We use the property that nCa = nCb implies either a = b or a + b = n. Since r is not equal to r + 2, we must have r + (r + 2) = 10, so 2r + 2 = 10, giving 2r = 8 and r = 4.

nCa = nCb => a = b or a + b = n

The binomial theorem states that (a + b)^n = nC0 a^n + nC1 a^(n-1) b + nC2 a^(n-2) b^2 + ... + nCn b^n = sum over r from 0 to n of nCr a^(n-r) b^r. The general term (the (r+1)th term) is T(r+1) = nCr a^(n-r) b^r.

T(r+1) = nCr a^(n-r) b^r

Here n = 6 is even, so there is one middle term, the (n/2 + 1) = 4th term. T4 = 6C3 x^3 (2y)^3 = 20 x^3 x 8 y^3 = 160 x^3 y^3.

middle term = (n/2 + 1)th term when n is even

Here the first term a = 3 and common difference d = 7 - 3 = 4. The nth term is a_n = a + (n - 1)d. So a_20 = 3 + (20 - 1) x 4 = 3 + 76 = 79.

a_n = a + (n - 1)d

Here the first term a = 2 and common ratio r = 6/2 = 3. The nth term of a GP is a_n = a r^(n-1). So a_8 = 2 x 3^7 = 2 x 2187 = 4374.

a_n = a r^(n-1)

AM = (4 + 16)/2 = 20/2 = 10. GM = sqrt(4 x 16) = sqrt64 = 8. Since 10 > 8, we have AM > GM, which agrees with the general result AM >= GM for positive numbers.

AM = (a + b)/2; GM = sqrt(ab)

The slope of a line through (x1, y1) and (x2, y2) is m = (y2 - y1)/(x2 - x1). Here m = (9 - 3)/(5 - 2) = 6/3 = 2.

m = (y2 - y1)/(x2 - x1)

Using the point-slope form y - y1 = m(x - x1) with m = 2 and (x1, y1) = (1, -3): y - (-3) = 2(x - 1), so y + 3 = 2x - 2, giving y = 2x - 5 or 2x - y - 5 = 0.

y - y1 = m(x - x1)

For parallel lines Ax + By + C1 = 0 and Ax + By + C2 = 0, the distance is d = |C1 - C2|/sqrt(A^2 + B^2). Here A = 3, B = -4, C1 = 7, C2 = 5. d = |7 - 5|/sqrt(9 + 16) = 2/5.

d = |C1 - C2|/sqrt(A^2 + B^2)

Compare with x^2 + y^2 + 2gx + 2fy + c = 0: 2g = -4 so g = -2, 2f = 6 so f = 3, c = -12. Centre = (-g, -f) = (2, -3). Radius = sqrt(g^2 + f^2 - c) = sqrt(4 + 9 + 12) = sqrt25 = 5.

centre (-g, -f), radius sqrt(g^2 + f^2 - c)

The equation of a circle with centre (h, k) and radius r is (x - h)^2 + (y - k)^2 = r^2. Here (h, k) = (2, -3) and r = 5, so (x - 2)^2 + (y + 3)^2 = 25. Expanding: x^2 + y^2 - 4x + 6y - 12 = 0.

(x - h)^2 + (y - k)^2 = r^2

The focus is on the positive y-axis and the directrix is y = -3, so the parabola opens upwards with vertex at the origin and form x^2 = 4ay. Here a = 3 (distance from vertex to focus), so x^2 = 4(3)y = 12y.

x^2 = 4ay

The distance between (x1, y1, z1) and (x2, y2, z2) is sqrt[(x2 - x1)^2 + (y2 - y1)^2 + (z2 - z1)^2]. Here = sqrt[(4 - 2)^2 + (3 - 3)^2 + (1 - 5)^2] = sqrt[4 + 0 + 16] = sqrt20 = 2 sqrt5.

d = sqrt[(x2-x1)^2 + (y2-y1)^2 + (z2-z1)^2]

The distance of a point (x, y, z) from the origin is sqrt(x^2 + y^2 + z^2). Here = sqrt(1^2 + (-2)^2 + 3^2) = sqrt(1 + 4 + 9) = sqrt14 units.

d = sqrt(x^2 + y^2 + z^2)

The midpoint of the segment joining (x1,y1,z1) and (x2,y2,z2) is ((x1+x2)/2, (y1+y2)/2, (z1+z2)/2). = ((2+6)/2, (3+7)/2, (4+8)/2) = (4, 5, 6).

M = ((x1+x2)/2, (y1+y2)/2, (z1+z2)/2)

Direct substitution gives 0/0, so factorise the numerator: x^2 - 4 = (x - 2)(x + 2). Then (x^2 - 4)/(x - 2) = x + 2 for x not equal to 2. Taking the limit as x approaches 2 gives 2 + 2 = 4.

a^2 - b^2 = (a - b)(a + b)

The standard limit is lim as x approaches 0 of (sin x)/x = 1 (with x in radians). This is a fundamental result used throughout limits and derivatives of trigonometric functions.

lim x->0 (sin x)/x = 1

By definition f'(x) = lim as h approaches 0 of [f(x + h) - f(x)]/h = lim [ (x + h)^2 - x^2 ]/h = lim [x^2 + 2xh + h^2 - x^2]/h = lim [2xh + h^2]/h = lim (2x + h) = 2x. So f'(x) = 2x.

f'(x) = lim h->0 [f(x+h) - f(x)]/h

Using the power rule d/dx(x^n) = n x^(n-1) term by term: dy/dx = 3x^2 + 3(2x) - 5 + 0 = 3x^2 + 6x - 5.

d/dx(x^n) = n x^(n-1)

Using the product rule with u = sin x (u' = cos x) and v = cos x (v' = -sin x): dy/dx = cos x cos x + sin x (-sin x) = cos^2 x - sin^2 x = cos 2x. (Alternatively, y = (1/2) sin 2x, whose derivative is cos 2x.)

cos^2 x - sin^2 x = cos 2x