Consider the system of linear equations x+y+z=5, x+2 y+ ^2 z=9, x… - Vidyadip

MCQ

Consider the system of linear equations $x+y+z=5, x+2 y+\lambda^2 z=9$, $x+3 y+\lambda z=\mu$ where $\lambda, \mu \in R$. Then, which of the following statement is $\text{NOT}$ correct?

A
System has infinite number of solution if $\lambda=1$ and $\mu=13$
B
System is inconsistent if $\lambda=1$ and $\mu \neq 13$
C
System is consistent if $\lambda \neq 1$ and μ = 13
D
System has unique solution if $\lambda \neq 1$ and $\mu \neq 13$

✓

Answer

$\left|\begin{array}{ccc}1 & 1 & 1 \\ 1 & 2 & \lambda^2 \\ 1 & 3 & \lambda\end{array}\right|=0$
$\Rightarrow 2 \lambda^2-\lambda-1=0 \\ \lambda=1,-\frac{1}{2}$
$\left|\begin{array}{ccc}1 & 1 & 5 \\ 2 & \lambda^2 & 9 \\ 3 & \lambda & \mu\end{array}\right|=0$
$\Rightarrow \mu=13$
Infinite solution $\lambda=1$ & $\mu=13$
For unique $\operatorname{sol}^{ n } \lambda \neq 1$
If $\lambda \neq 1$ and $\mu \neq 13$
Considering the case when $\lambda=-\frac{1}{2}$ and $\mu \neq 13$ this will generate no soluton case

Need a full question paper?

Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.

Start Generating Free

Explore more

JEE STD 11 Science question papers→Gujarat Board STD 11 Science question papers→Gujarat Board question paper generator→

Similar questions

If $A$ and $B$ are two independent events such that $P\,(A) = \frac{1}{2},\,\,P(B) = \frac{1}{5},$ then

A student is allowed to select at most $n$ books from a collection of $(2n + 1)$ books. If the total number of ways in which he can select one book is $63$, then the value of $n$ is

Let $L$ be the line $y = 2x,$ in the two dimensional plane.

Statement $1:$ The image of the point $(0, 1)$ in $L$ is the point $\left( {\frac{4}{5},\frac{3}{5}} \right).$

Statement $2:$ The points $(0, 1)$ and $\left( {\frac{4}{5},\frac{3}{5}} \right)$ lie on opposite sides of the line $L$ and are at equal distance from it.

Let $g(x)=3 f\left(\frac{x}{3}\right)+f(3-x)$ and $f^{\prime \prime}(x)>0$ for all $x \in(0,3)$. If $g$ is decreasing in $(0, \alpha)$ and increasing in $(\alpha, 3)$, then $8 \alpha$ is

Let $f$ be a differentiable function $R$ to $R$ such that $\left| {f\,(x)\, - \,f(y)} \right|\, \le \,2\,{\left| {x - y} \right|^{\frac{3}{2}}},$ for all $x,y\,\in R .$ If $f\,(0)=1$ then $\int\limits_0^1 {{f^2}\,(x)\,dx} $ is equal to

Let $p =99$ and $q =101$. Define $p _1=\log \left(\frac{ p + q }{2}\right)$ and $q _1=\frac{1}{2}(\log p+\log q)$ and $p _2=\log \left(\frac{ p _1+ q _1}{2}\right), \quad q _2=\frac{1}{2}\left(\log p _1+\log q _1\right).$ Where all logarithms have base $10$ . Then

If $y = {x^2} + {1 \over {{x^2} + {1 \over {{x^2} + {1 \over {{x^2} + ......\infty }}}}}},$ then ${{dy} \over {dx}} = $

In an ellipse, its foci and ends of its major axis are equally spaced. If the length of its semi-minor axis is $2 \sqrt{2}$, then the length of its semi-major axis is

Find adjoint of each of the matrices $\left[\begin{array}{ccc}1 & -1 & 2 \\ 2 & 3 & 5 \\ -2 & 0 & 1\end{array}\right]$

Let $\text{ABC}$ be an equilateral triangle, let $\text{KLMN}$ be a rectangle with $\text{K, L}$ on $\text{B C, M}$ on $\text{AC}$ and $N$ on $\text{AB}$. Suppose $\text{AN / NB}=2$ and the area of $\triangle \text{BKN}$ is $6$ .The area of the $\triangle \text{ABC}$ is