Slope m = (6 - 2)/(3 - 1) = 4/2 = 2. Using point-slope form with point (1, 2): y - 2 = 2(x - 1), so y - 2 = 2x - 2, giving y = 2x, or 2x - y = 0.

y - y1 = m(x - x1)

Slope of first line m1 = -1/sqrt3; slope of second line m2 = -sqrt3. The angle between them satisfies tan(theta) = |(m1 - m2)/(1 + m1 m2)| = |(-1/sqrt3 + sqrt3)/(1 + (-1/sqrt3)(-sqrt3))| = |((-1 + 3)/sqrt3)/(1 + 1)| = |(2/sqrt3)/2| = 1/sqrt3. So theta = 30 degrees.

tan theta = |(m1 - m2)/(1 + m1 m2)|

The distance of point (x1, y1) from line Ax + By + C = 0 is d = |A x1 + B y1 + C|/sqrt(A^2 + B^2). Here A = 3, B = -4, C = -26 and (x1, y1) = (3, -5). d = |3(3) - 4(-5) - 26|/sqrt(9 + 16) = |9 + 20 - 26|/sqrt25 = |3|/5 = 3/5.

d = |A x1 + B y1 + C|/sqrt(A^2 + B^2)

The intercept form of a line is x/a + y/b = 1, where a and b are the x- and y-intercepts. Here a = -3 and b = 2, so x/(-3) + y/2 = 1. Multiplying throughout by 6: -2x + 3y = 6, i.e. 2x - 3y + 6 = 0.

x/a + y/b = 1

Midpoint of the segment = ((1 + 3)/2, (1 + 5)/2) = (2, 3). Slope of the segment = (5 - 1)/(3 - 1) = 2, so the slope of the perpendicular bisector = -1/2. Using point-slope form through (2, 3): y - 3 = -1/2 (x - 2), so 2(y - 3) = -(x - 2), 2y - 6 = -x + 2, giving x + 2y - 8 = 0.

product of perpendicular slopes = -1

Compare y^2 = 12x with y^2 = 4ax: 4a = 12, so a = 3. The parabola opens to the right. Focus = (a, 0) = (3, 0). Directrix: x = -a, i.e. x = -3. Axis: the x-axis, i.e. y = 0. Length of latus rectum = 4a = 12.

y^2 = 4ax: focus (a,0), directrix x = -a

Here a^2 = 25 (a = 5) and b^2 = 9 (b = 3), with a > b so the major axis is along the x-axis. c = sqrt(a^2 - b^2) = sqrt(25 - 9) = 4. Vertices = (+/-5, 0); foci = (+/-4, 0); eccentricity e = c/a = 4/5; length of latus rectum = 2b^2/a = 2(9)/5 = 18/5.

e = c/a; latus rectum = 2b^2/a

Here a^2 = 16 (a = 4) and b^2 = 9 (b = 3). For a hyperbola, c = sqrt(a^2 + b^2) = sqrt(16 + 9) = sqrt25 = 5. Vertices = (+/-4, 0); foci = (+/-5, 0); eccentricity e = c/a = 5/4. (Length of latus rectum = 2b^2/a = 9/2.)

hyperbola: c = sqrt(a^2 + b^2), e = c/a

The vertices and foci are on the x-axis, so the ellipse is x^2/a^2 + y^2/b^2 = 1 with a > b. From vertices, a = 5; from foci, c = 4. Then b^2 = a^2 - c^2 = 25 - 16 = 9. Hence the equation is x^2/25 + y^2/9 = 1.

b^2 = a^2 - c^2

Latus rectum = 2b^2/a and major axis = 2a. Given 2b^2/a = (1/2)(2a) = a, so 2b^2 = a^2. Since b^2 = a^2(1 - e^2), we get 2a^2(1 - e^2) = a^2, so 2(1 - e^2) = 1, giving 1 - e^2 = 1/2, e^2 = 1/2 and e = 1/sqrt2.

b^2 = a^2(1 - e^2)

By the section formula for internal division in ratio m : n, the coordinates are ((m x2 + n x1)/(m + n), (m y2 + n y1)/(m + n), (m z2 + n z1)/(m + n)) with m = 2, n = 3. x = (2(3) + 3(1))/5 = 9/5; y = (2(4) + 3(-2))/5 = 2/5; z = (2(-5) + 3(3))/5 = -1/5. The point is (9/5, 2/5, -1/5).

((m x2 + n x1)/(m+n), ...)

The centroid of a triangle with vertices (x1,y1,z1), (x2,y2,z2), (x3,y3,z3) is ((x1+x2+x3)/3, (y1+y2+y3)/3, (z1+z2+z3)/3). x = (1+3+4)/3 = 8/3; y = (2-1+0)/3 = 1/3; z = (3+5-2)/3 = 6/3 = 2. Centroid = (8/3, 1/3, 2).

G = ((x1+x2+x3)/3, (y1+y2+y3)/3, (z1+z2+z3)/3)

Write (sin 3x)/(sin 5x) = [(sin 3x)/(3x) x 3x] / [(sin 5x)/(5x) x 5x]. As x approaches 0, (sin 3x)/(3x) -> 1 and (sin 5x)/(5x) -> 1, so the limit = (1 x 3x)/(1 x 5x) = 3/5.

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

Direct substitution gives 0/0. Factorise: x^2 - 9 = (x - 3)(x + 3) and x^2 - x - 6 = (x - 3)(x + 2). Cancelling (x - 3): the expression becomes (x + 3)/(x + 2). Taking x approaching 3 gives (3 + 3)/(3 + 2) = 6/5.

Direct substitution gives 0/0. Rationalise by multiplying numerator and denominator by (sqrt(1 + x) + 1): [(sqrt(1+x) - 1)(sqrt(1+x) + 1)]/[x(sqrt(1+x) + 1)] = (1 + x - 1)/[x(sqrt(1+x) + 1)] = x/[x(sqrt(1+x) + 1)] = 1/(sqrt(1+x) + 1). As x approaches 0 this equals 1/(1 + 1) = 1/2.

(sqrt a - sqrt b)(sqrt a + sqrt b) = a - b

Let u = x^2 + 1 and v = x - 3, so u' = 2x and v' = 1. By the product rule dy/dx = u'v + uv' = (2x)(x - 3) + (x^2 + 1)(1) = 2x^2 - 6x + x^2 + 1 = 3x^2 - 6x + 1.

(uv)' = u'v + uv'

Let u = x + 1 and v = x - 1, so u' = 1 and v' = 1. By the quotient rule dy/dx = (u'v - uv')/v^2 = [1(x - 1) - (x + 1)(1)]/(x - 1)^2 = (x - 1 - x - 1)/(x - 1)^2 = -2/(x - 1)^2.

(u/v)' = (u'v - uv')/v^2

f'(x) = lim h->0 [sin(x + h) - sin x]/h. Expand sin(x + h) = sin x cos h + cos x sin h, so the numerator = sin x cos h + cos x sin h - sin x = sin x(cos h - 1) + cos x sin h. Thus f'(x) = sin x . lim h->0 (cos h - 1)/h + cos x . lim h->0 (sin h)/h. Since lim (cos h - 1)/h = 0 and lim (sin h)/h = 1, we get f'(x) = sin x . 0 + cos x . 1 = cos x.

d/dx(sin x) = cos x

There are 8 observations. Mean = (6+7+10+12+13+4+8+12)/8 = 72/8 = 9. Absolute deviations from the mean: |6-9|=3, |7-9|=2, |10-9|=1, |12-9|=3, |13-9|=4, |4-9|=5, |8-9|=1, |12-9|=3. Sum of absolute deviations = 3+2+1+3+4+5+1+3 = 22. Mean deviation = 22/8 = 2.75.

MD = (sum of |xi - mean|)/n

Mean = (2+4+6+8+10)/5 = 30/5 = 6. Squared deviations: (2-6)^2=16, (4-6)^2=4, (6-6)^2=0, (8-6)^2=4, (10-6)^2=16; their sum = 40. Variance = 40/5 = 8. Standard deviation = sqrt(variance) = sqrt8 = 2 sqrt2 (approximately 2.83).

variance = (sum of (xi - mean)^2)/n; SD = sqrt(variance)