Explanation

If a real root of f(x) = 0 lies in the interval, then the graph of f(x) must cross the x-axis between a and b.[a][b]

To cross the x-axis:

At one end the function value will be positive, f(a) > 0

At the other end the function value will be negative, f(b) < 0

When signs are opposite, their product is negative.

So f(a) × f(b) < 0

Last verified on 05-06-2026

Explanation

Matrix rows are dependent ⇒ Rank = 2

Determinant of this matrix = 0

So, R − D = 2 − 0 = 2

Last verified on 05-06-2026

Explanation

Equation 3x² + 7xy + 2y² = 0 represents a pair of lines through origin.

Factor it:

3x² + 6xy + xy + 2y² = 0  

3x(x + 2y) + y(x + 2y) = 0  

(3x + y)(x + 2y) = 0

So lines are: y = -3x and y = -x/2  

Slopes: m₁ = -3, m₂ = -1/2

Angle between lines:

tan θ = |(m₁ - m₂)/(1 + m₁m₂)|  

= |(-3 + 1/2) / (1 + 3/2)|  

= |(-5/2) / (5/2)| = | -1 | = 1

tan θ = 1 → θ = 45°

So the acute angle between the two lines is 45°.

Last verified on 05-06-2026

Explanation

∂²u/∂x² + ∂²u/∂y² = f(x,y) is the 2D Poisson equation.  

If f(x,y) = 0, then it becomes Laplace's equation: ∇²u = 0

Last verified on 05-06-2026

Explanation

Jacobian J = ∂(x,y)/∂(r,θ) = |∂x/∂r ∂x/∂θ|

                               |∂y/∂r ∂y/∂θ|

∂x/∂r = cosθ, ∂x/∂θ = -r sinθ  

∂y/∂r = sinθ, ∂y/∂θ = r cosθ

J = cosθ × r cosθ - (-r sinθ) × sinθ  

  = r cos²θ + r sin²θ  

  = r(cos²θ + sin²θ) = r

Last verified on 05-06-2026