Matrices & DeterminantsmediumPYQ · JEE Main · 2 Apr 2026 · Shift 2 (Afternoon)Free

System with Infinite Solutions: Point (a,b) on x - y = 3 | JEE 2026

JEE Maths question with a full step-by-step solution.

Question
If the system of equations
x+5y+6z=4,2x+3y+4z=7,x+6y+az=bx+5y+6z=4,\quad 2x+3y+4z=7,\quad x+6y+az=b
 has infinitely many solutions, then the point (a,b)(a,b) lies on the line
Ay−x=3y-x=3
Bx−y=3x-y=3correct
Cx+y=11x+y=11
Dx+y=12x+y=12
Solution
Step 1: For infinitely many solutions the determinant of coefficients must vanish:
D=∣15623416a∣=0.D=\begin{vmatrix}1&5&6\\2&3&4\\1&6&a\end{vmatrix}=0.
 Expanding gives
a=507.a=\frac{50}{7}.
 Step 2: Also Dz=0D_z=0 (replacing the third column by the constants):
Dz=∣15423716b∣=0 ⇒ b=297.D_z=\begin{vmatrix}1&5&4\\2&3&7\\1&6&b\end{vmatrix}=0\ \Rightarrow\ b=\frac{29}{7}.
 Step 3: Then
a−b=507−297=217=3,a-b=\frac{50}{7}-\frac{29}{7}=\frac{21}{7}=3,
 so the point (a,b)(a,b) lies on the line x−y=3x-y=3. Correct answer: (2)
Still stuck on this question?Ask your doubt on WhatsApp
Similar questions
Matrices & Determinants · easy
If AA is a 3×33 \times 3 matrix and det⁡A=5\det A = 5, then det⁡(Adj(Adj A))\det\big(\text{Adj}(\text{Adj}\,A)\big) equals:
Matrices & Determinants · easy
The equation (12213434k)(xyz)=(000)\begin{pmatrix} 1 & 2 & 2 \\ 1 & 3 & 4 \\ 3 & 4 & k \end{pmatrix}\begin{pmatrix} x \\ y \\ z \end{pmatrix} = \begin{pmatrix} 0 \\ 0 \\ 0 \end{pmatrix} has a solution (x,y,z)(x, y, z) besides (0,0,0)(0, 0, 0). The value of kk equals:
Matrices & Determinants · easy
AA is an involutory matrix given by A=(01−14−343−34)A = \begin{pmatrix}0 & 1 & -1\\ 4 & -3 & 4\\ 3 & -3 & 4\end{pmatrix}, then the inverse of A2\dfrac{A}{2} will be:
Matrices & Determinants · easy
A square matrix PP satisfies P2=I−PP^2 = I - P, where II is the identity matrix. If Pn=5I−8PP^n = 5I - 8P, then nn is:

Solve more, learn faster

Sign up free to solve more JEE Maths questions and explore doMath — timed drills, mastery sprints, bookmarks, and chapter-wise progress tracking.

Sign up freeExplore doMath