Given AB || CD and EF is a transversal, find the values of x and y. (b) Simplify the expression 2(4-3xy)-5(6-xy) and multiply 0.00834 by 0.02 in standard form.

1. (a)

i) In the diagram, AB is parallel to CD, and EF is a transversal. The angles $5x$ and $x+78^\circ$ are alternate exterior angles, which are equal.

$$5x = x+78^\circ$$

Subtract $x$ from both sides:

$$4x = 78^\circ$$

Divide by 4:

$$x = \frac{78}{4}$$ $$x = \frac{39}{2}$$ $$x = \boxed{19.5^\circ}$$

ii) The angles $5x$ and $4x+y$ are vertically opposite angles, which are equal.

$$5x = 4x+y$$

Subtract $4x$ from both sides:

$$x = y$$

Since $x = 19.5^\circ$:

$$y = \boxed{19.5^\circ}$$

1. (b)

i) Simplify the expression:

$$2(4-3xy)-5(6-xy)$$

Distribute the numbers into the parentheses:

$$8 - 6xy - 30 + 5xy$$

Combine like terms:

$$(8 - 30) + (-6xy + 5xy)$$ $$-22 - xy$$

The simplified expression is $\boxed{-22 - xy}$.

ii) Multiply 0.00834 by 0.02 and express the answer in standard form.
First, convert the numbers to standard form:

$$0.00834 = 8.34 \times 10^{-3}$$ $$0.02 = 2 \times 10^{-2}$$

Now, multiply them:

$$(8.34 \times 10^{-3}) \times (2 \times 10^{-2})$$ $$$$