Algebra PYQ — Page 55

The value of log₂8 is:

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 :

Let ${R}_{1}$ and ${R}_{2}$ be two relations defined as follows :${R}_{1}={(a,b)\in {R}^{2}:{a}^{2}+{b}^{2}\in Q}$ and ${R}_{2}={(a,b)\in {R}^{2}:{a}^{2}+{b}^{2}\notin Q}$, where $Q$ is the set of all rational numbers, then

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.....

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 number of real roots of the equation, ${e}^{4x}+{e}^{3x}-4{e}^{2x}+{e}^{x}+1=0$ is:

The minimum value of ${2}^{\mathrm{sin}x}+{2}^{\mathrm{cos}x}$ is :

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 :

If the first term of an $A.P.$ is $3$ and the sum of its first $25$ terms is equal to the sum of its next $15$ terms, then the common difference of this $A.P.$is

The number of terms common to the two A.P.’s $3,7,11,\ldots ,407$ and $2,9,16,\ldots ,709$ is ____________.

Let $[t]$ denote the greatest integer$\leq t$. Then the equation in $x,{[x]}^{2}+2[x+2]-7=0$ has :

The region represented by ${z=x+iy\in C:|z|-Re(z)\leq 1}$ is also given by the inequality

If the number of five digit numbers with distinct digits and $2$ at the ${10}^{th}$ place is $336k$ , then $k$ 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 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

If $\frac{3+isin\theta }{4-icos\theta },\theta \in [0,2\pi ],$ is a real number, then an argument of $sin\theta +icos\theta$ is

If $a,b$ and $c$ are the greatest values of ${C}_{p}19,{C}_{q}20$ and ${C}_{r}21$ respectively, then:

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

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

Suppose that a function $f:R\rightarrow R$ satisfies $f(x+y)=f(x)f(y)$ for all $x,y\epsilon R$ and $f(1)=3$. If $\sum _{i=1}^{n}f(i)=363$, then $n$ is equal to ..... .

The number of words, with or without meaning, that can be formed by taking 4 letters at a time from the letters of the word 'SYLLABUS' such that two letters are distinct and two letters are alike, is

If for some $\alpha$ and $\beta$ in $R$ , the intersection of the following three planes $x+4y-2z=1$ $x+7y-5z=\beta$ $x+5y+\alpha z=5$ is a line in ${R}^{3}$ , then $\alpha +\beta$ is equal to: