If the system of equations
$ x + y + z = 6 $,
$ 2x + 5y + \alpha z = \beta $,
$ x + 2y + 3z = 14 $
has infinitely many solutions, then $ \alpha + \beta $ is equal to:
If the system of equations
$ x + y + z = 6 $,
$ 2x + 5y + \alpha z = \beta $,
$ x + 2y + 3z = 14 $
has infinitely many solutions, then $ \alpha + \beta $ is equal to:
- 8
- 36
- 44
- 48
The Correct Option is C
Solution and Explanation
\(\begin{vmatrix}1&1&1\\2&5&α\\1&2&3\end{vmatrix}\)=1(15−2α)–1(6−α)+1(−1)
=15–2α–6+α−1
=8–α
For infinite solutions,~Δ=0 ⇒α=8
\(\triangle_x\)=\(\begin{vmatrix}6&1&1\\β&5&8\\14&2&3\end{vmatrix}\)= 6(−1)−1(3β–112)+1(2β−70)
=−6–3β+112+2β–70
=36–β
\(\triangle_x\)=0
⇒ For β=36
α+β=44
So, the correct option is (C): 44
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Concepts Used:
Differential Equations
A differential equation is an equation that contains one or more functions with its derivatives. The derivatives of the function define the rate of change of a function at a point. It is mainly used in fields such as physics, engineering, biology and so on.
Orders of a Differential Equation
First Order Differential Equation
The first-order differential equation has a degree equal to 1. All the linear equations in the form of derivatives are in the first order. It has only the first derivative such as dy/dx, where x and y are the two variables and is represented as: dy/dx = f(x, y) = y’
Second-Order Differential Equation
The equation which includes second-order derivative is the second-order differential equation. It is represented as; d/dx(dy/dx) = d2y/dx2 = f”(x) = y”.
Types of Differential Equations
Differential equations can be divided into several types namely
- Ordinary Differential Equations
- Partial Differential Equations
- Linear Differential Equations
- Nonlinear differential equations
- Homogeneous Differential Equations
- Nonhomogeneous Differential Equations