Algebra PYQ — Page 76

If $P(S)$ denotes the set of all subsets of a given set $S$, then the number of one-to-one functions from the set $S=\{1,2,3\}$ to the $\operatorname{set} P(S)$ is

The sum of the series $$ \frac{1}{1+\sqrt{2}}+\frac{1}{\sqrt{2}+\sqrt{3}}+\frac{1}{\sqrt{3}+\sqrt{4}}+\ldots $$ upto 15 terms is

The area of the triangle whose vertices are complex numbers $z, i z, z+i z$ in the Argand diagram is

The sum of the series $1+\frac{4}{3}+\frac{10}{9}+\frac{28}{27}+\ldots$ upto $n$ terms is

The number of arrangements that can be formed from the letters $a, b, c, d, e, f$ taken 3 at a time without repetition and each arrangement containing at least one vowel, is

Let $X=\{1,2,3,4,5\}$. The number of different ordered pairs $(Y, Z)$ that can be formed such that $Y \subseteq X, Z$ $\subseteq \mathrm{X}$ and $\mathrm{Y} \cap \mathrm{Z}$ is empty, is

The number of terms in the expansion of $\left(y^{1 / 5}+x^{1 / 10}\right)^{55}$, in which powers of $x$ and $y$ are free from radical signs are

The sum of the series $1^2+2.2^2+3^2+2.4^2+5^2+2.6^2+\ldots . .+2(2 m)^2$ is

Statement $1$: The sum of the series $1+(1+2+4)+(4+6+9)+(9+12+16)+\ldots \ldots+(361+380+ 400)$ is $8000$. Statement $2$: $\sum_{k=1}^n\left(k^3-(k-1)^3\right)=n^3$ for any natural number $n$.

If $f(y)=1-(y-1)+(y-1)^2-(y-1)^3$ $+\ldots-(y-1)^{17}$ then the coefficient of $y^2$ in it is

The value of $\mathrm{k}$ for which the equation $(K-2) x^2+8 x+K+4=0$ has both roots real, distinct and negative is

If seven women and seven men are to be seated around a circular table such that there is a man on either side of every woman, then the number of seating arrangements is

Let $Z_1$ and $Z_2$ be any two complex number. Statement 1: $\left|Z_1-Z_2\right| \geq\left|Z_1\right|-\left|Z_2\right|$ Statement 2: $\left|Z_1+Z_2\right| \leq\left|Z_1\right|+\left|Z_2\right|$

Let $P$ and $Q$ be $3 \times 3$ matrices with $P \neq Q$. If $P^3=Q^3$ and $P^2 Q=Q^2 P$, then determinant of $\left(P^2+Q^2\right)$ is equal to

$\left|z_1+z_2\right|^2+\left|z_1-z_2\right|^2$ is equal to

If $A^T$ denotes the transpose of the matrix $A=\left[\begin{array}{lll}0 & 0 & a \\ 0 & b & c \\ d & e & f\end{array}\right]$, where $a, b, c, d, e$ and $f$ are integers such that $a b d \neq 0$, then the number of such matrices for which $A^{-1}=A^T$ is

Let $f(x)=\sin x, g(x)=x$. Statement 1: $f(x) \leqslant g x($ for $) \mathrm{x}$ in $(0, \infty)$ Statement 2: $f(x) \leq 1$ for $x$ in $(0, \infty)$ but $g(x) \rightarrow \infty$ as $x \rightarrow \infty$.

If $\left|\begin{array}{ccc}-2 a & a+b & a+c \\ b+a & -2 b & b+c \\ c+a & b+c & -2 c\end{array}\right|$ $$ =\alpha(a+b() b+c() c+a) \neq 0 $$ then $\alpha$ is equal to

Let $p, q, r \in R$ and $r>p>0$. If the quadratic equation $p x^2+q x+r=0$ has two complex roots $\alpha$ and $\beta$, then $|\alpha|+|\beta|$ is

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 $\operatorname{Re} z=1$, then it is necessary that

The number of values of $\mathrm{k}$ for which the linear equations $4 x+k y+2 z=0 ; k x+4 y+z=0 ; 2 x+2 y+z=0$ possess a non-zero solution is

Let $A$ and $B$ be two symmetric matrices of order 3 . This question has Statement $-1$ and Statement $-2$. Of the four choices given after the statements, choose the one that best describes the two statements. Statement $-1$ : $\mathrm{A}(\mathrm{BA})$ and $(\mathrm{AB}) \mathrm{A}$ are symmetric matrices. Statement - 2 : $\quad A B$ is symmetric matrix if matrix multiplication of $A$ and $B$ is commutative.

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

This question has Statement $-1$ and Statement $-2$. Of the four choices given after the statements, choose the one that best describes the two statements. Statement - 1 : The number of ways of distributing 10 identical balls in 4 distinct boxes such that no box is empty is ${ }^9 \mathrm{C}_3$ Statement-2: The number of ways of choosing any 3 places from 9 different places is ${ }^9 \mathrm{C}_3$.