Further Mathematics Questions and Answers

d = \(\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\)
x1 = x, x2 = -x, y1 = 4, y2 = 3, d = 7
7 =  \(\sqrt{(-x -x)^2 + (3 - 4)^2}\)
7 = \(\sqrt{(-2x)^2 + (-1)^2}\)
7 = ( \(\sqrt{4x^2 + 1}\)
square both sides
7\(^2\) = 4x\(^2\) + 1
collect like terms
4x\(^2\) = 49 - 1
4x\(^2\) = 48

x\(^2\) = \(\frac{48}{4}\)

x\(^2\) = 12
x = √12
x = 2√3

x - 1 = 2
x = 3
f(2) = (3)\(^3\) + 3(3)\(^2\) + 4(3) - 5
f(2) = 27 + 27 + 12 - 5

= 61

Converting the forces to their rectangular forms
F = (10N, 060º)
Fx = 10cos60 = 5i
fy = 10sin60 = 8.66j
F = 5i + 8.66j
P = (15N, 120º)
Px = 15cos120 = -7.5i
py = 15sin120 = 12.99j
P = -7.5i + 12.99j
Q = (12N, 200º)
Qx = 12cos200 = -11.28i
Qy = 12sin200 = -4.1j
Q = -11.28i -4.1j
The resultant force = F + P + Q
R = 5i + 8.66j + (-7.5i + 12.99j) + (-11.28i -4.1j)
R = -13.78i + 17.55j

x\(^2\) + mx - n = 0

a = 1, b = m, c = -n

α + β = \(\frac{-b}{a}\) = \(\frac{-m}{1}\) = -m

αβ = \(\frac{c}{a}\) = \(\frac{-n}{1}\) = -n

the roots are = \(\frac{1}{α}\) and \(\frac{1}{β}\)

sum of the roots = \(\frac{1}{α}\) + \(\frac{1}{β}\)

\(\frac{1}{α}\) + \(\frac{1}{β}\) = \(\frac{α+β}{αβ}\)

α + β = -m
αβ = -n

\(\frac{α+β}{αβ}\) = \(\frac{-m}{-n}\) → \(\frac{m}{n}\)

product of the roots = \(\frac{1}{α}\) * \(\frac{1}{β}\)

\(\frac{1}{α}\) + \(\frac{1}{β}\) = \(\frac{1}{αβ}\) → \(\frac{1}{-n}\)

x\(^2\) - (sum of roots)x + (product of roots)
x\(^2\) - ( m/n )x + ( 1/-n ) = 0
multiply through by n
nx\(^2\) - mx - 1 = 0

F = m * a

d = t\(^2\) + 3t.

a = \(\frac{d^2d}{dt^2}\)

\(\frac{d[d]}{dt}\) = 2t + 3

\(\frac{d^2d}{dt^2}\) = 2m/s\(^2\)

a = 2m/s\(^2\)

F = m * a

F = 3 × 2 = 6N