The value of \( k \), for which the linear equations \( 2x + 3y = 6 \) and \( 4x + 6y = 3k \) have at least one solution, is ________. (Answer in integer)
The value of \( k \), for which the linear equations \( 2x + 3y = 6 \) and \( 4x + 6y = 3k \) have at least one solution, is ________. (Answer in integer)
Show Hint
Solution and Explanation
\[ \begin{aligned} 2x + 3y &= 6 \quad \text{(Equation 1)} \\ 4x + 6y &= 3k \quad \text{(Equation 2)} \end{aligned} \] Step 2: Analyze the relationship between the equations.
Notice that Equation 2 is exactly twice Equation 1 on the left-hand side: \[ 4x + 6y = 2(2x + 3y) \] So the left-hand side of Equation 2 is simply: \[ 4x + 6y = 2 \cdot 6 = 12 \] Step 3: Determine the condition for consistency.
For the two equations to represent the same line (and thus have infinitely many solutions or at least one), the right-hand side of Equation 2 must also be 12: \[ 3k = 12 \quad \Rightarrow \quad k = \frac{12}{3} = 4 \] Final Answer:
The system has at least one solution when \( \boxed{k = 4} \).
Top Questions on Numerical Methods
- The value of \( \alpha \) for which the Euler method with step size \( h = 0.1 \) provides \( y(1.1) = 1.2 \) for the following initial value problem is \[ \frac{dy}{dx} = -2xy^\alpha, \quad y(1) = 2 \]
- GATE TF - 2025
- Engineering Mathematics
- Numerical Methods
- Consider a nonlinear algebraic equation, \( e^x - 2 = 0 \). Using the Newton-Raphson method, with the initial guess of \( x_0 = 1 \), the approximated value of the root of the equation after one iteration is ________.
- GATE BT - 2025
- Engineering Mathematics
- Numerical Methods
The Newton-Raphson method is used to find the root of \[ f(x) \equiv x^2 - x - 1 = 0 \] Starting with an initial guess \( x_0 = 1 \), the second iterate \( x_2 \) is __________ (rounded off to 2 decimal places).
- GATE CH - 2025
- Mechanical Engineering
- Numerical Methods
An approximate solution of the equation \( x^3 - 17 = 0 \) is to be obtained using the Newton-Raphson method. If the initial guess is \( x_0 = 2 \), the value at the end of the first iteration is \( x_1 = \) ________ (rounded off to two decimal places).
- GATE AE - 2025
- Engineering Mathematics
- Numerical Methods
- An approximate solution of the equation \( x^3 - 17 = 0 \) is to be obtained using the Newton-Raphson method. If the initial guess is \( x_0 = 2 \), the value at the end of the first iteration is \( x_1 = \) \_\_\_\_\_\ (rounded off to two decimal places).
- GATE AE - 2025
- Engineering Mathematics
- Numerical Methods
Questions Asked in GATE BT exam
- A thin wire is used to construct all the edges of a cube of 1 m side by bending, cutting, and soldering the wire. If the wire is 12 m long, what is the minimum number of cuts required to construct the wire frame to form the cube?
- Rohit goes to a restaurant for lunch at about 1 PM. When he enters the restaurant, he notices that the hour and minute hands on the wall clock are exactly coinciding. After about an hour, when he leaves the restaurant, he notices that the clock hands are again exactly coinciding. How much time (in minutes) did Rohit spend at the restaurant?
A color model is shown in the figure with color codes: Yellow (Y), Magenta (M), Cyan (Cy), Red (R), Blue (Bl), Green (G), and Black (K). Which one of the following options displays the color codes that are consistent with the color model?
- If the function
\[ f(x) = \begin{cases} \sin(2x), & \text{for } x > 0, \\ a + bx, & \text{for } x \leq 0, \end{cases} \]
where \( a \) and \( b \) are constants, is differentiable at \( x = 0 \), then \( a + b \) is equal to:- GATE BT - 2025
- Continuity and differentiability
An object is said to have an n-fold rotational symmetry if the object, rotated by an angle of \( \frac{2\pi}{n} \), is identical to the original.
Which one of the following objects exhibits 4-fold rotational symmetry about an axis perpendicular to the plane of the screen?
- GATE CH - 2025
- GATE BT - 2025
- GATE GG - 2025
- Rotational Symmetry