The maximum length of the third side = sum of the other two sides - 1 = 7 + 10 - 1 = 16 cm.

The coefficient of x^3 in (2x - 3)^4 is given by 4C1 * (2)^3 * (-3)^1 = 4 * 8 * (-3) = -96. The coefficient is -108.

Using the binomial theorem, the coefficient of x^5 in (3x - 4)^7 is given by 7C5 * (3)^5 * (-4)^2 = 21 * 243 * 16 = 21 * 3888 = 81588.

Using the binomial theorem, the coefficient of x^4 in (x + 3)^6 is given by 6C4 * (3)^2 = 15 * 9 = 135.

The sum of angles in a triangle is 180 degrees. Therefore, angle C = 180 - (30 + 60) = 90 degrees.

Since two angles are equal (45 degrees), triangle DEF is an isosceles triangle.

The longest side of triangle GHI is GI, which measures 26 cm.

Perimeter = PQ + QR + PR = 8 + 6 + 10 = 24 cm.

The function f(x) = sqrt(x) is continuous at x = 0 as it is defined and the limit exists.

Subtract 2x from both sides: x - 7 = 5. Then add 7: x = 12.

Circumference = 2πr => r = circumference / (2π) = 31.4 / (2 * 3.14) = 5 cm

Centroid = ((2+4+6)/3, (3+5+7)/3) = (4, 5).

Using point-slope form: y - 3 = 2(x - 1) => y = 2x + 1.

The distance from the vertex to the focus is 3, so the equation is x^2 = 12y.

Factoring the equation gives (x - 2)(x - 5) = 0, so the roots are 2 and 5.

The number 5 appears most frequently (3 times), so the mode is 5.

The function f(x) = 1/(x-1) is discontinuous at x = 1, hence it is continuous on (1, ∞).

The left limit as x approaches 0 is 0, but the right limit is 1. Hence, it is not continuous at x = 0.

At x = 0, lim x→0- f(x) = 0 and lim x→0+ f(x) = 1, hence it is discontinuous at x = 0.

Using Heron's formula, s = (5 + 12 + 13)/2 = 15. Area = √[s(s-a)(s-b)(s-c)] = √[15(15-5)(15-12)(15-13)] = √[15*10*3*2] = √[900] = 30 cm².

Using the Pythagorean theorem, 7^2 + 24^2 = 49 + 576 = 625 = 25^2. Therefore, triangle ABC is a right triangle.

Mean = (5 + 15 + 25 + x) / 4 = 20. Solving gives x = 30.

Mean = (7 + 9 + 11 + 13) / 4 = 40 / 4 = 10.

Mean = (8 + 12 + 16 + 20) / 4 = 56 / 4 = 14.

Standard deviation is the square root of variance. Therefore, SD = √4 = 2.

Mean = (4 + 8 + 10 + 12 + x) / 5 = 10. Solving gives x = 14.

Let the other two numbers be a and b. Then, (10 + a + b) / 3 = 15. Solving gives (a + b) = 35, so Mean of a and b = 35 / 2 = 17.5.

The discriminant must be less than zero: (-4)^2 - 4*2*k < 0 leads to k > 8.

This is a perfect square trinomial: (x + 3)(x + 3) = 0.

For both roots to be negative, k must be greater than the square of half the coefficient of x, hence k > 9.