Algebra PYQ — Page 78

The first two terms of a geometric progression add up to 12. The sum of the third and the fourth terms is 48. If the terms of the geometric progression are alternately positive and negative, then the first term is

Let $f: N \rightarrow Y$ be a function defined as $f(x)=4 x+3$, where $Y=\{y \in N: y=4 x+3$ for some $x \in N\}$. Show that $\mathrm{f}$ is invertible and its inverse is

Let $\mathbf{A}$ be a square matrix all of whose entries are integers. Then which one of the following is true?

Let $\mathrm{A}$ be a $2 \times 2$ matrix with real entries. Let I be the $2 \times 2$ identity matrix. Denote by $\operatorname{tr}(\mathrm{A})$, the sum of diagonal entries of $A$. Assume that $A^2=1$. Statement -1: If $A \neq 1$ and $A \neq-1$, then $\operatorname{det} A=-1$. Statement $-2$ : If $A \neq 1$ and $A \neq-1$, then $\operatorname{tr}(A) \neq 0$.

How many different words can be formed by jumbling the letters in the word MISSISSIPPI in which no two $\mathrm{S}$ are adjacent?

Statement - 1: For every natural number $n \geq 2, \frac{1}{\sqrt{1}}+\frac{1}{\sqrt{2}}+\ldots+\frac{1}{\sqrt{n}}>\sqrt{n}$. Statement $-2$ : For every natural number $n \geq 2, \sqrt{n(n+1)} < n+1$.

Let $a, b, c$ be any real numbers. Suppose that there are real numbers $x, y, z$ not all zero such that $x=$ $c y+b z, y=a z+c x$ and $z=b x+a y$. Then $a^2+b^2+c^2+2 a b c$ is equal to

In a shop there are five types of ice-creams available. A child buys six ice-creams. Statement -1: The number of different ways the child can buy the six ice-creams is ${ }^{10} \mathrm{C}_5$. Statement $-2$ : The number of different ways the child can buy the six ice-creams is equal to the number of different ways of arranging $6 \mathrm{~A}^{\text {'s }}$ and 4 B's in a row.

The quadratic equations $x^2-6 x+a=0$ and $x^2-c x+6=0$ have one root in common. The other roots of the first and second equations are integers in the ratio $4: 3$. Then the common root is

Let $R$ be the real line. Consider the following subsets of the plane $R \times R$. $S=\{(x, y): y=x+1$ and $0 < x < 2\}, T=\{(x, y): x-y$ is an integer $\}$. Which one of the following is true?

In a geometric progression consisting of positive terms, each term equals the sum of the next two terms. Then the common ratio of this progression equals

If $|z+4| \leq 3$, then the maximum value of $|z+1|$ is

If $D=\left|\begin{array}{ccc}1 & 1 & 1 \\ 1 & 1+x & 1 \\ 1 & 1 & 1+y\end{array}\right|$ for $x \neq 0, y \neq 0$ then $D$ is

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

The set $S=\{1,2,3, \ldots, 12)$ is to be partitioned into three sets $A, B, C$ of equal size. Thus, $A \cup B \cup C=S, A \cap B=B \cap C=A \cap C=\phi$. The number of ways to partition $S$ is

Let $A=\left[\begin{array}{ccc}5 & 5 \alpha & \alpha \\ 0 & \alpha & 5 \alpha \\ 0 & 0 & 5\end{array}\right]$. If $\left|A^2\right|=25$, then $|\alpha|$ equals

The largest interval lying in $\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$ for which the function $\left[f(x)=4^{-x^2}+\cos ^{-1}\left(\frac{x}{2}-1\right)+\log (\cos x)\right]$ is defined, is

If the difference between the roots of the equation $x^2+a x+1=0$ is less than $\sqrt{5}$, then the set of possible values of $a$ is

If $p$ and $q$ are positive real numbers such that $p^2+q^2=1$, then the maximum value of $(p+q)$ is

If $a_1, a_2, \ldots, a_n$ are in H.P., then the expression $a_1 a_2+a_2 a_3+\ldots+a_{n-1} a_n$ is equal to

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?

Let $A=\left(\begin{array}{ll}1 & 2 \\ 3 & 4\end{array}\right)$ and $B=\left(\begin{array}{ll}a & 0 \\ 0 & b\end{array}\right), a, b \in N$. Then

Let W denote the words in the English dictionary. Define the relation R by : $R=\{(x, y) \in W \times W \mid$ the words $x$ and $y$ have at least one letter in common $\}$. Then $R$ is

All the values of $m$ for which both roots of the equations $x^2-2 m x+m^2-1=0$ are greater than $-2$ but less than 4 , lie in the interval