Algebra PYQ — Page 17

Let the set $S=\{2,4,8,16, \ldots, 512\}$ be partitioned into 3 sets $A, B, C$ with equal number of elements such that $\mathrm{A} \cup \mathrm{B} \cup \mathrm{C}=\mathrm{S}$ and $\mathrm{A} \cap \mathrm{B}=\mathrm{B} \cap \mathrm{C}=\mathrm{A} \cap \mathrm{C}=\phi$. The maximum number of such possible partitions of $S$ is equal to:

For a differentiable function $f: \mathbb{R} \rightarrow \mathbb{R}$, suppose $f^{\prime}(x)=3 f(x)+\alpha$, where $\alpha \in \mathbb{R}$, $f(0)=1$ and $\lim _{x \rightarrow-\infty} f(x)=7$. Then $9 f\left(-\log _{\mathrm{e}} 3\right)$ is equal to_________

There are 4 men and 5 women in Group A, and 5 men and 4 women in Group B. If 4 persons are selected from each group, then the number of ways of selecting 4 men and 4 women is _____

If the system of equations $x+4 y-z=\lambda, 7 x+9 y+\mu z=-3,5 x+y+2 z=-1$ has infinitely many solutions, then $(2 \mu+3 \lambda)$ is equal to :

Let $A$ be a $2\times 2$ real matrix and $I$ be the identity matrix of order $2.$ If the roots of the equation $|A-\mathrm{xI}|=0$ be $-1$ and $3,$ then the sum of the diagonal elements of the matrix ${A}^{2}$ is _____.

Let for any three distinct consecutive terms $a,b,c$ of an A.P, the lines $ax+by+c=0$ be concurrent at the point $P$ and $Q(\alpha ,\beta )$ be a point such that the system of equations $x+y+z=6$, $2x+5y+\alpha z=\beta$ and $x+2y+3z=4$, has infinitely many solutions. Then $(PQ{)}^{2}$ is equal to _______.

In an A.P., the sixth term ${a}_{6}=2$. If the ${a}_{1}{a}_{4}{a}_{5}$ is the greatest, then the common difference of the A.P., is equal to

If the system of equations $\begin{aligned} & x+(\sqrt{2} \sin \alpha) y+(\sqrt{2} \cos \alpha) z=0 \\ & x+(\cos \alpha) y+(\sin \alpha) z=0 \\ & x+(\sin \alpha) y-(\cos \alpha) z=0 \end{aligned}$ has a non-trivial solution, then $\alpha \in\left(0, \frac{\pi}{2}\right)$ is equal to :

All the letters of the word $GTWENTY$ are written in all possible ways with or without meaning and these words are written as in a dictionary. The serial number of the word $GTWENTY$ IS

If the system of equations $\begin{aligned} & 2 x+7 y+\lambda z=3 \\ & 3 x+2 y+5 z=4 \\ & x+\mu y+32 z=-1 \end{aligned}$ has infinitely many solutions, then $(\lambda-\mu)$ is equal to________

The values of $m, n$, for which the system of equations $\begin{aligned} & x+y+z=4, \\ & 2 x+5 y+5 z=17, \\ & x+2 y+\mathrm{m} z=\mathrm{n} \end{aligned}$ has infinitely many solutions, satisfy the equation:

Consider the function $f: \mathbb{R} \rightarrow \mathbb{R}$ defined by $f(x)=\frac{2 x}{\sqrt{1+9 x^2}}$. If the composition of $f, \underbrace{(f \circ f \circ f \circ \cdots \circ f)}_{10 \text { times }}(x)=\frac{2^{10} x}{\sqrt{1+9 \alpha x^2}}$, then the value of $\sqrt{3 \alpha+1}$ is equal to ______

Let $\alpha \beta \gamma=45 ; \alpha, \beta, \gamma \in \mathbb{R}$. If $x(\alpha, 1,2)+y(1, \beta, 2)+z(2,3, \gamma)=(0,0,0)$ for some $x, y, z \in \mathbb{R}, x y z \neq 0$, then $6 \alpha+4 \beta+\gamma$ is equal to _______

Let $f(x)=\frac{1}{7-\sin 5 x}$ be a function defined on $\mathbf{R}$. Then the range of the function $f(x)$ is equal to ;

Let a relation $\mathrm{R}$ on $\mathrm{N} \times N$ be defined as: $\left(x_1, y_1\right) \mathrm{R}\left(x_2, y_2\right)$ if and only if $x_1 \leq x_2$ or $y_1 \leq y_2$. Consider the two statements: (I) $\mathrm{R}$ is reflexive but not symmetric. (II) $R$ is transitive Then which one of the following is true?

If the term independent of $x$ in the expansion of $\left(\sqrt{\mathrm{a}} x^2+\frac{1}{2 x^3}\right)^{10}$ is 105 , then $\mathrm{a}^2$ is equal to :

If the coefficients of $x^4, x^5$ and $x^6$ in the expansion of $(1+x)^n$ are in the arithmetic progression, then the maximum value of $n$ is:

If $f(x)=|\begin{matrix}2{\mathrm{cos}}^{4}x & 2{\mathrm{sin}}^{4}x & 3+{\mathrm{sin}}^{2}2x \\ 3+2{\mathrm{cos}}^{4}x & 2{\mathrm{sin}}^{4}x & {\mathrm{sin}}^{2}2x \\ 2{\mathrm{cos}}^{4}x & 3+2{\mathrm{sin}}^{4}x & {\mathrm{sin}}^{2}2x\end{matrix}|$ then $\frac{1}{5}{f}^{'}(0)$ is equal to ________.

The values of $\alpha$, for which $|\begin{matrix}1 & \frac{3}{2} & \alpha +\frac{3}{2} \\ 1 & \frac{1}{3} & \alpha +\frac{1}{3} \\ 2\alpha +3 & 3\alpha +1 & 0\end{matrix}|=0$, lie in the interval

For $\alpha, \beta \in \mathbb{R}$ and a natural number $n$, let $A_r=\left|\begin{array}{ccc}r & 1 & \frac{n^2}{2}+\alpha \\ 2 r & 2 & n^2-\beta \\ 3 r-2 & 3 & \frac{n(3 n-1)}{2}\end{array}\right|$. Then

Let $B=\left[\begin{array}{ll}1 & 3 \\ 1 & 5\end{array}\right]$ and $A$ be a $2 \times 2$ matrix such that $A B^{-1}=A^{-1}$. If $B C B^{-1}=A$ and $C^4+\alpha C^2+\beta I=O$, then $2 \beta-\alpha$ is equal to

Consider the matrix $f(x)=[\begin{matrix}\mathrm{cos}x & -\mathrm{sin}x & 0 \\ \mathrm{sin}x & \mathrm{cos}x & 0 \\ 0 & 0 & 1\end{matrix}]$. Given below are two statements : Statement I:$f(-x)$ is the inverse of the matrix $f(x)$. Statement II: $f(x)f(y)=f(x+y)$. In the light of the above statements, choose the correct answer from the options given below

Consider the matrices : $A=\left[\begin{array}{ll}2 & -5 \\ 3 & m\end{array}\right], B=\left[\begin{array}{l}20 \\ m\end{array}\right]$ and $X=\left[\begin{array}{l}x \\ y\end{array}\right]$. Let the set of all $m$, for which the system of equations $A X=B$ has a negative solution (i.e., $x < 0$ and $y < 0$ ), be the interval $(a, b)$. Then $8 \int_a^b|A| d m$ is equal to_________

If $f(x)={\begin{matrix}2+2x,-1\leq x<0 \\ 1-\frac{x}{3},0\leq x\leq 3\end{matrix};g(x)={\begin{matrix}-x,-3\leq x\leq 0 \\ x,0<x\leq 1\end{matrix}$, then range of $(f\circ g(x))$ is