Explanation

Skew lines are lines that are not parallel and do not intersect, and they exist in three-dimensional space.

They lie in different planes, unlike parallel or intersecting lines.

Explanation

To find the slope of the tangent to the curve y³x + y²x² = 6 at (2, 1), we'll use implicit differentiation:

Given: y³x + y²x² = 6

Differentiating both sides with respect to x:

y³ + x_3y²(dy/dx) + y²_2x + x²*2y(dy/dx) = 0

Rearranging terms to isolate dy/dx:

(3xy² + 2x²y)(dy/dx) = -y³ - 2xy²

dy/dx = (-y³ - 2xy²) / (3xy² + 2x²y)

Now, evaluating dy/dx at (2, 1):

dy/dx = (-(1)³ - 2_2_(1)²) / (3_2_(1)² + 2*(2)²*1)

= (-1 - 4) / (6 + 8)

= -5 / 14

Explanation

Given that 1 - i is a root of the equation x² + ax + b = 0, and the coefficients are real, the complex conjugate 1 + i must also be a root.

The sum of the roots = (1 - i) + (1 + i) = 2 = -a

So, a = -2

The product of the roots = (1 - i)(1 + i) = 1² - i² = 1 + 1 = 2 = b

So, b = 2

Now, a - b = -2 - 2 = -4

Explanation

To find ∫ln(x)dx, we use integration by parts:

Let u = ln(x) and dv = dx

Then du = 1/x dx and v = x

∫ln(x)dx = uv - ∫v du

= x_ln(x) - ∫x * 1/x dx

= x_ln(x) - ∫1 dx

= x*ln(x) - x + C

The answer is xlnx – x + c.

Explanation

For matrices A × B ≠ B × A in general, so matrix multiplication is not commutative.

However, matrix addition is commutative and associative, and matrix multiplication is associative, but not commutative.

Explanation

Lagrange's Mean Value Theorem is a special case of Cauchy's Mean Value Theorem when one function is f(x) and the other is g(x) = x.

Cauchy’s theorem is more general and applies to two functions.

Explanation

To find the fraction equivalent to 2/5%, first convert the percentage to a decimal:

2/5% = (2/5) × (1/100) = 2/500 = 1/250

Explanation

The given partial differential equation:

∂²U/∂x² + ∂²U/∂y² = f(x,y)

is known as the Poisson equation.

If f(x,y) = 0, it would reduce to the Laplace equation.

Explanation

Skew lines are non-coplanar, do not intersect, and are not parallel.

They exist in 3D geometry, not lying in the same plane.

Explanation

The Maclaurin series expansion of cos(x) is derived from its Taylor series expansion around x = 0:

cos(x) = 1 - x²/2! + x⁴/4! - x⁶/6! + ...

This series alternates signs and involves even powers of x, with factorials in the denominators.

The answer is cosx = 1 - x²/2! + x⁴/4! - x⁶/6! + ...