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SSAT Upper Level Math : Geometry

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Example Questions

Example Question #391 : Geometry

Simplify:

$\left ( 8 - 3i\right ) ^{2}$

Possible Answers:

55 - 48i

55 +48i

73- 48i

Correct answer:

55 - 48i

Explanation:

Use the square of a binomial pattern to multiply this:

$\left ( 8 - 3i\right ) ^{2}$

$= 8 ^{2}- 2 \cdot8\cdot3i + \left ( 3i\right )^{2}$

$= 64- 48i + 3^{2}i^{2}$

= 64- 48i + 9 (-1)

= 64-9 - 48i

= 55 - 48i

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Example Question #392 : Geometry

Multiply:

(1.1 + 0.6i )(1.1 - 0.6i )

Possible Answers:

0.85

1.57

1.57 - 1.32i

0.85 + 1.32i

1.21 - 0.36i

Correct answer:

1.57

Explanation:

This is a product of an imaginary number and its complex conjugate, so it can be evaluated using this formula:

$\left (a + bi \right )\left (a -bi \right ) = a^{2} + b ^{2}$

$\left (1.1 + 0.6i \right )\left (1.1 -0.6 i \right ) = 1.1^{2} + 0.6 ^{2} = 1.21 + 0.36 = 1.57$

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Example Question #393 : Geometry

Multiply:

$\left (5 - 3i \right )\left (5 +3i \right )$

Possible Answers:

34 +30i

25 -9i

16 -30i

Correct answer:

Explanation:

This is a product of an imaginary number and its complex conjugate, so it can be evaluated using this formula:

$\left (a - bi \right )\left (a +bi \right ) = a^{2} + b ^{2}$

$\left (5 - 3i \right )\left (5 +3i \right ) = 5 ^{2} + 3 ^{2}= 25 + 9 = 34$

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Example Question #5 : How To Graph Complex Numbers

Define an operation as follows:

For all complex numbers a,b ,

$a*b = a ^{5}-b^{3}$

Evaluate $\left (2 i \right )* \left (3i \right )$

Possible Answers:

-27+32i

27 -32i

32 - 27i

59i

Correct answer:

59i

Explanation:

$a*b = a ^{5}-b^{3}$

$\left (2 i \right )* \left (3i \right ) = \left (2 i \right ) ^{5}-\left (3i \right ) ^{3}$

$= 2^{5} i^{5} - 3^{3} i ^{3}$

$= 32\cdot i^{4} \cdot i - 27 \cdot (-i )$

$= 32\cdot 1 \cdot i +27 i$

= 32 i +27 i

= 59 i

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Example Question #5 : How To Graph Complex Numbers

Define an operation as follows:

For all complex numbers a,b ,

a*b = a+3ib

Evaluate (4+7i) * (3-8i)

Possible Answers:

28 +16 i

13+31i

35-35i

13-17i

-20 + 16i

Correct answer:

28 +16 i

Explanation:

a*b = a+3ib

(4+7i) * (3-8i) = (4+7i) + 3i (3-8i)

$= 4+7i + 3i \cdot 3- 3i \cdot 8i$

$= 4+7i + 9i - 24i ^{2}$

= 4+7i + 9i - 24 (-1)

= 4+7i + 9i- (-24)

= 4+7i + 9i +24

= 28 +16i

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Example Question #394 : Geometry

Evaluate $(3i)^{-5}$ .

Possible Answers:

$\frac{1}{243} i$

The expression is undefined.

-243 i

243i

$- \frac{1}{243} i$

Correct answer:

$- \frac{1}{243} i$

Explanation:

$(3i)^{-5} = \frac{1}{(3i)^{5} }= \frac{1}{3^{5}i^{5} } = \frac{1}{243 i^{4} \cdot i } = \frac{1}{243 \cdot 1 \cdot i }$

$= \frac{1}{243 i } \cdot = \frac{1\cdot i}{243 i \cdot i } = \frac{i}{243 (-1)} = - \frac{1}{243} i$

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Example Question #8 : How To Graph Complex Numbers

Define an operation $\wedge$ as follows:

For all complex numbers a,b ,

$a \wedge b = a \div (b+ 2i)$

Evaluate $8 \wedge 4$

Possible Answers:

$-\frac{8}{3} + \frac{ 4}{3}i$

$-\frac{8}{5} + \frac{ 4}{5}i$

$\frac{8}{3} - \frac{ 4}{3}i$

$\frac{8}{5} + \frac{ 4}{5}i$

$\frac{8}{5} - \frac{ 4}{5}i$

Correct answer:

$\frac{8}{5} - \frac{ 4}{5}i$

Explanation:

$a \wedge b = a \div (b+ 2i) = \frac{a}{b+2i}$

$8 \wedge 4 = \frac{8}{4+2i}$

Multiply both numerator and denominator by the conjugate of the denominator, 4 - 2i , to rationalize the denominator:

$8 \wedge 4 = \frac{8(4-2i)}{(4+2i)(4-2i)}$

$= \frac{8\cdot 4-8\cdot 2i}{4^{2}+2^{2}}$

$= \frac{32-16i}{16+4}$

$= \frac{32-16i}{20}$

$= \frac{32 }{20} - \frac{ 16}{20}i$

$= \frac{8}{5} -\frac{ 4}{5}i$

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Example Question #7 : How To Graph Complex Numbers

Define an operation $\bigstar$ as follows:

For all complex numbers ,

$\bigstar a = a^{2}i + a$

Evaluate $\bigstar (7i)$ .

Possible Answers:

49+7i

-56i

-42i

-49+7i

None of the other choices gives the correct answer.

Correct answer:

-42i

Explanation:

$\bigstar a = a^{2}i + a$

$\bigstar (7i) = (7i)^{2}i + 7i$

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

$= 49\cdot (-1) \cdot i + 7i$

= -49 i + 7i

=-42i

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Example Question #10 : How To Graph Complex Numbers

Define an operation $\bigstar$ as follows:

For all complex numbers ,

$\bigstar a = a^{2}i - 4 a$

If $\bigstar N = 0$ and $N \ne 0$ , evaluate .

Possible Answers:

N = -4i

N = -2i

$N = - \frac{1}{4}i$

$N = \frac{1}{4}i$

N = 4i

Correct answer:

N = -4i

Explanation:

If $\bigstar N = 0$ , then

$N^{2}i - 4 N= 0$ .

Distribute out to yield

N (Ni - 4 )= 0

Either N = 0 or N i - 4 = 0 . However, we are given that $N \ne 0$ , so

N i - 4 = 0

N i = 4

$\frac{N i }{i}= \frac{4}{i}$

$N= \frac{4}{i} = \frac{4 \cdot i}{i\cdot i}= \frac{4i}{-1} = -4i$

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Example Question #11 : How To Graph Complex Numbers

Give the number which, when multiplied by , yields the same result as if it were increased by 6.

Possible Answers:

-3-3i

No such number exists.

3 -3i

3 +3i

-3+3i

Correct answer:

-3-3i

Explanation:

Let be the number in question. The statement "[a number] multiplied by yields the same result as if it were increased by 6" can be written as

Ni = N + 6

We can solve this for as follows:

Ni - N= N + 6 - N

-N + Ni = 6

N(-1+i)= 6

$N(-1+i) \div (-1+i) = 6 \div (-1+i)$

$N = \frac{6}{-1+i}$

Rationalize the denominator by multiplying both numerator and denominator by the conjugate of the denominator, which is -1-i :

$N = \frac{6(-1- i)}{(-1+i)(-1- i)}$

$= \frac{6(-1)- 6\cdot i}{(-1)^{2}+1^{2}}$

$= \frac{-6-6i}{1+1}$

$= \frac{-6-6i}{2}$

=-3-3i

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SSAT Upper Level Math : Geometry

Study concepts, example questions & explanations for SSAT Upper Level Math

All SSAT Upper Level Math Resources

Example Questions

Example Question #391 : Geometry

Example Question #392 : Geometry

Example Question #393 : Geometry

Example Question #5 : How To Graph Complex Numbers

Example Question #5 : How To Graph Complex Numbers

Example Question #394 : Geometry

Example Question #8 : How To Graph Complex Numbers

Example Question #7 : How To Graph Complex Numbers

Example Question #10 : How To Graph Complex Numbers

Example Question #11 : How To Graph Complex Numbers

All SSAT Upper Level Math Resources

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