The equation y^2 = 4ax represents a parabola that opens to the right.

The equation y = a^x represents an exponential function where a is a constant.

Q1 = 3, Q3 = 6. Interquartile Range = Q3 - Q1 = 6 - 3 = 3.

Since 7^2 + 24^2 = 25^2, triangle ABC is a right triangle.

To determine the nature of the lines, we can analyze the discriminant of the quadratic equation.

The lines are perpendicular if the product of their slopes is -1. We can find the slopes from the equation and check this condition.

The discriminant is negative, indicating that the lines are perpendicular.

The lines intersect at the origin (0,0) as derived from the equation.

The lines intersect at the origin, which can be verified by substituting x = 0 and y = 0 into the equation.

The lines are perpendicular if the product of their slopes is -1, which can be verified from the equation.

The lines intersect at the origin and are not parallel, hence they are intersecting.

Completing the square shows that the lines intersect at two distinct points.

The equation can be factored as (x - 3y)^2 = 0, indicating that the lines are coincident.

To find the maximum, we calculate f'(x) = -2x + 4. Setting f'(x) = 0 gives x = 2. Since f''(x) = -2 < 0, this is a maximum point.

The vertex form of a parabola gives the maximum value at x = -b/(2a) = 2. Evaluating f(2) = -2^2 + 4*2 + 1 = 9.

The vertex of the parabola given by f(x) = -x^2 + 4x + 1 occurs at x = -b/(2a) = -4/(-2) = 2, which gives the maximum value.

Coefficient of Variation = (Standard Deviation / Mean) * 100 = (5 / 50) * 100 = 10%.

Arranging the numbers: 2, 3, 5, 7, 11, 13. Median = (5 + 7) / 2 = 6.

Finding the derivative f'(x) = 4x^3 - 16x. Setting f'(x) = 0 gives x = 0, ±2. Evaluating f(0) = 16, f(2) = 0, and f(-2) = 0, the minimum value is 0.

Mode is the number that appears most frequently, which is 4.

The slopes can be found by solving the quadratic equation in terms of m, yielding slopes -1 and -2.

The discriminant of the quadratic equation is positive, indicating two distinct real roots.

The product of the slopes of the lines is given by m1*m2 = c/a = 1/2 = -2.

Factoring gives (2x - 3y)(2x - 3y) = 0, indicating the lines are coincident.

The discriminant of the quadratic equation is positive, indicating two distinct real roots.

The discriminant of the quadratic equation is negative, indicating imaginary lines.

Rearranging gives (x-2)^2 + (y-2)^2 = 0, which represents two intersecting lines.

To determine the nature of the lines, we can rewrite the equation in the form of (x - a)^2 + (y - b)^2 = r^2 and analyze the discriminant.

Rearranging gives (x-2)^2 + (y-3)^2 = 0, which represents a single point, hence the lines are coincident.

To determine the nature of the lines, we can find the slopes from the equation. The product of the slopes will help us conclude if they are perpendicular.