Let λ, μ ∈ R . If system of equations 3 x+5 y+λ z=3 7 x+11 y-9 z=2 97 x+155 y-189 z=μ has infi

English

Gujarati

Hindi

3 and 4 .Determinants and Matrices

hard

Let $\lambda, \mu \in R$. If the system of equations

$ 3 x+5 y+\lambda z=3 $

$ 7 x+11 y-9 z=2 $

$ 97 x+155 y-189 z=\mu$

has infinitely many solutions, then $\mu+2 \lambda$ is equal to :

$25$

$24$

$27$

$22$

(JEE MAIN-2024)

Solution

$ 3 x+5 y+\lambda z=3 $

$ 7 x+11 y-9 z=2 $

$ 97 x+155 y-189 z=\mu $

$ 93 x+155 y+31 \lambda z=93 $

$ 97 x+155 y-189 z=\mu $

$ -\quad-\quad+\quad- $

$ -4 x+(31 \lambda+189) z=93-\mu $

$ 1085 x+1705 y-1395 z=310 $

$ 1067 x+1705 y-2079 z=11 \mu $

$ -\quad+\quad-\quad-\quad $

$ 18 x+684 z=310-11 \mu $

$ -36 x+9(31 \lambda+189) z=9(93-\mu) $

$ 36 x+1368 z=2(310-11 \mu) $

$ (279 \lambda+3069) z=1457-31 \mu $

$ \text { for infinite solutions – } $

$ \lambda=\frac{-3069}{279}=\frac{-341}{31} $

$ \mu=\frac{1457}{31}$

$\mu+2 \lambda=\frac{1457-682}{31}=\frac{775}{31}=25$

Standard 12

Mathematics

Let $\alpha, \beta(\alpha \neq \beta)$ be the values of $m$, for which the equations $x+y+z=1 ; x+2 y+4 z=m$ and $x+4 y+10 z=m^2$ have infinitely many solutions. Then the value of $\sum_{n=1}^{10}\left(n^\alpha+n^\beta\right)$ is equal to :

hard

(JEE MAIN-2025)

If $a, b, c$ are three complex numbers such that $a^2 + b^2 + c^2 = 0$ and $\left| {\begin{array}{*{20}{c}}
{\left( {{b^2} + {c^2}} \right)}&{ab}&{ac}\\
{ab}&{\left( {{c^2} + {a^2}} \right)}&{bc}\\
{ac}&{bc}&{\left( {{a^2} + {b^2}} \right)}
\end{array}} \right| = K{a^2}{b^2}{c^2}$ then value of $K$ is

normal

The equation $\left| {\begin{array}{{20}{c}}{{{(1 + x)}^2}}&{{{(1 – x)}^2}}&{ – \,(2 + {x^2})}\\{2x + 1}&{3x}&{1 – 5x}\\{x + 1}&{2x}&{2 – 3x}\end{array}} \right|$ $+$ $\left| {\begin{array}{{20}{c}}{{{(1 + x)}^2}}&{2x + 1}&{x + 1}\\{{{(1 – x)}^2}}&{3x}&{2x}\\{1 – 2x}&{3x – 2}&{2x – 3}\end{array}} \right|$ $= 0$

normal

For all values of $A,B,C$ and $P,Q,R$, the value of $\left| {\,\begin{array}{*{20}{c}}{\cos (A – P)}&{\cos (A – Q)}&{\cos (A – R)}\\{\cos (B – P)}&{\cos (B – Q)}&{\cos (B – R)}\\{\cos (C – P)}&{\cos (C – Q)}&{\cos (C – R)}\end{array}\,} \right|$ is

hard

(IIT-1994)

The value of the determinant $\left| {\begin{array}{*{20}{c}}{{a^2}}&a&1\\{\cos \,(nx)}&{\cos \,(n\, + \,1)\,x}&{\cos \,(n\, + \,2)\,x}\\{\sin \,(nx)}&{\sin \,(n\, + \,1)\,x}&{\sin \,(n\, + \,2)\,x}\end{array}} \right|$ is independent of :

normal

Let $\lambda, \mu \in R$. If the system of equations $ 3 x+5 y+\lambda z=3 $ $ 7 x+11 y-9 z=2 $ $ 97 x+155 y-189 z=\mu$ has infinitely many solutions, then $\mu+2 \lambda$ is equal to :

Solution

Similar Questions

Let $\alpha, \beta(\alpha \neq \beta)$ be the values of $m$, for which the equations $x+y+z=1 ; x+2 y+4 z=m$ and $x+4 y+10 z=m^2$ have infinitely many solutions. Then the value of $\sum_{n=1}^{10}\left(n^\alpha+n^\beta\right)$ is equal to :

For all values of $A,B,C$ and $P,Q,R$, the value of $\left| {\,\begin{array}{*{20}{c}}{\cos (A – P)}&{\cos (A – Q)}&{\cos (A – R)}\\{\cos (B – P)}&{\cos (B – Q)}&{\cos (B – R)}\\{\cos (C – P)}&{\cos (C – Q)}&{\cos (C – R)}\end{array}\,} \right|$ is

The value of the determinant $\left| {\begin{array}{*{20}{c}}{{a^2}}&a&1\\{\cos \,(nx)}&{\cos \,(n\, + \,1)\,x}&{\cos \,(n\, + \,2)\,x}\\{\sin \,(nx)}&{\sin \,(n\, + \,1)\,x}&{\sin \,(n\, + \,2)\,x}\end{array}} \right|$ is independent of :

Start a Free Trial Now

Let $\lambda, \mu \in R$. If the system of equations

$ 3 x+5 y+\lambda z=3 $

$ 7 x+11 y-9 z=2 $

$ 97 x+155 y-189 z=\mu$

has infinitely many solutions, then $\mu+2 \lambda$ is equal to :