Mathematics Questions and Answers

[\(\frac{1}{0.03}\) + \(\frac{1}{0.024}\)]

= [\(\frac{1}{0.03 \times 0.024}\)]^-1

= [\(\frac{0.024}{0.003}\)]^-1

= \(\frac{0.03}{0.024}\)

= \(\frac{30}{24}\) = 1.25

1011₂ + x₇ = 25₁₀ = 1011₂ = 1 x 2³ + 0 x 2² + 1 x 2¹ + 1 x 2^o

= 8 + 0 + 2 + 1

= 11₁₀

x₇ = 25₁₀ - 11₁₀

= 14₁₀

\(\begin{array}{c|c}
7 & 14 \\ 7 & 2 R 0 \\ & 0 R 2
\end{array}\)

X = 20₇

Equation: x\(^3\) - 2x\(^2\) - 5x + 6 = 0.

First, bring out a\(_n\) which is the coefficient of x\(^3\) = 1.

Then, a\(_0\) which is the coefficient void of x = 6.

The factors of a\(_n\) = 1; The factors of a\(_0\) = 1, 2, 3 and 6.

The numbers to test for the roots are \(\pm (\frac{a_0}{a_n})\).

= \(\pm (1, 2, 3, 6)\).

Test for +1: 1\(^3\) - 2(1\(^2\)) - 5(1) + 6 = 1 - 2 - 5 + 6 = 0.

Therefore x = 1 is a root of the equation.

Using long division method, \(\frac{x^3 - 2x^2 - 5x + 6}{x - 1}\) = x\(^2\) - x - 6.

x\(^2\) - x - 6 = (x - 3)(x + 2).

x = -2, 3.

\(\therefore\) The roots of the equation = 1, -2 and 3.

2(1) - (-1) = 3      2(2) - (0) = 4          2(-1) - (2) = -4

2(0) - (1) = -1      2(-1) - (-1) = -1      2(0) - (-1) = 1

2(1) - 2 = 0         2(0) - (-1) = 1         2(1) - (1) = 1

\(\begin{pmatrix} 3 & -1 & 0 \\ 4 & -1 & 1 \\ -4 & 1 & 1\end{pmatrix}\)

2p x 1 + 8 x 2 = 24

\(\to\) 4p = 24 - 8 = 16,

p = 4

3 x 1 + -5q x 2 = -17

\(\to\) -10q = -17 - 3

-10q = -20

q = 2