Algebra PYQ — Page 37

The probability that a randomly chosen $2\times 2$ matrix with all the entries from the set of first $10$ primes, is singular, is equal to

The positive value of the determinant of the matrix $A$, whose $Adj(Adj(A))=[\begin{matrix}14 & 28 & -14 \\ -14 & 14 & 28 \\ 28 & -14 & 14\end{matrix}]$, is ______.

The ordered pair $(a,b)$, for which the system of linear equations $3x-2y+z=b$ $5x-8y+9z=3$ $2x+y+az=-1$ has no solution, is

The number of ways to distribute $30$ identical candies among four children ${C}_{1},{C}_{2},{C}_{3}$ and ${C}_{4}$ so that ${C}_{2}$ receives atleast $4$ and atmost $7$ candies, ${C}_{3}$ receives atleast $2$ and atmost $6$ candies, is equal to

The number of ways, $16$ identical cubes, of which $11$ are blue and rest are red, can be placed in a row so that between any two red cubes there should be at least $2$ blue cubes, is ______.

The number of values of $\alpha$ for which the system of equations $x+y+z=\alpha$ $\alpha x+2\alpha y+3z=-1$ $x+3\alpha y+5z=4$ is inconsistent, is

The number of real values of $\lambda$, such that the system of linear equations $2x-3y+5z=9$ $x+3y-z=-18$ $3x-y+({\lambda }^{2}-|\lambda |)z=16$ has no solutions, is

The number of real solutions of the equation ${e}^{4x}+4{e}^{3x}-58{e}^{2x}+4{e}^{x}+1=0$ is _____.

The number of positive integers $k$ such that the constant term in the binomial expansion of ${(2{x}^{3}+\frac{3}{{x}^{k}})}^{12},x\neq 0$ is ${2}^{8}\cdot l$, where $l$ is an odd integer, is ______.

The number of points of intersection $|z-(4+3i)|=2|$ and $|z|+|z-4|=6,z\in C$ is

The number of one-one functions $f:{a,b,c,d}\rightarrow {0,1,2,\ldots ,10}$ such that $2f(a)-f(b)+3f(c)+f(d)=0$ is _____

The number of natural numbers lying between $1012$ and $23421$ that can be formed using the digits $2,3,4,5,6$ (repetition of digits is not allowed) and divisible by $55$ is _____.

The number of matrices $A=[\begin{matrix}a & b \\ c & d\end{matrix}]$, where $a,b,c,d\in {-1,0,1,2,3,\ldots \ldots ,10}$, such that $A={A}^{-1}$, is ______.

The number of matrices of order $3\times 3$, whose entries are either $0$ or $1$ and the sum of all the entries is a prime number, is _______.

The number of functions $f$, from the set $A={x\in N:{x}^{2}-10x+9\leq 0}$ to the set $B={{n}^{2}:n\in N}$ such that $f(x)\leq {(x-3)}^{2}+1$, for every $x\in A$, is _______.

The number of $\theta \in (0,4\pi )$ for which the system of linear equations $3(\mathrm{sin}3\theta )x-y+z=2$ $3(\mathrm{cos}2\theta )x+4y+3z=3$ $6x+7y+7z=9$ has no solution is

The number of elements in the set {$z=a+ib\in \mathbb{C}:a,b\in \mathbb{Z}$ and $1<|z-3+2i|<4$} is _____.

The number of bijective function $f(1,3,5,7,\cdots ,99)\rightarrow (2,4,6,8,\cdots ,100)$ if $f(3)>f(5)>f(7)\cdots >f(99)$ is

The number of $3$-digit odd numbers, whose sum of digits is a multiple of $7$, is _____.

The number of $7$-digit numbers which are multiples of $11$ and are formed using all the digits $1,2,3,4,5,7$ and $9$ is _____.

The number of $5$-digit natural numbers, such that the product of their digits is $36$, is

The minimum value of the sum of the squares of the roots of ${x}^{2}+(3-a)x=2a-1$ is

The letters of the word 'MANKIND' are written in all possible orders and arranged in serial order as in an English dictionary. Then the serial number of the word 'MANKIND' is _____ .

The greatest integer less than or equal to the sum of first $100$ terms of the sequence $\frac{1}{3},\frac{5}{9},\frac{19}{27},\frac{65}{81},\ldots$ is equal to ______