Algebra PYQ — Page 55

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