Algebra PYQ — Page 62

The value of $\lambda$ such that sum of the squares of the roots of the quadratic equation, ${x}^{2}+(3-\lambda ) x+2=\lambda$ has the least value is:

If the system of linear equations $x+y+z=5$, $x+2y+2z=6$, $x+3y+\lambda z=\mu , (\lambda , \mu \in R)$ , has infinitely many solutions, then the value of $\lambda +\mu$ is:

Let $a,b$ and $c$ be in $G.P.$ with common ratio $r,$ where $a\neq 0$ and $0<r\leq \frac{1}{2}.$ If $3a,7b$ and $15c$ are the first three terms of an $A.P.,$ then the ${4}^{th}$ term of this $A.P.$ is :

The number of natural numbers less than $7000$ which can be formed by using the digits $0, 1, 3, 7, 9$ (repetition of digits allowed) is equal to:

If $\lambda$ be the ratio of the roots of the quadratic equation in $x, 3{m}^{2}{x}^{2}+m(m-4)x+2=0$, then the least value of $m$ for which $\lambda +\frac{1}{\lambda }=1$, is :

For $x\in R-{0, 1},$ let ${f}_{1}(x)=\frac{1}{x},{f}_{2}(x)=1-x$ and ${f}_{3}(x)=\frac{1}{1-x}$ be three given functions. If a function, $J(x)$ satisfies $({f}_{2}oJo{f}_{1})(x)={f}_{3}(x)$ then $J(x)$ is equal to:

Let $N$ be the set of natural numbers and two functions $f$ and $g$ be defined as $f,g:N\rightarrow N$ such that $f(n)={\begin{matrix}\frac{n+1}{2}, if n is odd \\ \frac{n}{2}, if n is even\end{matrix}$ and $g(n)=n-{(-1)}^{n}.$ Then $fog$ is:

If $\alpha$ and $\beta$ are the roots of the quadratic equation ${x}^{2}+xsin\theta -2sin\theta =0, \theta \in (0,\frac{\pi }{2})$ , then $\frac{{\alpha }^{12}+{\beta }^{12}}{({\alpha }^{-12}+{\beta }^{-12}){.(\alpha -\beta )}^{24}}$ is equal to :

If the third term in the binomial expansion of ${(1+{x}^{{\mathrm{log}}_{2}x})}^{5}$ equals $2560,$ then a possible value of $x$ is

Let $Z$ be the set of integers. If $A={x\in Z :{2}^{(x+2)({x}^{2}-5x+6)}=1}$ and $B={x\in Z :- 3< 2x-1<9}$, then the number of subsets of the set $A\times B$, is :

If $f(x)={\mathrm{log}}_{e}(\frac{1-x}{1+x}), |x|< 1,$ then $f(\frac{2x}{1+{x}^{2}})$ is equal to

Let $z$ be a complex number such that $|z|+z=3+i$ $($ where $i=\sqrt{-1})$ Then $|\mathrm{z}|$ is equal to :

The number of functions $f$ from $\{1,2,3, \ldots, 20\}$ onto $\{1,2,3, \ldots, 20\}$ such that $f(k)$ is a multiple of $3,$ whenever $k$ is a multiple of 4 is:

Let a function $f:(0, \infty) \rightarrow(0, \infty)$ be defined by $f(x)=\left|1-\frac{1}{x}\right| .$ Then $f$ is :

If ${}^{n}{C}_{4},{ }^{n}{C}_{5}$ and ${}^{n}{C}_{6}$ are in A.P., then $n$ can be

If both the roots of the quadratic equation ${x}^{2}-mx+4=0$ are real and distinct and they lie in the interval $(1, 5),$ then $m$ lies in the interval: Note: In the actual JEE paper interval was $[1, 5]$

If $[\begin{matrix}1 & 1 \\ 0 & 1\end{matrix}][\begin{matrix}1 & 2 \\ 0 & 1\end{matrix}][\begin{matrix}1 & 3 \\ 0 & 1\end{matrix}]\ldots .[\begin{matrix}1 & n-1 \\ 0 & 1\end{matrix}]=[\begin{matrix}1 & 78 \\ 0 & 1\end{matrix}],$ then the inverse of $[\begin{matrix}1 & n \\ 0 & 1\end{matrix}]$ is:

The sum of the solutions of the equation $|\sqrt{x}-2|+\sqrt{x}(\sqrt{x}-4)+2=0,(x>0)$ is equal to

Let $A=\left(\begin{array}{ccc}0 & 2 q & r \\ p & q & -r \\ p & -q & r\end{array}\right)$. If $\mathrm{AA}^{\mathrm{T}}=\mathrm{I}_{3},$ then $|\mathrm{p}|$ is:

Let $d\in R,$ and $A=[\begin{matrix}-2 & 4+d & (\mathrm{sin}\theta )-2 \\ 1 & (\mathrm{sin}\theta )+2 & d \\ 5 & (2\mathrm{sin}\theta )-d & (-\mathrm{sin}\theta )+2+2d\end{matrix}],$ $\theta \in [0, 2\pi ].$ If the minimum value of $det(A)$ is $8,$ then a value of $d$ is:

The number of integral values of $m$ for which the equation, $(1+{m}^{2}){x}^{2}-2(1+3m)x+(1+8m)=0$ has no real root, is

The system of linear equations $x+y+z=2$ $2x+3y+2z=5$ $2x+3y+({a}^{2}-1)z=a+1$

Consider the quadratic equation $(c-5){x}^{2}-2cx+(c-4)=0,c\neq 5.$ Let $S$ be the set of all integral values of $c$ for which one root of the equation lies in the interval $(0, 2)$ and its other root lies in the interval $(2, 3).$ Then the number of elements in $S$ is

If $m$ is chosen in the quadratic equation $({m}^{2}+1){x}^{2}-3x+{({m}^{2}+1)}^{2}=0$ such that the sum of its roots is greatest, then the absolute difference of the cubes of its roots is: