Algebra PYQ — Page 31

Let ${x}_{1},{x}_{2},\ldots ,{x}_{100}$ be in an arithmetic progression, with ${x}_{1}=2$ and their mean equal to $200$ . If ${y}_{i}=i({x}_{i}-i),1\leq i\leq 100$, then the mean of ${y}_{1},{y}_{2},\ldots$, ${y}_{100}$ is

Let $A={1,2,3,4,5}$ and $B={1,2,3,4,5,6}$. Then the number of functions $f:A\rightarrow B$ satisfying $f(1)+f(2)=f(4)-1$ is equal to........

If the domain of the function $f(x)=\frac{[x]}{1+{x}^{2}}$, where $[x]$ is greatest integer $\leq x$, is $[2,6)$, then its range is

Let $[t]$ denote the greatest integer $\leq t$. if the constant term in the expansion of ${(3{x}^{2}-\frac{1}{2{x}^{5}})}^{7}$ is $\alpha$ then $[\alpha ]$ is equal to $_____$

The number of elements in the set ${n\in \mathbb{N}:10\leq n\leq 100$ and ${3}^{n}-3$ is a multiple of $7}$ is _______.

For $\alpha ,\beta ,z\in \mathbb{C}$ and $\lambda >1$, if $\sqrt{\lambda -1}$ is the radius of the circle ${|z-\alpha |}^{2}+{|z-\beta |}^{2}=2\lambda ,$ then $|\alpha -\beta |$ is equal to _____.

The remainder when ${(2023)}^{2023}$ is divided by $35$ is

The number of elements in the set ${n\in \mathbb{Z}:|{n}^{2}-10n+19|<6}$ is _______ .

The number of integers, greater than $7000$ that can be formed, using the digits $3,5,6,7,8$ without repetition is

The number of symmetric matrices of order 3, with all the entries from the set ${0,1,2,3,4,5,6,7,8,9}$ is

If $\frac{1}{n+1}{}^{n}{C}_{n}+\frac{1}{n}{}^{n}{C}_{n-1}+...+\frac{1}{2}{}^{n}{C}_{1}{+}^{n}{C}_{0}=\frac{1023}{10}$ then $n$ is equal to

Suppose $f:R\rightarrow (0,\infty )$ be a differentiable function such that $5f(x+y)=f(x)\cdot f(y),\forall x,y\in R$, If $f(3)=320$, then $\sum _{n=0}^{5}f(n)$ 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 the sum of first n terms of an AP is 3n² + 5n, then the common difference is:

If α and β are roots of x² - 5x + 6 = 0, then α³ + β³ is equal to:

If $a$ and $b$ are the roots of the equation ${x}^{2}-7x-1=0$, then the value of $\frac{{a}^{21}+{b}^{21}+{a}^{17}+{b}^{17}}{{a}^{19}+{b}^{19}}$ is equal to

The number of relations, on the set ${1,2,3}$ containing $(1,2)$ and $(2,3)$ which are reflexive and transitive but not symmetric, is _________.

If $f(x)={x}^{3}-{x}^{2}{f}^{'}(1)+x{f}^{"}(2)-{f}^{'''}(3)$, $x\in R$, then

For the two positive numbers $a,b$, if $a,b$ and $\frac{1}{18}$ are in a geometric progression, while $\frac{1}{a},10$ and $\frac{1}{b}$ are in an arithmetic progression, then, $16a+12b$ is equal to _____ .

The number of arrangements of the letters of the word "INDEPENDENCE" in which all the vowels always occur together is

If $P$ is a $3\times 3$ real matrix such that ${P}^{T}=aP+(a-1)I$, where $a>1$, then

Let $f:R-{2,6}\rightarrow R$ be real valued function defined as $f(x)=\frac{x+2x+1}{{x}^{2}-8x+12}$. Then range of $f$ is

The number of ways of selecting two numbers $a$ and $b$, $a\in {2,4,6,\ldots \ldots ,100}$ and $b\in {1,3,5,\ldots \ldots ,99}$ such that $2$ is the remainder when $a+b$ is divided by $23$ is

If a point $P(\alpha ,\beta ,\gamma )$ satisfying $(\begin{matrix}\alpha & \beta & \gamma \end{matrix})(\begin{matrix}2 & 10 & 8 \\ 9 & 3 & 8 \\ 8 & 4 & 8\end{matrix})=(\begin{matrix}0 & 0 & 0\end{matrix})$ lies on the plane $2x+4y+3z=5$, then $6\alpha +9\beta +7\gamma$ is equal to