Algebra PYQ — Page 67

If $b$ is the first term of an infinite geometric progression whose sum is five, then $b$ lies in the interval

$n$-digit numbers are formed using only three digits $2,5$ and $7$. The smallest value of $n$ for which $900$ such distinct numbers can be formed is :

If $\lambda \in R$ is such that the sum of the cubes of the roots of the equation ${x}^{2}+(2-\lambda )x+(10-\lambda )=0$ is minimum, then the magnitude of the difference of the roots of this equation is :

If $|z-3+2 i| \leq 4$ then the difference between the greatest value and the least value of $|z|$ is

If $\alpha , \beta \in C$ are the distinct roots of the equation ${x}^{2}-x+1=0,$ then ${\alpha }^{101}+{\beta }^{107}$ is equal to

The set of all $\alpha \in R$, for which $w=\frac{1+(1-8\alpha )z}{1-z}$ is a purely imaginary number, for all $z\in C$ satisfying $|z|=1$ and $Re(z)\neq 1$, is :

The number of four letter words that can be formed using the letters of the word BARRACK is

The coefficient of $x^{10}$ in the expansion of $(1+x)^2$ $\left(1+x^2\right)^3\left(1+x^3\right)^4$ is equal to

If $\mathrm{tan}⁡A$ and $\mathrm{tan}⁡B$ are the roots of the quadratic equation $3{x}^{2}-10x-25=0$, then the value of $3{\mathrm{sin}}^{2}⁡(A+B)-10\mathrm{sin}⁡(A+B)\mathrm{cos}⁡(A+B)-25{\mathrm{cos}}^{2}⁡(A+B)$ is :

The coefficient of ${x}^{2}$ in the expansion of the product $(2-{x}^{2}){{(1+2x+3{x}^{2})}^{6}+{(1-4{x}^{2})}^{6}}$ is

If the system of linear equations $x+ky+3z=0$ $3x+ky-2z=0$ $2x+4y-3z=0$ has a non-zero solution $(x, y, z),$ then $\frac{xz}{{y}^{2}}$ is equal to:

The set of all $\alpha \in R$, for which $w=\frac{1+(1-8 \alpha) z}{1-z}$ is a purely imaginary number, for all $z \in C$ satisfying $|z|=1$ and $\operatorname{Re} z \neq 1$, is

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

If ${x}_{1},{x}_{2},\ldots ..,{x}_{n}$ and $\frac{1}{{h}_{1}},\frac{1}{{h}_{2}},\ldots ..,\frac{1}{{h}_{n}}$ are two A.P.s such that ${x}_{3}={h}_{2}=8&{x}_{8}={h}_{7}=20$, then ${x}_{5}\cdot {h}_{10}$ is equal to

The least positive integer $n$ for which ${(\frac{1+i\sqrt{3}}{1-i\sqrt{3}})}^{n}=1$ is

Let $f: \mathrm{A} \rightarrow$ B be a function defined as $f(x)=$ $\frac{x-1}{x-2}$, where $A=R-\{2\}$ and $B=R-\{1\}$. Then $f$ is

If $x_1, x_2, \ldots ., x_n$ and $\frac{1}{h_1}, \frac{1}{h^2}, \ldots . . \frac{1}{h_n}$ are two A.P's such that $x_3=h_2=8$ and $x_8=h_7=20$, then $x_5 . h_{10}$ equals.

If an angle $A$ of a $\Delta ABC$ satisfies $5\mathrm{cos}A+3=0$, then the roots of the quadratic equation $9{x}^{2}+27x+20=0$ are

If $f(x)=\left|\begin{array}{ccc}\cos x & x & 1 \\ 2 \sin x & x^2 & 2 x \\ \tan x & x & 1\end{array}\right|$, then $\lim _{x \rightarrow 0} \frac{f^{\prime}(x)}{x}$

Let $\frac{1}{{x}_{1}},\frac{1}{{x}_{2}},\ldots ,\frac{1}{{x}_{n}}({x}_{i}\neq 0$ for $i=1,2,\ldots .,n)$ be in A.P. such that ${x}_{1}=4$ and ${x}_{21}=20$$.$ If $n$ is the least positive integer for which ${x}_{n}>50$, then $\sum _{i=1}^{n}(\frac{1}{{x}_{i}})$ is equal to

If $|\begin{matrix}x-4 & 2x & 2x \\ 2x & x-4 & 2x \\ 2x & 2x & x-4\end{matrix}|=(A+Bx) {(x-A)}^{2},$ then the ordered pair $(A,B)$ is equal to

If all the words, with or without meaning, are written using the letters of the word QUEEN and are arranged as in English dictionary, then the position of the word QUEEN is:

If $A=[\begin{matrix}2 & -3 \\ -4 & 1\end{matrix}]$ , then $\mathrm{Adj}(3{A}^{2}+12A)$ is equal to:

A man $X$ has $7$ friends, $4$ of them are ladies and $3$ are men. His wife $Y$ also has $7$ friends, $3$ of them are ladies and $4$ are men. Assume $X$ and $Y$ have no common friends. Then the total number of ways in which $X$ and $Y$ together can throw a party inviting $3$ ladies and $3$ men, so that $3$ friends of each of $X$ and $Y$ are in this party is: