Algebra PYQ — Page 47

If $1,{\mathrm{log}}_{10}({4}^{x}-2)$ and ${\mathrm{log}}_{10}({4}^{x}+\frac{18}{5})$ are in arithmetic progression for a real number $x$ then the value of the determinant $|\begin{matrix}2(x-\frac{1}{2}) & x-1 & {x}^{2} \\ 1 & 0 & x \\ x & 1 & 0\end{matrix}|$ is equal to:

In an increasing geometric series, the sum of the second and the sixth term is $\frac{25}{2}$ and the product of the third and fifth term is $25.$ Then, the sum of ${4}^{th},{6}^{th}$ and ${8}^{th}$ terms is equal to:

A man is walking on a straight line. The arithmetic mean of the reciprocals of the intercepts of this line on the coordinate axes is $\frac{1}{4}$. Three stones $A,B$ and $C$ are placed at the points $(1,1),(2,2)$ and $(4,4)$ respectively. Then which of these stones is$/$are on the path of the man?

If the sum of the coefficients in the expansion of $(x+y{)}^{n}$ is $4096,$ then the greatest coefficient in the expansion is _____.

Consider an arithmetic series and a geometric series having four initial terms from the set ${11,8,21,16,26,32,4}$. If the last terms of these series are the maximum possible four digit numbers, then the number of common terms in these two series is equal to _______.

$3\times {7}^{22}+2\times {10}^{22}-44$ when divided by $18$ leaves the remainder

If $f(x)$ and $g(x)$ are two polynomials such that the polynomial $P(x)=f({x}^{3})+xg({x}^{3})$ is divisible by ${x}^{2}+x+1$, then $P(1)$ is equal to ___ .

Consider the following system of equations: $x+2y-3z=a$ $2x+6y-11z=b$ $x-2y+7z=c$ where $a,b$ and $c$ are real constants. Then the system of equations :

The total number of $4$-digit numbers whose greatest common divisor with $18$ is $3$ is _____.

The values of $\lambda$ and $\mu$ such that the system of equations $x+y+z=6,3x+5y+5z=26$ and $x+2y+\lambda z=\mu$ has no solution, are:

If the system of equations$kx+y+2z=1$ $3x-y-2z=2$ $-2x-2y-4z=3$ has infinitely many solutions, then $k$ is equal to ______ .

The equation $arg(\frac{z-1}{z+1})=\frac{\pi }{4}$ represents a circle with:

A scientific committee is to be formed from $6$ Indians and $8$ foreigners, which includes at least $2$ Indians and double the number of foreigners as Indians. Then the number of ways, the committee can be formed, is:

The sum of all the elements in the set ${n\in {1,2,\ldots \ldots ,100}\mid$ H.C.F. of $n$ and $2040$ is $1$} is equal to __________.

The number of elements in the set ${n\in {1,2,3,\ldots ,100}\mid (11{)}^{n}>(10{)}^{n}+(9{)}^{n}}$ is ___________.

Consider function $f:A\rightarrow B$ and $g:B\rightarrow C(A,B,C\subseteq R)$ such that ${(gof)}^{-1}$ exists, then:

If $[x]$ be the greatest integer less than or equal to $x,$ then $\sum _{n=8}^{100}[\frac{{(-1)}^{n}n}{2}]$ is equal to:

Let ${S}_{n}$ denote the sum of first $n$-terms of an arithmetic progression. If ${S}_{10}=530,{S}_{5}=140,$ then ${S}_{20}-{S}_{6}$ is equal to:

Let $N$ be the set of natural numbers and a relation $R$ on $N$ be defined by $R={(x,y)\in N\times N:{x}^{3}-3{x}^{2}y-x{y}^{2}+3{y}^{3}=0}$. Then the relation $R$ is

Let $P=[\begin{matrix}-30 & 20 & 56 \\ 90 & 140 & 112 \\ 120 & 60 & 14\end{matrix}]$ and $A=[\begin{matrix}2 & 7 & {\omega }^{2} \\ -1 & -\omega & 1 \\ 0 & -\omega & -\omega +1\end{matrix}]$ where $\omega =\frac{-1+i\sqrt{3}}{2}$, and ${I}_{3}$ be the identity matrix of order $3$. If the determinant of the matrix ${({P}^{-1}AP-{I}_{3})}^{2}$ is $\alpha {\omega }^{2}$, then the value of $\alpha$ is equal to _________.

Let $A$ be a $3\times 3$ real matrix. If $det(2Adj(2Adj(Adj(2A))))={2}^{41},$ then the value of $det({A}^{2})$ equals ______.

If $z$ and $\omega$ are two complex numbers such that $|z\omega |=1$ and $\mathrm{arg}(z)-\mathrm{arg}(\omega )=\frac{3\pi }{2}$, then $\mathrm{arg}(\frac{1-2\bar{z}\omega }{1+3\bar{z}\omega })$ is: (Here $\mathrm{arg}(z)$ denotes the principal argument of complex number $z$)

Consider a rectangle $ABCD$ having $5,6,7,9$ points in the interior of the line segments $AB,BC,CD,DA$ respectively. Let $\alpha$ be the number of triangles having these points from different sides as vertices and $\beta$ be the number of quadrilaterals having these points from different sides as vertices. Then $(\beta -\alpha )$ is equal to

Let $f:R-{3}\rightarrow R-{1}$ be defined by $f(x)=\frac{x-2}{x-3}$. Let $g:R\rightarrow R$ be given as $g(x)=2x-3$. Then, the sum of all the values of $x$ for which ${f}^{-1}(x)+{g}^{-1}(x)=\frac{13}{2}$ is equal to