Algebra PYQ — Page 57

Let $S$ be the set of all integer solutions $(x,y,z)$ of the system of equations $x-2y+5z=0$ $-2x+4y+z=0$ $-7x+14y+9z=0$ such that $15\leq {x}^{2}+{y}^{2}+{z}^{2}\leq 150$. Then, the number of elements in the set $S$ is equal to ..........

The number of words (with or without meaning) that can be formed from all the letters of the word$"LETTER"$ in which vowels never come together is.....

The least positive value of ‘ $a$ ’ for which the equation, $2{x}^{2}+(a-10)x+\frac{33}{2}=2a$ has real roots is ___________.

If $|x|<1,|y|<1$ and $x\neq 1$, then the sum to infinity of the following series $(x+y)+({x}^{2}+xy+{y}^{2})+({x}^{3}+{x}^{2}y+x{y}^{2}+{y}^{3})+.....$ is

If the determinant |1 2 3; 4 5 6; 7 8 k| = 0, then k equals:

The inverse function of $f(x)=\frac{{8}^{2x}-{8}^{-2x}}{{8}^{2x}+{8}^{-2x}},x\in (-1,1),$ is __________.

If the ${10}^{th}$, term of an A.P. is $\frac{1}{20}$, and its ${20}^{th}$, term is $\frac{1}{10}$, then the sum of its first $200$, terms is.

Let $S$, be the set of all real roots of the equation, ${3}^{x}({3}^{x}-1)+2=|{3}^{x}-1|+|{3}^{x}-2|$, then

Let $u=\frac{2z+i}{z-ki},z=x+iy$ and $k>0$. If the curve represented by$Re(u)+Im(u)=1$ intersects the $y$-axis at points $P$ and $Q$ where $\mathrm{PQ}=5$ then the value of $k$ is

Let $S$ be the sum of the first $9$ term of the series : ${x+ka}+{{x}^{2}+(k+2)a}+{{x}^{3}+(k+4)a}+{{x}^{4}+(k+6)a}+\ldots$ where $a\neq 0$ and $x\neq 1$. If $S=\frac{{x}^{10}-x+45a(x-1)}{x-1}$ , then $k$ is equal to

The product of the roots of the equation $9{x}^{2}-18|x|+5=0$ is :

The value of $0.{16}^{{\mathrm{log}}_{2.5}(\frac{1}{3}+\frac{1}{{3}^{2}}+\frac{1}{{3}^{3}}+\ldots .\infty )}$ is __________

Let $a,1{a}_{2},\ldots ,{a}_{n}$ be a given A.P. whose common difference is an integer and ${S}_{n}={a}_{1}+{a}_{2}+\ldots +{a}_{n}$. If ${a}_{1}=1,{a}_{n}=300$ and $15\leq n\leq 50,$ then the ordered pair $({S}_{n-4},{a}_{n-4})$ is equal to:

If the system of equations $x-2y+3z=9$ $2x+y+z=b$ $x-7y+az=24,$ has infinitely many solutions, then $a-b$ is equal to ______

Let ${(2{x}^{2}+3x+4)}^{10}=\sum _{r=0}^{20}{a}_{r}{x}^{r}.$ Then $\frac{{a}_{7}}{{a}_{13}}$ is equal to ______

Let $X={n\in N:1\leq n\leq 50}$. If $A={n\in X:n is a multiple of2}$ and $B={n\in X:n is a multiple of 7}$, then the number of elements in the smallest subset of $X$, containing both $A$ and $B$, is.

The sum of the first three terms of $G.P$ is $S$and their products is $27$. Then all such $S$ lie in

The number of distinct solutions of the equation, ${\mathrm{log}}_{\frac{1}{2}}⁡|\mathrm{sin}⁡x|=2-{\mathrm{log}}_{\frac{1}{2}}⁡|\mathrm{cos}⁡x|$ in the interval $[0,2\pi ],$ is ________

Five numbers are in $A.P.,$ whose sum is $25$ and product is $2520.$ If one of these five numbers is $-\frac{1}{2},$ then the greatest number amongst them is

The greatest positive integer $k,$ for which ${49}^{k}+1$ is a factor of the sum ${49}^{125}+{49}^{124}+\ldots +{49}^{2}+49+1,$ is

If $\alpha$ and $\beta$ are the roots of the equation $2x(2x+1)=1$, then $\beta$ is equal to :

Let $\lambda \neq 0$ be in $R$. If $\alpha$ and $\beta$ are the roots of the equation, ${x}^{2}-x+2\lambda =0$ and $\alpha$ and $\gamma$ are the roots of the equation, $3{x}^{2}-10x+27\lambda =0,$ then $\frac{\beta \gamma }{\lambda }$ is equal to:

If $\alpha$ and $\beta$ are the roots of the equation ${x}^{2}+px+2=0$ and $\frac{1}{\alpha }$ and $\frac{1}{\beta }$ are the roots of the equation $2{x}^{2}+2qx+1=0,$ then $(\alpha -\frac{1}{\alpha })(\beta -\frac{1}{\beta })(\alpha +\frac{1}{\beta })(\beta +\frac{1}{\alpha })$ is equal to :

Let $\alpha$ and $\beta$ be the roots of the equation, $5{x}^{2}+6x-2=0$. If ${S}_{n}={\alpha }^{n}+{\beta }^{n},n=1,2,3,....,$ then