Explanation

The rank of a matrix is the number of non-zero rows in its row echelon form.

It represents the maximum number of linearly independent rows or columns.

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

The 3rd June Plan announced the partition of British India into 2 states:

• India
• Pakistan

Punjab and Bengal to be divided according to 3rd June Plan

This Plan also known as the Mountbatten Plan or Partition Plan 

It was the last plan for independence.

 Lord Mountbatten assassinated: August 1979

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

Explanation

The pattern of the sequence appears to be formed by adding consecutive increments:

1 + 4 = 5

5 + 9 = 14

14 + 16 = 30

30 + 25 = 55

55 + 36 = 91

The next term in the sequence would be 91.

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.