Using the distance formula: d = √[(3 - 3)² + (1 - 7)²] = √[0 + 36] = √36 = 6.

Using the distance formula: d = √[(1 - 5)² + (1 - 5)²] = √[16 + 16] = √32 = 4√2.

The standard form of a circle's equation is (x-h)² + (y-k)² = r². Here, h=3, k=-2, r=5, so (x-3)² + (y+2)² = 25.

Standard form of a circle: (x-h)² + (y-k)² = r². Here, h=2, k=-3, r=5.

Parallel lines have the same slope. Using point-slope form: y - 2 = 3(x - 1) gives y = 3x - 1.

The equation of a line through the origin with slope m is y = mx. Thus, y = -3x.

Using the slope-intercept form y = mx + b, with m = -4 and b = 0, the equation is y = -4x.

The latus rectum of a parabola y^2 = 4px is given by 4p. Here, 4p = 12, so p = 3, and the latus rectum is 4p = 12.

The length of the diagonal d of a rectangle can be found using the Pythagorean theorem: d = √(length² + width²) = √(6² + 8²) = √(36 + 64) = √100 = 10 cm.

Diagonal = √(length² + width²) = √(6² + 8²) = √(36 + 64) = √100 = 10 cm

Area = πr². Given area = 50π, r² = 50, r = √50 = 5√2. Diameter = 2r = 10√2 units.

Using the Taylor series expansion for cos(x), we find that lim (x -> 0) (cos(x) - 1)/x^2 = -1/2.

Using the derivative of e^x at x = 0, we find that lim (x -> 0) (e^x - 1)/x = 1.

The measure of each interior angle in a regular hexagon can be calculated using the formula (n-2) * 180/n, where n is the number of sides. For a hexagon, n=6, so the measure is (6-2) * 180/6 = 120 degrees.

A regular quadrilateral is a square, and each angle in a square measures 90 degrees.

The exterior angle of a triangle is equal to the sum of the two opposite interior angles. Therefore, the exterior angle = 50 + 60 = 110 degrees.

The exterior angle of a triangle is equal to the sum of the two opposite interior angles. Therefore, the exterior angle = 50 + 60 = 110 degrees.

The number 4 appears most frequently (3 times). Hence, mode = 4.

The order of a matrix is given by the number of rows followed by the number of columns. Therefore, a matrix with 3 rows and 4 columns is of order 3x4.

The perimeter of a rectangle is given by P = 2(length + width). Here, P = 2(10 + 5) = 30 cm.

There are 13 hearts in a deck of 52 cards. Probability = 13/52 = 1/4.

The product of the roots of the equation ax^2 + bx + c = 0 is given by c/a. Here, c = 6 and a = 2, so the product is 6/2 = 3.

Using Vieta's formulas, the product of the roots is c/a = 10/1 = 10.

The rank of E is 2 because the rows are linearly dependent (the third row is a linear combination of the first two).

The real part of z = 4 + 5i is 4.

In an equilateral triangle, all three angles are equal and measure 60 degrees each.

When two parallel lines are cut by a transversal, the corresponding angles are equal.

The sum of the interior angles of a triangle is 180 degrees, and the angles formed by a transversal cutting through two parallel lines are supplementary, meaning they add up to 180 degrees.

(a - b)(a + b) = a² - b², which is the difference of squares.

(x + y)(x - y) = x² - y², which is the difference of squares.