# High School Math : Algebra II

## Example Questions

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### Example Question #1 : Algebra Ii

Without using a calculator, which of the following is the best estimate for ?

Explanation:

We know that and .

Because 90 falls is approximately halfway between 81 and 100, the square root of 90 is approximately halfway between 9 and 10, or 9.5.

### Example Question #2 : Algebra Ii

Place in order from smallest to largest:

Explanation:

To place in order, first we must find a common denominator and convert all fractions to that denominator.

have a common denominator of .

have a common denominator of .

have a common denominator of .

Therefore we can use common denominators to make all of the fractions look similar.  Then the ordering becomes trivial.

### Example Question #3 : Algebra Ii

What number is of ?

Explanation:

For percent problems there are verbal cues:

"IS" means equals and "OF" means multiplication.

Then the equation to solve becomes:

### Example Question #4 : Algebra Ii

Which of the following is NOT a real number?

Explanation:

We are looking for a number that is not real.

, and  are irrational numbers, but they are still real.

Then,  is equivalent to  by the rules of complex numbers. Thus, it is also real.

That leaves us with:  which in fact is imaginary (since no real number multiplied by itself yields a negative number) and simplifies to

### Example Question #5 : Algebra Ii

Which of the following are considered real numbers?

Explanation:

Real numbers can be found anywhere on a continuous number line ranging from negative infinity to positive infinity; therefore, all of the numbers are real numbers.

### Example Question #6 : Algebra Ii

If a card is drawn randomly from a regular shuffled 52 card deck, what is the probability that the card is either a spade or a 3?

Explanation:

How many cards in the deck are either a spade or a 3?

There are four 3's, including a 3 of spades.

Since we are counting the same card (3 of spades) twice, there are actually

distinct cards that fit the criteria of being either a spade or a 3.

Since any of the 52 cards is equally likely to be drawn, the probability that it is a spade or a 3, is

### Example Question #1 : Algebra Ii

Find the distance between  and  on a number line.

Explanation:

To find the distance on a number line:

### Example Question #1 : Adding And Subtracting Fractions

Simplify

Explanation:

Find the least common denominator (LCD) and convert each fraction to the LCD and then add.  Simplify as necessary.

The result is an improper fraction because the numerator is larger than the denominator and can be simplified and converted to a mix numeral.

### Example Question #9 : Algebra Ii

Simplify .

Explanation:

Chenge the mixed numbers into improper fractions by multiplying the whole number by the denominator and adding the numerator to get

.

### Example Question #10 : Algebra Ii

All of the following matrix products are defined EXCEPT:

Explanation:

Every matrix has a dimension, which is represented as the number of rows and columns. For example, a matrix with three rows and two columns is said to have dimension 3 x 2.

The matrix

has two rows and three columns, so its dimension is 2 x 3. (Remember that rows go from left to right, while columns run up and down.)

Matrix multiplication is defined only if the number of columns in the first matrix is equal to the number of rows on the second matrix. The easiest way to determine this is to write the dimension of each matrix. For example, let's say that one matrix has dimension a x b, and the second matrix has dimension c x d. We can only multiply the first matrix by the second matrix if the values of b and c are equal. It doesn't matter what the values of a and d are, as long as b (the number of columns in the first matrix) matches c (the number of rows in the second matrix).

Let's go back to the problem and analyze the choice .

The dimension of the first matrix is 2 x 3, because it has two rows and three columns. The second matrix has dimension 2 x 2, because it has two rows and two columns.

We can't multiply these matrices because the number of columns in the first matrix (3) is not equal to the number of rows in the second matrix (2). Thus, this product is not defined.