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 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 $\mu=13$
✓
System has unique solution if $\lambda \neq 1$ and $\mu \neq 13$

✓

Answer

Correct option: D.

System has unique solution if $\lambda \neq 1$ and $\mu \neq 13$

d
$\begin{aligned} & \left|\begin{array}{ccc}1 & 1 & 1 \\ 1 & 2 & \lambda^2 \\ 1 & 3 & \lambda\end{array}\right|=0 \\ & \Rightarrow 2 \lambda^2-\lambda-1=0 \\ & \lambda=1,-\frac{1}{2} \\ & \left|\begin{array}{ccc}1 & 1 & 5 \\ 2 & \lambda^2 & 9 \\ 3 & \lambda & \mu\end{array}\right|=0 \Rightarrow \mu=13\end{aligned}$

Infinite solution $\lambda=1 \& \mu=13$

For unique $\operatorname{sol}^{\mathrm{n}} \lambda \neq 1$

For no $\operatorname{sol}^{\mathrm{n}} \lambda=1 \& \mu \neq 13$

If $\lambda \neq 1$ and $\mu \neq 13$

Considering the case when $\lambda=-\frac{1}{2}$ and $\mu \neq 13$ this will generate no solution 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

Circles $x^2 + y^2 + 4x + d = 0, x^2 + y^2 + 4fy + d = 0$ touch each other, if

Let two circles $C_1$ and $C_2$ of radii $2$ and $4$ be tangent at point $P$ and tangent to a common straight line (not passing through $P$ ) at points $Q$ and $R$ , then value of $PQ^2 + QR^2 + RP^2$ is

If ${C_0},{C_1},{C_2},.......,{C_n}$ are the binomial coefficients, then $2.{C_1} + {2^3}.{C_3} + {2^5}.{C_5} + ....$ equals

If $P$ is a $3 \times 3$ matrix such that $P^{\top}=2 P+I$, where $P^{\top}$ is the transpose of $P$ and $I$ is the $3 \times 3$ identity matrix, then there exists a column matrix $X=\left[\begin{array}{l}x \\ y \\ z\end{array}\right] \neq\left[\begin{array}{l}0 \\ 0 \\ 0\end{array}\right]$ such that

Differential coefficient of ${\tan ^{ - 1}}\sqrt {{{1 - {x^2}} \over {1 + {x^2}}}} $ w.r.t. ${\cos ^{ - 1}}({x^2})$ is

A student appeared in an examination consisting of $8$ true - false type questions. The student guesses the answers with equal probability. The smallest value of $\mathrm{n}$, so that the probability of guessing at least $'n'$ correct answers is less than $\frac{1}{2}$, is:

Solution of the equation $(1 - {x^2})dy + xydx = x{y^2}dx$ is

The number of values of $'r'$ satisfying $^{69}C_{3r-1} - ^{69}C_{r^2}=^{69}C_{r^2-1} - ^{69}C_{3r}$ is :-

If $\mathrm{R}=\left\{(\mathrm{x}, \mathrm{y}): \mathrm{x}, \mathrm{y} \in \mathrm{Z}, \mathrm{x}^{2}+3 \mathrm{y}^{2} \leq 8\right\}$ is a relation on the set of integers $\mathrm{Z},$ then the domain of $\mathrm{R}^{-1}$ is

If in a right angled triangle $ABC$, the hypotenuse $AB = p,$ then $\overrightarrow {AB} \,\,.\,\,\overrightarrow {AC} + \overrightarrow {BC} \,.\,\,\overrightarrow {BA} + \overrightarrow {CA} \,\,.\,\,\overrightarrow {CB} $ is equal to