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SSAT Upper Level Math : SSAT Upper Level Quantitative (Math)

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

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All SSAT Upper Level Math Resources

6 Diagnostic Tests 311 Practice Tests Question of the Day Flashcards Learn by Concept

Example Questions

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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Example Question #342 : Coordinate Geometry

Define an operation $\wedge$ as follows:

For all complex numbers a,b ,

$a \wedge b = a \div (b+ 4)$ .

If $(8i )\wedge N = 16$ , evaluate .

Possible Answers:

4-2i

$N =- 4 + \frac{1}{2}i$

N =- 4 +128i

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

None of the other choices gives the correct answer.

Correct answer:

$N =- 4 + \frac{1}{2}i$

Explanation:

$(8i )\wedge N = 16$ ,

by our definition, can be rewritten as

$8i \div (N+ 4) = 16$

$\frac{8i }{N+ 4}= 16$

Taking the reciprocal of both sides, then multiplying:

$\frac{N+ 4}{8i }= \frac{1}{16}$

$\frac{N+ 4}{8i } \cdot 8i = \frac{1}{16}\cdot 8i$

$N+ 4 = \frac{1}{2}i$

$N+ 4 - 4 = \frac{1}{2}i- 4$

$N =- 4 + \frac{1}{2}i$

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Example Question #343 : Coordinate Geometry

Raise to the power of 4.

Possible Answers:

-256 i

256i

256

The expression is undefined.

Correct answer:

256

Explanation:

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

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Example Question #14 : How To Graph Complex Numbers

Define an operation $\bigstar$ as follows:

For all complex numbers ,

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

Evaluate $\bigstar (6i)$ .

Possible Answers:

-36+6i

30i

None of the other choices gives the correct answer.

42 i

36+6i

Correct answer:

-36+6i

Explanation:

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

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

$= 6^{2} i^{2} + 6i$

= 36(-1)+ 6i

= -36+ 6i

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Example Question #15 : How To Graph Complex Numbers

Give the number which, when added to 20, yields the same result as if it were subtracted from 14i .

Possible Answers:

40 +28i

No such number exists.

-10 + 7i

-40-28i

10 - 7i

Correct answer:

-10 + 7i

Explanation:

Let be the number in question. The statement "[a number] added to 20 yields the same result as if it were aubtracted from 14i " can be written as

N + 20 = 14 i - N

Solve for :

N + 20+N - 20 = 14 i - N +N - 20

2N = -20 + 14 i

$\frac{1}{2} \cdot 2N = \frac{1}{2} \cdot (-20 + 14 i)$

N = -10 + 7i

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Example Question #16 : How To Graph Complex Numbers

Define an operation $\wedge$ as follows:

For all complex numbers a,b ,

$a \wedge b = a \div (b+ 3)$

Evaluate $7i \wedge 5i$

Possible Answers:

$-\frac{ 35 }{16} + \frac{ 21}{16} i$

$\frac{ 35 }{34}-\frac{ 21}{34} i$

$\frac{ 35 }{16} - \frac{ 21}{16} i$

$\frac{ 35 }{34} + \frac{ 21}{34} i$

$-\frac{ 35 }{16} - \frac{ 21}{16} i$

Correct answer:

$\frac{ 35 }{34} + \frac{ 21}{34} i$

Explanation:

$a \wedge b = a \div (b+ 3) = \frac{a}{b+3}$

$7i \wedge 5i = \frac{7i }{5i+3} = \frac{7i }{3+ 5i}$

Multiply both numerator and denominator by the conjugate of the denominator, 3-5i , to rationalize the denominator:

$7i \wedge 5i = \frac{7i (3-5i)}{(3+ 5i)(3-5i)}$

$= \frac{7i \cdot 3-7i \cdot 5i}{3^{2}+ 5^{2}}$

$= \frac{21i -35 \cdot i^{2}}{9+25}$

$= \frac{21i-35 (-1)}{34}$

$= \frac{21i+35}{34}$

$= \frac{ 35+21i}{34}$

$= \frac{ 35 }{34} + \frac{ 21}{34} i$

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Example Question #17 : How To Graph Complex Numbers

Subtract 7 - 9i from its complex conjugate. What is the result?

Possible Answers:

18i

2 +16i

2 - 2i

Correct answer:

18i

Explanation:

The complex conjugate of a complex number a - bi is a + bi , so the complex conjugate of 7 - 9i is 7 + 9i . Subtract the former from the latter:

$\left ( 7+9i\right ) - \left ( 7-9i\right ) = 7 + 9i - 7 + 9i = 7 - 7 +9i + 9i = 18i$

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Example Question #91 : Graphing

Give the product of 5+ 4 i and its complex conjugate.

Possible Answers:

25-16i

16+25i

25+16i

16 -25i

The correct answer is not given among the other responses.

Correct answer:

The correct answer is not given among the other responses.

Explanation:

The product of a complex number a + bi and its conjugate a - bi is

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

which will always be a real number. Therefore, all four given choices, all of which are imaginary, can be immediately eliminated. The correct response is that the correct answer is not given among the other responses.

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All SSAT Upper Level Math Resources

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SSAT Upper Level Math : SSAT Upper Level Quantitative (Math)

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

All SSAT Upper Level Math Resources

Example Questions

Example Question #7 : How To Graph Complex Numbers

Example Question #10 : How To Graph Complex Numbers

Example Question #11 : How To Graph Complex Numbers

Example Question #342 : Coordinate Geometry

Example Question #343 : Coordinate Geometry

Example Question #14 : How To Graph Complex Numbers

Example Question #15 : How To Graph Complex Numbers

Example Question #16 : How To Graph Complex Numbers

Example Question #17 : How To Graph Complex Numbers

Example Question #91 : Graphing

All SSAT Upper Level Math Resources

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