Question
Which value(s) of $\lambda,$ do the pair of linear equations $\lambda\text{x}+\text{y}=\lambda^2$ and $\text{x}+\lambda\text{y}=1$ have:
No solution?
No solution?
✓
Answer
The given pair of linear equations is
$\lambda\text{x}+\text{y}=\lambda^2$ and $\text{x}+\lambda\text{y}=1$
Here,
$\text{a}_1=\lambda,\text{b}_1=1,\text{c}_1=-\lambda^2$
$\text{a}_2=1,\text{b}_2=\lambda,\text{c}_2=-1$
For no solution,
$\frac{\text{a}_1}{\text{a}_1}=\frac{\text{b}_1}{\text{b}_2}\neq\frac{\text{c}_1}{\text{c}_2}$
$\Rightarrow\frac{\lambda}{1}=\frac{1}{\lambda}\neq\frac{-\lambda^2}{-1}$
$\Rightarrow\lambda^2-1=0$
$\Rightarrow(\lambda-1)(\lambda+1)=0$
$\Rightarrow\lambda=1,-1$
Here, we take only $\lambda=-1$ because at $\lambda=1$ the system of linear equations has infinitely many solutions.
$\lambda\text{x}+\text{y}=\lambda^2$ and $\text{x}+\lambda\text{y}=1$
Here,
$\text{a}_1=\lambda,\text{b}_1=1,\text{c}_1=-\lambda^2$
$\text{a}_2=1,\text{b}_2=\lambda,\text{c}_2=-1$
For no solution,
$\frac{\text{a}_1}{\text{a}_1}=\frac{\text{b}_1}{\text{b}_2}\neq\frac{\text{c}_1}{\text{c}_2}$
$\Rightarrow\frac{\lambda}{1}=\frac{1}{\lambda}\neq\frac{-\lambda^2}{-1}$
$\Rightarrow\lambda^2-1=0$
$\Rightarrow(\lambda-1)(\lambda+1)=0$
$\Rightarrow\lambda=1,-1$
Here, we take only $\lambda=-1$ because at $\lambda=1$ the system of linear equations has infinitely many solutions.
Need a full question paper?
Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.
Explore more
Similar questions
A box contains cards numbered from 1 to 30 . Write the sample space $S$ and no. of sample points $n(S)$ and if a card is drawn at random, write $A$ and $n(A)$ if the card is divisible by 5.
→Solve the following equations by using the method of completing the square:
$x^2 - 4x + 1 = 0$
→$x^2 - 4x + 1 = 0$
Find the value of k for which each of the following system of equations have infinitely many solutions:
→$kx + 3y = 2k + 1$
$2(k + 1)x + 9y = 7k + 1$
Show that the points $A(5, 6), B(1, 5), C(2, 1)$ and $D(6, 2)$ are the vertices of a square.
→Show that A(1, 2), B(4, 3), C(6, 6) and D(3, 5) are the vertices of a Parallelogram. Show that ABCD is not a rectangle.
Prove the following:$\frac{\cos(90^\circ-\text{A})\sin(90^\circ-\text{A})}{\tan(90^\circ-\text{A})}-\sin^2\text{A}=0$
→construct tangents to a circle from a point outside the circle.
A bucket is in the form of a frustum of a cone and holds 15.25 litres of water. The diameters of the top and bottom are 25cm and 20cm respectively. Find its height and area of tin used in its construction.
In figure, PQ || BC and AP : PB = 1 : 2. Find: $\frac{\text{area}(\triangle\text{APQ})}{\text{area}(\triangle\text{ABC})}$
→In the following, determine whether the given quadratic equation have real root and if so, find the root:
$2\text{x}^2+5\sqrt{3}\text{x}+6=0$
→$2\text{x}^2+5\sqrt{3}\text{x}+6=0$