Live practice

Translate every condition into an equation and retain exact values throughout the conic calculation.

The last line satisfies the original conditions and supplies every requested part.

Write 101=100+1.

Square twice.

In quadrant III, sine and cosine are negative while tangent is positive. Use sin²x+cos²x=1.

Let \[\begin{aligned}p &= (1,3),q\\&= (1,4),r\\&= (2,3),s\\&= (2,4)\end{aligned}\]. Then \(A\times B=\{p,q,r,s\}\). A four-element set has \(2^4=16\) subsets:

\(\varnothing;\{p\},\{q\},\{r\},\{s\};\{p,q\},\{p,r\},\{p,s\},\{q,r\},\{q,s\},\{r,s\};\{p,q,r\},\{p,q,s\},\{p,r,s\},\{q,r,s\};\{p,q,r,s\}\).

1. Finite: there are 12 months.
2. Infinite: natural numbers have no greatest element.
3. Finite: exactly 100 elements.
4. Infinite: adding 1 always gives another member.
5. Finite: the set is bounded above by 99.

Grass, flowers, leather, fruit peels, cake, lime juice and wood are derived from living matter and can be decomposed biologically. Plastic is non-biodegradable. Hence the groups containing plastic must be rejected, while the remaining three groups qualify.

Magnetic force is perpendicular to velocity, so it changes direction but does no work. Speed and mass remain constant. Velocity changes because direction changes, and momentum \(\vec p=m\vec v\) changes with velocity.

A thick wire has greater area. Since \(R=\rho \frac{l}{A}\), its resistance is smaller. From \(I=\frac{V}{R}\), the same voltage therefore produces a larger current in the thick wire.

Ciliary muscles alter tension in suspensory ligaments and change lens curvature.