Algebra PYQ — Page 49

The system of linear equations$3x-2y-kz=10$ $2x-4y-2z=6$ $x+2y-z=5m$ is inconsistent if :

Let $A={2,3,4,5,\ldots .,30}$ and $'\simeq '$ be an equivalence relation on $A\times A,$ defined by $(a,b)\simeq (c,d),$ if and only if $ad=bc$. Then the number of ordered pairs which satisfy this equivalence relation with ordered pair $(4,3)$ is equal to :

Let $\alpha =\underset{x\in R}{\mathrm{max}}{{8}^{2\mathrm{sin}3x}\cdot {4}^{4\mathrm{cos}3x}}$ and $\beta =\underset{x\in R}{\mathrm{min}}{{8}^{2\mathrm{sin}3x}\cdot {4}^{4\mathrm{cos}3x}}.$ If $8{x}^{2}+bx+c=0$ is a quadratic equation whose roots are ${\alpha }^{1/5}$ and ${\beta }^{1/5},$ then the value of $c-b$ is equal to :

If $x,y,z$ are in arithmetic progression with common difference $d,x\neq 3d,$ and the determinant of the matrix $[\begin{matrix}3 & 4\sqrt{2} & x \\ 4 & 5\sqrt{2} & y \\ 5 & k & z\end{matrix}]$ is zero, then the value of ${k}^{2}$ is

The equation of a circle is $Re({z}^{2})+2{(\mathrm{Im}(z))}^{2}+2Re(z)=0,$ where $z=x+iy.$ A line which passes through the centre of the given circle and the vertex of the parabola, ${x}^{2}-6x-y+13=0,$ has $y$-intercept equal to _________.

The value of $k\in R,$ for which the following system of linear equations $3x-y+4z=3$ $x+2y-3z=-2$ $6x+5y+kz=-3$ has infinitely many solutions, is:

If the real part of the complex number $z=\frac{3+2i\mathrm{cos}\theta }{1-3i\mathrm{cos}\theta },\theta \in (0,\frac{\pi }{2})$ is zero, then the value of ${\mathrm{sin}}^{2}3\theta +{\mathrm{cos}}^{2}\theta$ is equal to ______.

Let $g:N\rightarrow N$ be defined as $g(3n+1)=3n+2$ $g(3n+2)=3n+3$ $g(3n+3)=3n+1$, for all $n\geq 0$ Then which of the following statements is true ?

Let ${(1+x+2{x}^{2})}^{20}={a}_{0}+{a}_{1}x+{a}_{2}{x}^{2}+\ldots +{a}_{40}{x}^{40},$ then ${a}_{1}+{a}_{3}+{a}_{5}+\ldots +{a}_{37}$ is equal to

Let $A={0,1,2,3,4,5,6,7}.$ Then the number of bijective functions $f:A\rightarrow A$ such that $f(1)+f(2)=3-f(3)$ is equal to

If the least and the largest real values of $\alpha ,$ for which the equation $z+\alpha |z-1|+2i=0$ $(z\in C\text{and}i=\sqrt{-1})$ has a solution, are $p$ and $q$ respectively; then $4({p}^{2}+{q}^{2})$ is equal to_______.

The area of the triangle with vertices $P(z),Q(iz)$ and $R(z+iz)$ is

Let $A=[\begin{matrix}[x+1] & [x+2] & [x+3] \\ [x] & [x+3] & [x+3] \\ [x] & [x+2] & [x+4]\end{matrix}]\begin{matrix} \\ \\ \end{matrix}$, where $[x]$ denotes the greatest integer less than or equal to $x$. If $det(A)$$=192$, then the set of values of $x$ is in the interval:

Let ${{{a}_{n}}}_{n=1}^{\infty }$ be a sequence such that ${a}_{1}=1,{a}_{2}=1$ and ${a}_{n+2}=2{a}_{n+1}+{a}_{n}$ for all $n\geq 1$. Then the value of $47\sum _{n=1}^{\infty }(\frac{{a}_{n}}{{2}^{3n}})$ is equal to ________.

Let ${P}_{1},{P}_{2}\ldots ,{P}_{15}$ be $15$ points on a circle. The number of distinct triangles formed by points ${P}_{i},{P}_{j},{P}_{k}$ such that $i+j+k\neq 15,$ is :

Which of the following is not correct for relation $R$ on the set of real numbers?

Let $\mathbb{C}$ be the set of all complex numbers. Let ${S}_{1}={z\in \mathbb{C}:|z-2|\leq 1}$ and ${S}_{2}={z\in \mathbb{C}:z(1+i)+\bar{z}(1-i)\geq 4}.$ Then, the maximum value of ${|z-\frac{5}{2}|}^{2}$ for $z\in {S}_{1}\cap {S}_{2}$ is equal to :

If for $x,y\in R,x>0,$ $y={\mathrm{log}}_{10}x+{\mathrm{log}}_{10}{x}^{1/3}+{\mathrm{log}}_{10}{x}^{1/9}+\ldots$upto $\infty$ terms and $\frac{2+4+6+\ldots +2y}{3+6+9+\ldots +3y}=\frac{4}{{\mathrm{log}}_{10}x},$ then the ordered pair $(x,y)$ is equal to

If the remainder when $x$ is divided by $4$ is $3$, then the remainder when ${(2020+x)}^{2022}$ is divided by $8$ is ___ .

Let $A=[\begin{matrix}i & -i \\ -i & i\end{matrix}],i=\sqrt{-1}$. Then, the system of linear equations ${A}^{8}[\begin{matrix}x \\ y\end{matrix}]=[\begin{matrix}8 \\ 64\end{matrix}]$ has :

If $\mathrm{tan}(\frac{\pi }{9}),x,\mathrm{tan}(\frac{7\pi }{18})$ are in arithmetic progression and $\mathrm{tan}(\frac{\pi }{9}),y,\mathrm{tan}(\frac{5\pi }{18})$ are also in arithmetic progression, then $|x-2y|$ is equal to :

Let$A=[\begin{matrix}1 & 2 \\ -1 & 4\end{matrix}].$ If ${A}^{-1}=\alpha I+\beta A,\alpha ,\beta \in R,I$ is a $2\times 2$ identity matrix, then $4(\alpha -\beta )$ is equal to :

Let $[x]$ denote greatest integer less than or equal to $x$. If for $n\in N,{(1-x+{x}^{3})}^{n}=\sum _{j=0}^{3n}{a}_{j}{x}^{j}$, then $\sum _{j=0}^{[\frac{3n}{2}]}{a}_{2j}+4\sum _{j=0}^{[\frac{3n-1}{2}]}{a}_{2j+1}$ is equal to :

Let $[\lambda ]$ be the greatest integer less than or equal to $\lambda$. The set of all values of $\lambda$ for which the system of linear equations $x+y+z=4,3x+2y+5z=3,9x+4y+(28+[\lambda ])z=[\lambda ]$ has a solution is: