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ISEE Upper Level Quantitative : Variables

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All ISEE Upper Level Quantitative Resources

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Example Questions

Example Question #61 : Variables

Expand: $(x + 1)^{10}$

Which is the greater quantity?

(a) The coefficient of $x^{3}$

(b) The coefficient of $x^{7}$

Possible Answers:

(b) is greater.

It is impossible to tell from the information given.

(a) is greater.

The two quantities are equal.

Correct answer:

The two quantities are equal.

Explanation:

By the Binomial Theorem, if $(x + 1)^{n}$ is expanded, the coefficient of $x^{r}$ is

$_{n}\textrm{C} _{r} = \frac{n!}{(n-r)!r!}$ .

(a) Substitute n = 10, r = 3 : The coerfficient of $x^{3}$ is

$_{10}\textrm{C} _{3} = \frac{10!}{(10-3)!3!} = \frac{10!}{7!3!}$ .

(b) Substitute n = 10, r = 7 : The coerfficient of $x^{7}$ is

$_{10}\textrm{C} _{7} = \frac{10!}{(10-7)!7!} = \frac{10!}{3!7!} = \frac{10!}{7!3!}$ .

The two are equal.

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Example Question #1 : How To Find The Exponent Of Variables

Which is greater?

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

(b) $\left ( \frac{1}{2} \right ) ^{3}$

Possible Answers:

(b) is greater.

(a) and (b) are equal.

It is impossible to tell from the information given.

(a) is greater.

Correct answer:

(b) is greater.

Explanation:

$\left (-2 \right )^{-3} = \left ( -\frac{1}{2} \right )^{3}$

A negative number to an odd power is negative, so the expression in (a) is negative. The expression in (b) is positive since the base is positive. (b) is greater.

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Example Question #3 : Variables And Exponents

x>0

Which is the greater quantity?

(a) $(x + 5)^{3}- (x - 5)^{3}$

(b) 250

Possible Answers:

(b) is greater.

It is impossble to tell from the information given.

(a) and (b) are equal.

(a) is greater.

Correct answer:

(a) is greater.

Explanation:

Simplify the expression in (a):

$(x + 5)^{3}- (x - 5)^{3}$

$=(x^{3}+ 3 \cdot x^{2} \cdot 5 + 3 \cdot x \cdot 5^{2}+ 5^{3}) -(x^{3}- 3 \cdotx^{2} \cdot 5 + 3 \cdot x \cdot 5^{2}- 5^{3})$

$=(x^{3}+15 x^{2}+75 x + 125) - (x^{3}-15 x^{2}+75 x- 125)$

$=x^{3}-x^{3}+15 x^{2}+15 x^{2}+75 x-75 x + 125+125$

$=30 x^{2} + 250$

Since x>0 ,

$(x + 5)^{3}- (x - 5)^{3}=30 x^{2} + 250 > 250$ ,

making (a) greater.

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Example Question #1 : How To Find The Exponent Of Variables

Expand: $(x + 2)^{10}$

Which is the greater quantity?

(a) The coefficient of

(b) The coefficient of $x^{2}$

Possible Answers:

(a) is greater.

It is impossible to tell from the information given.

(b) is greater.

The two quantities are equal.

Correct answer:

(b) is greater.

Explanation:

Using the Binomial Theorem, if $(x + 2)^{n}$ is expanded, the $x^{r}$ term is

$_{n}\textrm{C} _{r} \cdot x^{r} \cdot 2 ^{n-r}$

$= \frac{n!}{(n-r)!r!} \cdot 2 ^{n-r} \cdot x^{r}$ .

This makes $\frac{n!}{(n-r)!r!} \cdot 2 ^{n-r}$ the coefficient of $x^{r}$ .

We compare the values of this expression at n = 10 for both r = 1 and r = 2 .

(a) If n = 10 and r = 1 , the coefficient is

$\frac{10!}{(10-1)!1!} \cdot 2 ^{10-1} = \frac{10!}{9!} \cdot 2 ^{9} =10 \cdot 512 =5,120$ .

This is the coefficient of .

(b) If n = 10 and r = 2 , the coefficient is

$\frac{10!}{(10-2)!2!} \cdot 2 ^{10-2} = \frac{10!}{8!2!} \cdot 2 ^{8} = \frac{ 9 \cdot 10}{2} \cdot 256 =11,520$ .

This is the coefficient of $x^{2}$ .

(b) is the greater quantity.

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Example Question #4 : How To Find The Exponent Of Variables

Consider the expression $\left (x^{2} y \right )^{3} \left (x y ^{2} \right )^{3}$

Which is the greater quantity?

(a) The expression evaluated at x = 6, y = 10

(b) The expression evaluated at x = y = 8

Possible Answers:

(a) and (b) are equal

It is impossible to tell from the information given

(a) is greater

(b) is greater

Correct answer:

(b) is greater

Explanation:

Use the properties of powers to simplify the expression:

$\left (x^{2} y \right )^{3} \left (x y ^{2} \right )^{3}$

$= \left (x^{2} \right )^{3} y ^{3} \cdot x ^{3}\left ( y ^{2} \right )^{3}$

$= \left (x^{2} \right )^{3} \cdot x ^{3}\cdot y ^{3} \cdot \left ( y ^{2} \right )^{3}$

$= \left (x^{2 \cdot 3} \right ) \cdot x ^{3}\cdot y ^{3} \cdot \left ( y ^{2\cdot 3} \right )$

$= x^{6} \cdot x ^{3}\cdot y ^{3} \cdot y ^{6}$

$= x^{6+3} \cdot y ^{3+ 6}$

$= x^{9} y ^{9} = (xy)^{9}$

(a) If x = 6, y = 10 , then

$\left (x^{2} y \right )^{3} \left (x y ^{2} \right )^{3}= (xy)^{9} = ( 6 \cdot 10)^{9} = 60^{9}$

(b) If x = y = 8 , then

$\left (x^{2} y \right )^{3} \left (x y ^{2} \right )^{3}= (xy)^{9} = ( 8 \cdot 8)^{9} = 64^{9}$

(b) is greater.

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Example Question #7 : How To Find The Exponent Of Variables

Which of the following expressions is equivalent to

$\left (100 + \sqrt{x} \right )^{2}$ ?

Possible Answers:

10,000+x

200+x

200-x

None of the other answers is correct.

10,000-x

Correct answer:

None of the other answers is correct.

Explanation:

Use the square of a binomial pattern as follows:

$\left (100 + \sqrt{x} \right )^{2}$

$= 100^{2}+ 2 \cdot 100 \cdot \sqrt {x} +(\sqrt{x})^{2}$

$= 10,000+ 2 00 \sqrt {x} +x$

This expression is not equivalent to any of the choices.

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Example Question #3 : How To Find The Exponent Of Variables

$y = (x+ 3)^{2}$

y = 16z

z = w - 6

Express in terms of .

Possible Answers:

$w = 16 x^{2}+ 96x+ 138$

$w = \frac{1}{16}x^{2}+ \frac{3}{8}x+ \frac{105}{16}$

$w = 16 x^{2}+ 96x+ 150$

$w = \frac{1}{16}x^{2}+ \frac{3}{8}x+ \frac{25}{16}$

$w = 16 x^{2}+ 96x+ 105$

Correct answer:

$w = \frac{1}{16}x^{2}+ \frac{3}{8}x+ \frac{105}{16}$

Explanation:

$y = (x+ 3)^{2}$

$= x^{2}+ 2 \cdot 3 \cdot x + 3^{2}$

$= x^{2}+ 6 x + 9$

y = 16z , so

$16z = x^{2}+ 6 x + 9$

$\frac{16z}{16} = \frac{x^{2}+ 6 x + 9}{16}$

$z = \frac{1}{16}x^{2}+ \frac{3}{8}x+ \frac{9}{16}$

z = w - 6 , so

w = z+ 6

$= \frac{1}{16}x^{2}+ \frac{3}{8}x+ \frac{9}{16}+6$

$= \frac{1}{16}x^{2}+ \frac{3}{8}x+ \frac{9}{16} + \frac{96}{16}$

$= \frac{1}{16}x^{2}+ \frac{3}{8}x+ \frac{105}{16}$

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Example Question #11 : Variables And Exponents

$a ^{3} = -17$ . Which is the greater quantity?

(a) $a^{6}$

(b)

Possible Answers:

It is impossible to determine which is greater from the information given

(a) and (b) are equal

(b) is the greater quantity

(a) is the greater quantity

Correct answer:

(a) is the greater quantity

Explanation:

By the Power of a Power Principle,

$(a ^{3} )^{2} = a ^{3 \cdot 2 } = a ^{6}$

Therefore,

$a ^{6} = (a ^{3} )^{2} = (-17)^{2} = 17^{2} = 17 \cdot 17 = 289$

It follows that $a ^{6} > - 34$

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Example Question #62 : Variables

is a real number such that $t ^{6} = 121$ . Which is the greater quantity?

(a) $t^{3}$

(b) 11

Possible Answers:

(b) is the greater quantity

(a) is the greater quantity

(a) and (b) are equal

It is impossible to determine which is greater from the information given

Correct answer:

It is impossible to determine which is greater from the information given

Explanation:

By the Power of a Power Principle,

$(t ^{3} ) ^{2} = t^{3 \cdot 2 } = t ^{6} = 121$

Therefore, $t ^{3}$ is a square root of 121, of which there are two - 11 and . Since it is possible for a third (odd-numbered) power of a real number to be positive or negative, we cannot eliminate either possibility, so either

t = 11

t = -11 < 11 .

Therefore, we cannot determine whether is less than 11 or equal to 11.

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Example Question #64 : Variables

$x ^{4} = 12$

$y^{2} = 8$

$(xy) ^{4} = ?$

Possible Answers:

192

768

Correct answer:

768

Explanation:

By the Power of a Product Principle,

$(xy) ^{4} = x^{4} \cdot y ^{4}$

Also, by the Power of a Power Principle

$y^{4} = y^{2 \cdot 2} = (y^{2})^{2}$

Therefore,

$(xy) ^{4} = x^{4} \cdot (y^{2})^{2} = 12 \cdot 8 ^{2} = 12 \cdot 64 = 768$

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All ISEE Upper Level Quantitative Resources

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ISEE Upper Level Quantitative : Variables

Study concepts, example questions & explanations for ISEE Upper Level Quantitative

All ISEE Upper Level Quantitative Resources

Example Questions

Example Question #61 : Variables

Example Question #1 : How To Find The Exponent Of Variables

Example Question #3 : Variables And Exponents

Example Question #1 : How To Find The Exponent Of Variables

Example Question #4 : How To Find The Exponent Of Variables

Example Question #7 : How To Find The Exponent Of Variables

Example Question #3 : How To Find The Exponent Of Variables

Example Question #11 : Variables And Exponents

Example Question #62 : Variables

Example Question #64 : Variables

All ISEE Upper Level Quantitative Resources

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