Algebra PYQ — Page 6

The system of linear equations $x+y+z=6$ $2 x+5 y+a z=36$ $x+2 y+3 z=b$ has

Let $A$ be the set of first 101 terms of an A.P., whose first term is 1 and the common difference is 5 and let $B$ be the set of first 71 terms of an A.P., whose first term is 9 and the common difference is 7. Then the number of elements in $A \cap B$, which are divisible by 3, is :

The sum of the first ten terms of an A.P. is $160$ and the sum of the first two terms of a G.P. is $8$. If the first term of the A.P. is equal to the common ratio of the G.P. and the first term of the G.P. is equal to common difference of the A.P., then the sum of all possible values of the first term of the G.P. is:

Let $S=\{z \in \mathbb{C}: z^2+4z+16=0\}$. Then $\sum_{z \in S}|z+\sqrt{3}i|^2$ is equal to:

Let $a_1, a_2, a_3, \ldots$ be an A.P. and $g_1 = a_1, g_2, g_3, \ldots$ be an increasing G.P. If $a_1 = a_2 + g_2 = 1$ and $a_3 + g_3 = 4$, then $a_{10} + g_5$ is equal to:

The number of functions $f: \{1, 2, 3, 4\} \rightarrow \{a, b, c\}$, which are not onto, is:

Let $\mathrm{S}=\{1,2,3,4,5,6,7,8,9\}$. Let $x$ be the number of 9 -digit numbers formed using the digits of the set S such that only one digit is repeated and it is repeated exactly twice. Let $y$ be the number of 9-digit numbers formed using the digits of the set S such that only two digits are repeated and each of these is repeated exactly twice. Then,

The number of strictly increasing functions $f$ from the set $\{1,2,3,4,5,6\}$ to the set $\{1,2,3, \ldots., 9\}$ such that $f(i) \neq i$ for $1 \leq i \leq 6$, is equal to :

In the expansion of $\left(9x-\dfrac{1}{3\sqrt{x}}\right)^{18}$, $x>0$, if the term independent of $x$ is $(221)k$, then $k$ is equal to:

The number of ways of forming a queue of $4$ boys and $3$ girls such that all the girls are not together, is:

Let for some $\alpha \in \mathbb{R}$, $f:\mathbb{R}\rightarrow\mathbb{R}$ be a function satisfying $f(x+y)=f(x)+2y^2+y+\alpha xy$ for all $x,y \in \mathbb{R}$. If $f(0)=-1$ and $f(1)=2$, then the value of $\sum_{n=1}^{5}(\alpha+f(n))$ is:

Let $\mathrm{A}=\{2,3,5,7,9\}$. Let R be the relation on A defined by $x \mathrm{R} y$ if and only if $2 x \leq 3 y$. Let $l$ be the number of elements in R, and m be the minimum number of elements required to be added in R to make it a symmetric relation. Then $l+\mathrm{m}$ is equal to :

Let $\alpha$ and $\beta$ be the roots of the equation $x^{2}+2 a x+(3 a+10)=0$ such that $\alpha<1<\beta$. Then the set of all possible values of $a$ is :

If the system of linear equations: $x+y+z=6$, $x+2y+5z=10$, $2x+3y+\lambda z=\mu$ has infinitely many solutions, then the value of $\lambda+\mu$ equals:

If $X=\left[\begin{array}{l}x \\ y \\ z\end{array}\right]$ is a solution of the system of equations $A X=B$, where $\operatorname{adj} A=\left[\begin{array}{ccc}4 & 2 & 2 \\ -5 & 0 & 5 \\ 1 & -2 & 3\end{array}\right]$ and $\mathrm{B}=\left[\begin{array}{l}4 \\ 0 \\ 2\end{array}\right]$, then $|x+y+z|$ is equal to :

Let $\alpha=\dfrac{1}{4}+\dfrac{1}{8}+\dfrac{1}{16}+\ldots\infty$ and $\beta=\dfrac{1}{3}+\dfrac{1}{9}+\dfrac{1}{27}+\ldots\infty$. Then the value of $(0.2)^{\log_{\sqrt{5}}(\alpha)}+(0.04)^{\log_5(\beta)}$ is equal to:

Let $S = \left\{A = \begin{bmatrix} a & b \\ c & d \end{bmatrix} : a, b, c, d \in \{0, 1, 2, 3, 4\} \text{ and } A^2 - 4A + 3I = 0\right\}$ be a set of $2 \times 2$ matrices. Then the number of matrices in $S$, for which the sum of the diagonal elements is equal to $4$, is:

Let $\alpha, \alpha + 2, \alpha \in \mathbb{Z}$, be the roots of the quadratic equation $x(x+2) + (x+1)(x+3) + (x+2)(x+4) + \ldots + (x+n-1)(x+n+1) = 4n$ for some $n \in \mathbb{N}$. Then $n + \alpha$ is equal to :

Let R be a relation defined on the set $\{1,2,3,4\} \times\{1,2,3,4\}$ by $\mathrm{R}=\{((a, b),(c, d)): 2 a+3 b=3 c+4 d\}$. Then the number of elements in R is

Consider the matrices $A = \begin{bmatrix} 2 & -2 \\ 4 & -2 \end{bmatrix}$ and $B = \begin{bmatrix} 3 & 9 \\ 1 & 3 \end{bmatrix}$. If matrices P and Q are such that $PA = B$ and $AQ = B$, then the absolute value of the sum of the diagonal elements of $2(P + Q)$ is _______.

The value of $1^3 - 2^3 + 3^3 - \ldots + 15^3$ is:

The first term of an A.P. of $30$ non-negative terms is $\dfrac{10}{3}$. If the sum of this A.P. is the cube of its last term, then its common difference is:

Let $\sum_{k=1}^{n} a_{k}=\alpha n^{2}+\beta n$. If $a_{10}=59$ and $a_{6}=7 a_{1}$, then $\alpha+\beta$ is equal to

Let $A=\{(a,b,c): a,b,c \text{ are non-negative integers and } a+b+2c=22\}$. Then $n(A)$ is equal to: