Algebra Questions

If $A$ and $B$ are square matrices of size $n\times n$ such that $A^2-B^2=(A-B)(A+B)$, then which of the following will be always true?

If $x$ is real, the maximum value of $\frac{3x^2+9x+17}{3x^2+9x+7}$ is:

In the binomial expansion of $(a-b{)}^n,n\ge 5$, the sum of $5^{th}$ and $6^{th}$ terms is zero, then $\frac{a}{b}$ equals:

From $6$ different novels and $3$ different dictionaries, $4$ novels and $1$ dictionary are to be selected and arranged in a row on the shelf so that the dictionary is always in the middle. Then the number of such arrangements is:

If $\omega \ne 1$ is a cube root of unity, and $(1+{\omega }^2{)}^7=A+B\omega$. Then $(A,B)$ equals:

Let $A$ and $B$ be two symmetric matrices of order $3$.Statement-1 : $ABA$ and $A{B}^{2}A$ are symmetric matrices.Statement-2 : $AB$ is a symmetric matrix if matrix multiplication of $A$ and $B$ is commutative.

The number of values of $k$ for which the linear equations $4x+ky+2z=0$, $kx+4y+z=0$, and $2x+2y+z=0$ possess a non-zero solution is:

The vectors $\vec{a}$ and $\vec{b}$ are not perpendicular and $\vec{c}$ and $\vec{d}$ are two vectors satisfying: $\vec{b}\times \vec{c}=\vec{b}\times \vec{d}$ and $\vec{a}\cdot \vec{d}=0$. Then the vector $\vec{d}$ is equal to:

If $\vec{a}=\frac{1}{\sqrt{10}}(3\hat{i}+\hat{k})$ and $\vec{b}=\frac{1}{7}(2\hat{i}+3\hat{j}-6\hat{k})$, then the value of $(\overrightarrow{2a}-\vec{b})\cdot [(\vec{a}\times \vec{b})\times (\vec{a}+\overrightarrow{2b})]$ is:

A man saves Rs. $200$ in each of the first three months of his service. In each of the subsequent months his saving increases by Rs. $40$ more than the saving of immediately previous month. His total saving from the start of service will be Rs. $11040$ after:

Let $\alpha ,\beta$ be real and $z$ be a complex number. If $z^2+\alpha z+\beta =0$ has two distinct roots on the line $\text{Re}(z)=1$, then it is necessary that:

The coefficient of $x^7$ in the expansion of $(1-x-x^2+x^3{)}^6$ is:

If $\alpha$ and $\beta$ are the roots of the equation $x^2-x+1=0$, then ${\alpha }^{2009}+{\beta }^{2009}=$

1: $\text{adj}(\text{adj }A)=A$ where $A$ is a $2\times 2$ matrix.
2: $|\text{adj }A|=|A|$.

The sum to the infinity of the series $1+\frac{2}{3}+\frac{6}{3^2}+\frac{10}{3^3}+\frac{14}{3^4}+\dots$ is:

The remainder left out when $8^{2n}-(62{)}^{2n+1}$ is divided by $9$ is:

If $|z-4/z|=2$, then the maximum value of $|z|$ is equal to:

If the coefficients of $r^{\text{th}}$, $(r+1{)}^{\text{th}}$ and $(r+2{)}^{\text{th}}$ terms in the binomial expansion of $(1+y{)}^m$ are in A.P., then $m$ and $r$ satisfy the equation:

If the roots of the equation $b{x}^2+cx+a=0$ be imaginary, then for all real values of $x$, the expression $3b^2{x}^2+6bcx+2c^2$ is:

Let $a,b,c$ be such that $(b+a)c\ne 0$. If
$$\left|\begin{array}{ccc}a& a+1& a-1\\ b& b+1& b-1\\ c& c-1& c+1\end{array}\right|+\left|\begin{array}{ccc}a+1& b+1& c-1\\ a-1& b-1& c+1\\ (-1{)}^{n}a& (-1{)}^{n}b& (-1{)}^{n+1}c\end{array}\right|=0$$
then the value of '$n$ ' is:

If ${\left(\frac{1+i}{1-i}\right)}^x=1$, then:

A function $f$ from the set of natural numbers to integers defined by$f(n)=\{\begin{array}{ll}\frac{n-1}{2},& \text{when }n\text{ is odd}\\ -\frac{n}{2},& \text{when }n\text{ is even}\end{array}$ is:

If both the roots of the quadratic equation $x^2-2kx+k^2+k-5=0$ are less than $5$, then $k$ lies in the interval:

If $x$ is so small that $x^3$ and higher powers of $x$ may be neglected, then $\frac{(1+x{)}^{3/2}-(1+\frac{1}{2}x{)}^3}{(1-x{)}^{1/2}}$ may be approximated as:

The value of ${}^{50}{C}_4+{\sum }_{r=1}^6{}^{56-r}{C}_3$ is:

If roots of the equation $x^2-bx+c=0$ are two consecutive integers, then $b^2-4c$ equals:

The value of $a$ for which the sum of the squares of the roots of the equation $x^2-(a-2)x-a-1=0$ assumes the least value is:

If $z=x-iy$ and $z^{\frac{1}{3}}=p+iq,$ then $\frac{(\frac{x}{p}+\frac{y}{q})}{(p^2+q^2)}$ is equal to