Number Theory - SAT Subject Test in Math II

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Question

Define an operation $\vee$ on the set of real numbers as follows:

For any two real numbers ,

$a \vee b =\left | \left | a - 2b \right | + \left | 2a - b\right | \right |$ .

Evaluate $100 \vee\left ( -50 \right )$ .

Answer

Substitute in the expression:

$a \vee b =\left | \left | a - 2b \right | + \left | 2a - b\right | \right |$

$100 \vee (-50) =\left | \left | 100 - 2(-50)\right | + \left | 2 (100 ) - (-50)\right | \right |$

$=\left | \left | 100 - (-100)\right | + \left | 200 +50\right | \right |$

$=\left | \left | 200\right | + \left | 2 50\right | \right |$

$=\left | 200 + 2 50 \right |$

$=\left | 4 50 \right |$

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Question

Which of the following sets is closed under multiplication?

Answer

A set is closed under multiplication if and only if the product of any two (not necessarily distinct) elements of that set is itself an element of that set.

This can easily be disproved in the case of three of these sets:

$\left { 2, 12, 22, 32, 42...\right }$

$2 \times 12 = 24 otin \left { 2, 12, 22, 32, 42...\right }$

$\left { 3, 13, 23, 33, 43...\right }$

$3 \times 13 = 39 otin \left { 3, 13, 23, 33, 43...\right }$

$\left { 4, 14, 24, 34, 44...\right }$

$4 \times 14 = 96 otin \left { 4, 14, 24, 34, 44...\right }$

But closure can be proved to hold in the case of $\left { 1, 11, 21, 31, 41,...\right }$ . Each number takes the form of for some nonnegative integer . If we multiply two numbers in this form, we get

$\left (10N+1 \right )\left (10M+1 \right )$

which is an element of $\left { 1, 11, 21, 31, 41,...\right }$ , being of this form.

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Question

The above represents a Venn diagram. The universal set is the set of all positive integers.

Let be the set of all multiples of 2; let be the set of all multiples of 3; let be the set of all multiples of 5.

As you can see, the three sets divide the universal set into eight regions. Suppose each positive integer was placed in the correct region. Which of the following numbers would be in the same region as 873?

Answer

From the last digit, it can be immediately determined that 873 is not a multple of 2 or 5; since $873 \div 3 = 291$ , 873 is a multiple of 3. Therefore,

$873 \in A' \cap B \cap C'$

We are looking for an integer that is also in this set - that is, one that is also a multiple of 3 but not 2 or 5. From the last digits, we can immediately eliminate 366 and 368 as multiples of 2 and 365 as a multiple of 5. We test 367 and 369 to see which one is a multiple of 3:

$367 \div 3 = 122 \textrm{ R }1$

$369 \div 3 = 123$

369 is the correct choice.

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Question

What is the power set for the data set?

$X=\left {a,b,c}{ \right }$

Answer

The power set is the set of all subsets that can be created from the original set.

For the set $X=\left {a,b,c}{ \right }$ , you can create the subsets:

$\left { a\right }\left { b\right }\left { c\right }\left {a,b \right }\left { a,c\right }\left {b,c \right }\left { a,b,c\right }\left { \varnothing \right }$

This means that the power set is the set of all sets, so the power set is:

$P=\left { \left { a\right }\left { b\right }\left { c\right }\left {a,b \right }\left { a,c\right }\left {b,c \right }\left { a,b,c\right }\left { \varnothing \right } \right }$

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Question

Evaluate:

$\left ( 7+3i\right )^{2}$

Answer

Use the square of a sum pattern

$\left ( A+B\right ) = A^{2}+ 2AB + B^{2}$

where :

$\left ( 7+3i\right )^{2} = 7^{2}+ 2 \cdot 7 \cdot 3i + (3i)^{2}$

$= 7^{2}+ 2 \cdot 7 \cdot 3i + 3^{2}i^{2}$

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Question

Multiply: $\left ( 4+ 7i\right )(4 - 7i)$

Answer

This is the product of a complex number and its complex conjugate. They can be multiplied using the pattern

$\left (A + Bi \right )\left (A - Bi \right ) = A^{2} + B^{2}$

with

$\left (4 + 7i \right )\left (4 - 7i \right ) = 4^{2} + 7^{2} = 16 + 49 = 65$

This is not among the given responses.

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Question

Multiply:

$-3 i \cdot 8i \cdot 5i \cdot 6i$

Answer

$-3 i \cdot 8i \cdot 5i \cdot 6i$

$= \left (-3 \cdot 8 \cdot 5 \cdot 6 \right ) \cdot\left ( i \cdot i \cdot i \cdot i \right )$

$= \left (-720 \right ) \cdot\left ( i ^{4}\right )$

$-720 \cdot 1 = -720$

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Question

Which of the following is equal to $i^{37}$ ?

Answer

To raise to a power, divide the exponent by 4 and note the remainder.

$37 \div 4 = 9 \textrm{ R }1$

Raise to the power of that remainder:

$i^{37} = i^{1} = i$

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Question

Evaluate:

$(6 +5i)^{2}$

Answer

Use the square of a sum pattern

$\left ( A+B\right ) = A^{2}+ 2AB + B^{2}$

where :

$(6 +5i)^{2} = 6^{2} + 2 \cdot 6i \cdot 5 + (5i)^{2}$

$= 6^{2} + 2 \cdot 6 \cdot 5i + 5^{2}i^{2}$

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Question

Multiply:

Answer

Apply the distributive property:

$= 7i \cdot 5 + 7i \cdot 9i$

$= 35i + 63 i^{2}$

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Question

Which of the following is equal to $\left ( 4i\right )^{4}$ ?

Answer

By the power of a product property,

$\left ( 4i\right )^{4} = 4^{4} \cdot i^{4} = 256 \cdot 1 = 256$

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Question

Multiply:

Answer

Use the FOIL method:

$= 6 \cdot 7 +6 \cdot 6i - 7 \cdot 7i - 7i \cdot 6i$

$=42 +36i - 49i - 42i ^{2}$

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Question

Which of the following is equal to $i^{77}$ ?

Answer

To raise to a power, divide the exponent by 4 and note the remainder.

$77 \div 4 = 19\textrm{ R }1$

Raise to the power of that remainder:

$i^{77} = i^{1} = i$

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Question

Multiply:

Answer

Apply the distributive property:

$= 8i \cdot 5- 8i \cdot 7i$

$= 40i- 56i^{2}$

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Question

Multiply:

$\left ( -4i\right ) (-6i)(-7i)$

Answer

$\left ( -4i\right ) (-6i)(-7i)$

$= \left ( -4\right ) (-6)(-7) \cdot i \cdot i \cdot i$

$= -168 \cdot i ^{3} = -168 (-i) = 168i$

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Question

$\overline{x}$ denotes the complex conjugate of .

If $x = \frac{1}{10} - \frac{1}{5} i$ , then evaluate $\overline{x}^{2} - x^{2}$ .

Answer

By the difference of squares pattern,

$\overline{x}^{2} - x^{2} = (\overline{x}+ x) ( \overline{x} - x)$

If $x = \frac{1}{10} - \frac{1}{5} i$ , then $\overline{x} = \frac{1}{10} + \frac{1}{5} i$ .

Consequently:

$\overline{x} + x =\left ( \frac{1}{10} + \frac{1}{5} i \right ) + \left ( \frac{1}{10} - \frac{1}{5} i \right )$

$= \frac{1}{10}+ \frac{1}{10} + \frac{1}{5} i- \frac{1}{5} i$

$= \frac{1}{5}$

$\overline{x} - x =\left ( \frac{1}{10} + \frac{1}{5} i \right ) - \left ( \frac{1}{10} - \frac{1}{5} i \right )$

$= \frac{1}{10}- \frac{1}{10} + \frac{1}{5} i+ \frac{1}{5} i$

$= \frac{2}{5} i$

Therefore,

$\overline{x}^{2} - x^{2} = (\overline{x}+ x) ( \overline{x} - x) = \frac{1}{5} \cdot \frac{2}{5} i = \frac{2}{25} i$

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Question

is a complex number; $\overline{x}$ denotes the complex conjugate of .

$x \cdot \overline{x} = 84$

Which of the following could be the value of ?

Answer

The product of a complex number and its complex conjugate $\overline{x} = a- bi$ is

$x \cdot \overline{x} = (a+ bi)(a-bi) = a^{2} + b^{2}$

Setting and accordingly for each of the four choices, we want to find the choice for which $x \cdot \overline{x} = a^{2} + b^{2} = 84$ :

$6 - 4 i \sqrt{3}$

$a = 6, b = - 4 \sqrt{3}$

$x \cdot \overline{x} = a^{2} + b^{2} = 6^{2}+ (-4 \sqrt{3})^{2} = 36 + 4 ^{2} (\sqrt{3} )^{2} = 36 + 16 \cdot 3 = 36 + 48 = 84$

$7 + i \sqrt{35}$

$a = 7, b = \sqrt{35}$

$x \cdot \overline{x} = a^{2} + b^{2} = 7^{2} + (\sqrt{35})^{2} = 49 + 35 = 84$

$8 + 2 i \sqrt{5}$

$a = 8 , b = 2 \sqrt{5}$

$x \cdot \overline{x} = a^{2} + b^{2} = 8^{2} + (2 \sqrt{5})^{2} = 64 + 2^{2} \cdot (\sqrt{5})^{2} = 64 + 4 \cdot 5 = 64 + 20 = 84$

$9 - i \sqrt{3}$

$a = 9 , b = - \sqrt{3}$

$x \cdot \overline{x} = a^{2} + b^{2} = 9^{2} + (- \sqrt{3})^{2} = 81 + 3 = 84$

For each given value of , $x \cdot \overline{x} = 84$ .

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Question

$\overline{x}$ denotes the complex conjugate of .

If $x = 5 + i \sqrt{2}$ , then evaluate $x^{4} \cdot \overline{x} ^{4}$ .

Answer

Applying the Power of a Product Rule:

$x^{4} \cdot \overline{x} ^{4}=(x \overline{x})^{4}$

The complex conjugate of an imaginary number is $\overline{x} = a - bi$ ; the product of the two is

$x \overline{x} = a^{2} + b^{2}$

$x = 5 + i \sqrt{2}$ , so, setting $a = 5 , b = \sqrt{2}$ in the above pattern:

$x \overline{x} = 5^{2} + (\sqrt{2})^{2} = 25 + 2= 27$

Consequently,

$x^{4} \cdot \overline{x} ^{4}=(x \overline{x})^{4} = 27 ^{4} = 531, 441$

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Question

Let and be complex numbers. $\overline{x}$ and $\overline{y}$ denote their complex conjugates.

$\overline{x} \cdot \overline{y } = 7 + 8 i \sqrt{2}$

$y = 8 - i\sqrt{3}$

Evaluate .

Answer

Knowing the actual values of and is not necessary to solve this problem. The product of the complex conjugates of two numbers is equal to the complex conjugate of the product of the numbers; that is,

$\overline{xy }= \overline{x} \cdot \overline{y }$

We are given that $\overline{xy }= \overline{x} \cdot \overline{y } = 7 + 8 i \sqrt{2}$ . is therefore the conjugate of $\overline{xy }$ , or $7 - 8 i \sqrt{2}$ .

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Question

Let be a complex number. $\overline{x}$ denotes the complex conjugate of .

$x+\overline{ x} = 20$ and $x \overline{ x} = 200$ .

How many of the following expressions could be equal to ?

(a)

(b)

(c)

(d)

Answer

is a complex number, so for some real ; also, $\overline{x} = a-bi$ .

Therefore,

$x+\overline{ x} = (a+ bi) + (a-bi) = a+ a + bi - bi = 2a$

Substituting:

$x+\overline{ x} = 20$

Therefore, we can eliminate choices (c) and (d).

Also, the product

$x \overline{ x} = (a+bi)(a-bi) = a^{2} +b^{2}$

Setting $x \overline{ x} = 200$ and substituting 10 for , we get

$10^{2} +b^{2} = 200$

$100 +b^{2} = 200$

$b^{2} = 100$

$b = \pm 10$

Therefore, either or - making two the correct response.

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