{lim}\sum _{r=1}^{n}{\alpha }^{r}+\underset{n\rightarrow \infty }{lim}\sum _{r=1}^{n}{\beta }^{r}$ is equal to:

The sum of all two digit positive numbers which when divided by $7$ yield $2$ or $5$ as remainder is

Let ${z}_{0}$ be a root of quadratic equation, ${x}^{2}+x+1=0.$ If $z=3+6i{z}_{0}^{81}-3i{z}_{0}^{93}$ , then $arg$ $(z)$ is equal to:

The set of all values of $\lambda$ for which the system of linear equations $x-2y-2z=\lambda x$ $x+2y+z=\lambda y$ $-x-y=\lambda z$ has a non-trivial solution :

There are $m$ men and two women participating in a chess tournament. Each participant plays two games with every other participant. If the number of games played by the men between themselves exceeds the number of games played between the men and the women by $84$, then the value of $m$ is :

Let $f: R \rightarrow R$ be defined by $f(x)=\frac{x}{1+x^{2}}, x \in R .$ Then the range of $f$ is

Let $z\in C$ be such that $|z|<1.$ If $\omega =\frac{5+3z}{5(1-z)},$ then:

Let $S={1, 2, 3,\ldots .,100}$, then number of non-empty subsets $A$ of $S$ such that the product of elements in $A$ is even is :

The Number of ways of choosing $10$ objects out of $31$ objects of which $10$ are identical and the remaining $21$ are distinct, is:

If the fractional part of the number $\frac{{2}^{403}}{15}$ is $\frac{k}{15},$ then $k$ is equal to

A committee of $11$ member is to be formed from $8$ males and $5$ females. If $m$ is the number of ways the committee is formed with at least $6$ males and $n$ is the number of ways the committee is formed with at least $3$ females, then:

The domain of the definition of the function $f(x)=\frac{1}{4-{x}^{2}}+{log}_{10}({x}^{3}-x)$ is:

Let ${a}_{1},{a}_{2},{a}_{3}...$ be an $A.P.$ with ${a}_{6}=2.$ Then, the common difference of this $A.P.,$ which maximise the product ${a}_{1}\cdot {a}_{4}\cdot {a}_{5},$is :

If $A=[\begin{matrix}1 & \mathrm{sin}\theta & 1 \\ -\mathrm{sin}\theta & 1 & \mathrm{sin}\theta \\ -1 & -\mathrm{sin}\theta & 1\end{matrix}]$, then for all $\theta \in (\frac{3\pi }{4},\frac{5\pi }{4})$, $\mathrm{det}(A)$ lies in the interval :

Let $\left(-2-\frac{1}{3} i\right)^{3}=\frac{x+i y}{27}(i=\sqrt{-1}),$ where $x$ and $y$ are real numbers then $\mathrm{y}-\mathrm{x}$ equals

If the function $f:R-{1, -1}\rightarrow A$ defined by $f(x)=\frac{{x}^{2}}{1-{x}^{2}},$ is surjective, then $A$ is equal to

If the sum of the first $15$ terms of the series ${(\frac{3}{4})}^{3}+{(1\frac{1}{2})}^{3}+{(2\frac{1}{4})}^{3}+{3}^{3}+{(3\frac{3}{4})}_{ }^{3}+\ldots$ is equal to $225K$, then $K$ is equal to :

All possible numbers are formed using the digits $1, 1, 2, 2, 2, 2, 3, 4, 4$ taken all at a time. The number of such numbers in which the odd digits occupy even places is

If both the roots of the quadratic equation ${x}^{2}-mx+4=0$ are real and distinct and they lie in the interval $(1, 5),$ then $m$ lies in the interval: Note: In the actual JEE paper interval was $[1, 5]$

A group of students comprises of $5$ boys and n girls. If the number of ways, in which a team of $3$ students can randomly be selected from this group such that there is at least one boy and at least one girl in each team, is $1750,$ then n is equal to

If ${\Delta }_{1}=|\begin{matrix}x & sin\theta & cos\theta \\ -sin\theta & -x & 1 \\ cos\theta & 1 & x\end{matrix}|$ and ${\Delta }_{2}=|\begin{matrix}x & sin2\theta & cos2\theta \\ -sin2\theta & -x & 1 \\ cos2\theta & 1 & x\end{matrix}|,$ $x\neq 0;$ then for all $\theta \in (0,\frac{\pi }{2})$ :

A ratio of the ${5}^{th}$ term from the beginning to the ${5}^{th}$ term from the end in the binomial expansion of ${({2}^{\frac{1}{3}}+\frac{1}{2{(3)}^{\frac{1}{3}}})}^{10}$ is

The smallest natural number $n$ , such that the coefficient of $x$ in the expansion of ${({x}^{2}+\frac{1}{{x}^{3}})}^{n}$ is $C23 n$ , is

Let $A, B$ and $C$ be sets such that $\phi \neq A\cap B\subseteq C.$ Then which of the following statements is not true?

Consider the quadratic equation $(c-5){x}^{2}-2cx+(c-4)=0,c\neq 5.$ Let $S$ be the set of all integral values of $c$ for which one root of the equation lies in the interval $(0, 2)$ and its other root lies in the interval $(2, 3).$ Then the number of elements in $S$ is

Let the sum of the first $n$ terms of a non-constant $A.P.,{a}_{1},{a}_{2},{a}_{3},....,{a}_{n}$ be $50n+\frac{n(n-7)}{2}A,$ where $A$ is a constant. If $d$ is the common difference of this $A.P.$, then the ordered pair $(d,{a}_{50})$ is equal to

Let $A$ and $B$ be two invertible matrices of order $3 \times 3$. If $\operatorname{det}\left(\mathrm{ABA}^{\mathrm{T}}\right)=8$ and det $\left(\mathrm{AB}^{-1}\right)=8$, then det $\left(\mathrm{BA}^{-1} \mathrm{~B}^{\mathrm{T}}\right)$ is equal to

If ${a}_{1},{a}_{2},{a}_{3}.........,{a}_{n}$ are in $A.P.$ and ${a}_{1}+{a}_{4}+{a}_{7}.........+{a}_{16}=114$ , then ${a}_{1}+{a}_{6}+{a}_{11}+{a}_{16}$ is equal to :

The sum of the co-efficient of all even degree terms in $x$ in the expansion of ${(x+\sqrt{{x}^{3}-1})}^{6}{+(x-\sqrt{{x}^{3}-1})}^{6},(x>1)$ is equal to

Let $z\in C$ with $Im(z)=10$ and it satisfies $\frac{2 z-n}{2 z+n}=2i-1$ for some natural number $n.$ Then

Some identical balls are arranged in rows to form an equilateral triangle. The first row consists of one ball, the second row consists of two balls and so on. If $99$ more identical balls are added to the total number of balls used in forming the equilateral triangle, then all these balls can be arranged in a square, whose each side contains exactly $2$ balls less than the number of balls each side of the triangle contains. Then the number of balls used to form the equilateral triangle is

The sum of all natural numbers $n$ such that $100<n<200$ and $H.C.F.$ $(91,n)>1$ is

If $\frac{z - \alpha }{z + \alpha }(\alpha \in R)$ is a purely imaginary number and $|z|=2$, then a value of $\alpha$ is :

Let $f:[0,1]\rightarrow R$ be such that $f(xy)=f(x).f(y),$ for all $x,y\in [0,1],$ and $f(0)\neq 0.$ If $y=y(x)$ satisfies the differential equation, $\frac{dy}{dx}=f(x)$ with $y(0)=1$ then $y(\frac{1}{4})+y(\frac{3}{4})$ is equal to:

The number of$6$ digit number that can be formed using the digits $0, 1, 2, 5, 7$ and $9$ which are divisible by $11$ and no digit is repeated is:

The greatest value of $c\in R$ for which the system of linear equations $x-cy-cz=0$, $cx-y+cz=0$, $cx+cy-z=0$ has a non-trivial solution, is

For $x\in R,$ Let $[x]$ denotes the greatest integer $\leq x,$then the sum of the series $[-\frac{1}{3}]+[-\frac{1}{3}-\frac{1}{100}]+[-\frac{1}{3}-\frac{2}{100}]+.....+[-\frac{1}{3}-\frac{99}{100}]$ is

If one real root of the quadratic equation $81 x^{2}+k x+256=0$ is cube of the other root, then a value of $\mathrm{k}$ is :