Algebra PYQ — Page 15

Let $\left|\frac{\bar{z}-i}{2 \bar{z}+i}\right|=\frac{1}{3}, z \in C$, be the equation of a circle with center at $C$. If the area of the triangle, whose vertices are at the points $(0,0), \mathrm{C}$ and $(\alpha, 0)$ is 11 square units, then $\alpha^2$ equals:

The number of singular matrices of order 2 , whose elements are from the set $\{2,3,6,9\}$ is

The sum of all rational terms in the expansion of $(2+\sqrt{3})^8$ is

Let $\left\langle a_{\mathrm{n}}\right\rangle$ be a sequence such that $a_0=0, a_1=\frac{1}{2}$ and $2 a_{\mathrm{n}+2}=5 a_{\mathrm{n}+1}-3 a_{\mathrm{n}}, \mathrm{n}=0,1,2,3, \ldots$. Then $\sum_{\mathrm{k}=1}^{100} a_k$ is equal to

The number of ways, 5 boys and 4 girls can sit in a row so that either all the boys sit together or no two boys sit together, is $\qquad$ -

Let $S=\left\{p_1, p_2 \ldots ., p_{10}\right\}$ be the set of first ten prime numbers. Let $A=S \cup P$, where $P$ is the set of all possible products of distinct elements of $S$. Then the number of all ordered pairs ( $x, y$ ), $x \in S$, $y \in A$, such that $x$ divides $y$, is ______.

The number of terms of an A.P. is even; the sum of all the odd terms is 24 , the sum of all the even terms is 30 and the last term exceeds the first by $\frac{21}{2}$. Then the number of terms which are integers in the A.P. is :

Group A consists of 7 boys and 3 girls, while group B consists of 6 boys and 5 girls. The number of ways, 4 boys and 4 girls can be invited for a picnic if 5 of them must be from group $A$ and the remaining 3 from group $B$, is equal to :

If \(1^2 \cdot\left({ }^{15} C_1\right)+2^2 \cdot\left({ }^{15} C_2\right)+3^2 \cdot\left({ }^{15} C_3\right)+\ldots\) \(+15^2 \cdot\left({ }^{15} C_{15}\right)=2^m \cdot 3^n \cdot 5^k\), where \(m, n, k \in \mathbf{N}\), then \(\mathrm{m}+\mathrm{n}+\mathrm{k}\) is equal to :

Let $a_1, a_2, a_3, \ldots$ be a G. P. of increasing positive numbers. If $\mathrm{a}_3 \mathrm{a}_5=729$ and $\mathrm{a}_2+\mathrm{a}_4=\frac{111}{4}$, then $24\left(a_1+a_2+a_3\right)$ is equal to

The value of \(\lim _{n \rightarrow \infty}\left(\sum_{k=1}^n \frac{k^3+6 k^2+11 k+5}{(k+3)!}\right)\) is:

Line $L_1$ of slope 2 and line $L_2$ of slope $\frac{1}{2}$ intersect at the origin O . In the first quadrant, $\mathrm{P}_1, \mathrm{P}_2, \ldots . \mathrm{P}_{12}$ are 12 points on line $L_1$ and $Q_1, Q_2, \ldots . . Q_9$ are 9 points on line $L_2$. Then the total number of triangles, that can be formed having vertices at three of the 22 points $\mathrm{O}, \mathrm{P}_1, \mathrm{P}_2, \ldots \mathrm{P}_{12}$, $\mathrm{Q}_1, \mathrm{Q}_2, \ldots . \mathrm{Q}_9$, is:

If $\mathrm{A}, \mathrm{B}$, and $\left(\operatorname{adj}\left(\mathrm{A}^{-1}\right)+\operatorname{adj}\left(\mathrm{B}^{-1}\right)\right)$ are non-singular matrices of same order, then the inverse of $\mathrm{A}\left(\operatorname{adj}\left(\mathrm{A}^{-1}\right)+\operatorname{adj}\left(\mathrm{B}^{-1}\right)\right)^{-1} \mathrm{~B}$, is equal to

Number of functions $f:\{1,2, \ldots, 100\} \rightarrow\{0,1\}$, that assign 1 to exactly one of the positive integers less than or equal to 98 , is equal to ________.

The number of ways, in which the letters $\mathrm{A}, \mathrm{B}, \mathrm{C}$, D, E can be placed in the 8 boxes of the figure below so that no row remains empty and at most one letter can be placed in a box, is : 

If the number of seven-digit numbers, such that the sum of their digits is even, is $m \cdot n \cdot 10^{\mathrm{n}}$; $m, n \in\{1,2,3, \ldots, 9\}$, then $m+n$ is equal to _______

The product of the last two digits of $(1919)^{1919}$ is $\qquad$

Let $f:[0,3] \rightarrow$ A be defined by $f(x)=2 x^3-15 x^2+36 x+7$ and $g:[0, \infty) \rightarrow B$ be defined by $\mathrm{g}(x)=\frac{x^{2025}}{x^{2025}+1}$. If both the functions are onto and $\mathrm{S}=\{x \in \mathbf{Z}: x \in \mathrm{~A}$ or $x \in \mathrm{~B}\}$, then $\mathrm{n}(\mathrm{S})$ is equal to :

Let $\alpha_\theta$ and $\beta_\theta$ be the distinct roots of $2 x^2+(\cos \theta) x-1=0, \theta \in(0,2 \pi)$. If m and M are the minimum and the maximum values of $\alpha_\theta^4+\beta_\theta^4$, then $16(M+m)$ equals :

Consider an A. P. of positive integers, whose sum of the first three terms is 54 and the sum of the first twenty terms lies between 1600 and 1800. Then its \(11^{\text {th }}\) term is :

The number of complex numbers $z$, satisfying $|z|=1$ and $\left|\frac{z}{\bar{z}}+\frac{\bar{z}}{z}\right|=1$, is :

Let $z \in C$ be such that $\frac{z^2+3 i}{z-2+i}=2+3 i$. Then the sum of all possible values of $z^2$ is

Let $a_1, a_2, a_3, \ldots$ be a G.P. of increasing positive terms. If $a_1 a_5=28$ and $a_2+a_4=29$, then $a_6$ is equal to:

Let \(\mathrm{A}=\left[\mathrm{a}_{i j}\right]=\left[\begin{array}{cc}\log _5 128 & \log _4 5 \\ \log _5 8 & \log _4 25\end{array}\right]\). If \(\mathrm{A}_{i j}\) is the cofactor of \(\mathrm{a}_{i j}, \mathrm{C}_{i j}=\sum_{\mathrm{k}=1}^2 \mathrm{a}_{i \mathrm{k}} \mathrm{A}_{j \mathrm{k}}, 1 \leq i, j \leq 2\), and \(\mathrm{C}=\left[\mathrm{C}_{i j}\right]\), then \(8|\mathrm{C}|\) is equal to :