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ISEE Upper Level Quantitative : ISEE Upper Level (grades 9-12) Quantitative Reasoning

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

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All ISEE Upper Level Quantitative Resources

8 Diagnostic Tests 234 Practice Tests Question of the Day Flashcards Learn by Concept

Example Questions

Example Question #91 : Equations

Define $f (x) = \sqrt{4-x}$ and $g(x) = \sqrt{x+3}$ .

Which is the greater quantity?

(A) $\left (f+g \right ) (4)$

(B) $\left (f+g \right ) (-3)$

Possible Answers:

(A) and (B) are equal

(B) is greater

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

(A) is greater

Correct answer:

(A) and (B) are equal

Explanation:

$\left ( f + g \right ) (x) = f (x) + g(x) = \sqrt{4-x} + \sqrt{x+3}$

Substitute and to determine the values of the respective expressions:

$\left ( f + g \right ) (4) = \sqrt{4-4} + \sqrt{4+3}= \sqrt{0} + \sqrt{7} = \sqrt{7}$

$\left ( f + g \right ) (-3) = \sqrt{4-(-3)} + \sqrt{-3+3}= \sqrt{7} + \sqrt{0} = \sqrt{7}$

The expressions are equal.

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Example Question #93 : How To Find The Solution To An Equation

For all real numbers and , define an operation $\Leftrightarrow$ as follows:

$a\Leftrightarrow b = \frac{a + b}{ ab+1}$

For which value of is the expression $N \Leftrightarrow 5$ undefined?

Possible Answers:

N = - 5

$N = - \frac{1}{5}$

$N = \frac{1}{5}$

N = 0

N = 5

Correct answer:

$N = - \frac{1}{5}$

Explanation:

$a\Leftrightarrow b = \frac{a + b}{ ab+1}$

$N\Leftrightarrow 5 = \frac{N + 5}{ N \cdot 5 +1} = \frac{N + 5}{5 N +1}$

This expression is undefined if and only if the denominator is equal to 0, so

5N + 1 = 0

5N + 1 -1 = 0 -1

5N = -1

$5N \div 5 = -1 \div 5$

$N = - \frac{1}{5}$

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Example Question #92 : How To Find The Solution To An Equation

A line includes the points (6, 3.1) and (2, -1.5) . What is the -intercept of this line (-coordinate rounded to the nearest tenth)?

Possible Answers:

(3.3,0)

(0.2,0)

(1.2,0)

(-2.1,0)

(-3.8,0)

Correct answer:

(3.3,0)

Explanation:

Let $x_{1} = 2, y_{1} = -1.5, x_{2} =6, y_{2} = 3.1$

We calculate the slope as follows:

$m = \frac{y_{2}-y_{1}}{x_{2}-x_{1}} = \frac{3.1-(-1.5)}{6-2} = \frac{4.6}{4} = 1.15$

Apply the point-slope formula setting

$x_{1} = 2, y_{1} = -1.5, m = 1.15$ :

$y - y_{1} = m (x - x_{1})$

y - (-1.5)= 1.15 (x - 2)

y+1.5= 1.15 (x - 2)

y+1.5= 1.15 x - 2.3

Set $y \, = 0$ to find the -coordinate of the -intercept:

0+1.5= 1.15 x - 2.3

1.5= 1.15 x - 2.3

1.5+ 2.3= 1.15 x - 2.3 + 2.3

3.8= 1.15 x

$3.8 \div 1.15 = 1.15 x \div 1.15$

$x \approx 3.3$

The -intercept is (approximately at) (3.3,0)

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Example Question #95 : How To Find The Solution To An Equation

A line includes the points (5, 3.1) and (3, -1.7) . What is the -intercept of this line?

Possible Answers:

(0,-4.8)

(0,3.7)

(0,-2)

(0,-8.9)

(0,-6.5)

Correct answer:

(0,-8.9)

Explanation:

Let $x_{1} = 3, y_{1} = -1.7, x_{2} = 5, y_{2} = 3.1$

We calculate the slope as follows:

$m = \frac{y_{2}-y_{1}}{x_{2}-x_{1}} = \frac{3.1-(-1.7)}{5-3} = \frac{4.8}{2} = 2.4$

Apply the point-slope formula setting

$x_{1} = 5, y_{1} = 3.1, m= 2.4$

$y - y_{1} = m (x - x_{1})$

y - 3.1= 2.4 (x - 5)

Set x = 0 to find the -coordinate of the -intercept:

y - 3.1= 2.4 (0- 5)

y - 3.1= 2.4 (- 5)

y - 3.1= -12

y - 3.1+ 3.1 = -12 + 3.1

y = -8.9

The -intercept is (0,-8.9) .

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Example Question #91 : How To Find The Solution To An Equation

A line includes the points (5, -5.2) and (3, -1.8) . Which of these is the slope of that line?

Possible Answers:

$-\frac{10}{17}$

$-\frac{8}{7}$

$-\frac{17}{10}$

$-\frac{7}{8}$

The correct answer is not among the other choices

Correct answer:

$-\frac{17}{10}$

Explanation:

Let $x_{1} = 3, y_{1} = -1.8 , x_{2} = 5, y_{2} = -5.2$

We calculate the slope as follows:

$m = \frac{y_{2}-y_{1}}{x_{2}-x_{1}} = \frac{-5.2-(-1.8)}{5-3} = \frac{-3.4}{2} = -1.7 = -\frac{17}{10}$

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Example Question #91 : Equations

For all real numbers and , define an operation $\ast$ as follows:

$a \ast b = \frac{a + 2b }{ a + 3b}$

Let be a positive number. Then which is the greater quantity?

(A) $t \ast 4t$

(A) $4t \ast t$

Possible Answers:

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

(B) is greater

(A) and (B) are equal

(A) is greater

Correct answer:

(B) is greater

Explanation:

Substitute each pair of expressions:

$a \ast b = \frac{a + 2b }{ a + 3b}$

$t \ast 4t = \frac{t + 2 \cdot 4t }{ t + 3 \cdot 4t } = \frac{t + 8t }{ t + 12t } = \frac{9t }{ 13t } = \frac{9 }{ 13 }$

$4t \ast t = \frac{4t + 2 \cdot t }{4t + 3 \cdot t } = \frac{4t + 2t }{ 4t + 3t } = \frac{6t }{ 7t } = \frac{6 }{ 7 }$

We can compare these fractions by writing them with a common denominator:

$t \ast 4t = \frac{9 }{ 13 } = \frac{9 \times 7 }{ 13 \times 7 } = \frac{63}{91}$

$4t \ast t =\frac{6 }{ 7 }=\frac{6 \times 13 }{ 7 \times 13 } = \frac{78}{91}$

$4t \ast t > t \ast 4t$ regardless of the value of , making (B) greater.

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Example Question #101 : How To Find The Solution To An Equation

Define $f (x) = \frac{1}{x+ 6}$ and $g(x) = \frac{1}{x + 8 }$ .

Which is the greater quantity?

(A) $\left (f+g \right ) (1)$

(B) $\left (f+g \right ) (-1)$

Possible Answers:

(A) and (B) are equal

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

(A) is greater

(B) is greater

Correct answer:

(B) is greater

Explanation:

$\left (f+g \right ) (x) = f(x) + g(x) = \frac{1}{x+6}+ \frac{1}{x+8}$

$\left (f+g \right ) (1) = f(1) + g(1) = \frac{1}{1+6}+ \frac{1}{1+8} = \frac{1}{7}+ \frac{1}{9}$

$\left (f+g \right ) (-1) = f(-1) + g(-1) = \frac{1}{-1+6}+ \frac{1}{-1+8} = \frac{1}{5}+ \frac{1}{7}$

Since $\frac{1}{9} < \frac{1}{5}$ ,

$\frac{1}{7}+ \frac{1}{9} < \frac{1}{5} + \frac{1}{7}$

and

$\left (f+g \right ) (1) < \left (f+g \right ) (-1)$

making (B) greater.

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Example Question #102 : How To Find The Solution To An Equation

Define f (x) = 4x - 7 and g (x) = 3x - 8

Let be a positive number. Which is the greater quantity?

(A) $\left ( f \circ g\right ) (t)$

(B) $\left ( g \circ f\right ) (t)$

Possible Answers:

(A) and (B) are equal

(A) is greater

(B) is greater

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

Correct answer:

(B) is greater

Explanation:

$\left ( f \circ g\right ) (t) = f (g(t)) = f (3t-8) = 4(3t-8) - 7 = 12t -32 - 7 = 12t -39$

$\left ( g \circ f\right ) (t) = g (f(t)) =g (4t-7) = 3 (4t-7) -8 = 12 t - 21 - 8 = 12t - 29$

Regardless of the value of ,

$\left ( g \circ f\right ) (t) = \left ( f \circ g\right ) (t) + 10$ ,

which means that

$\left ( g \circ f\right ) (t) > \left ( f \circ g\right ) (t)$ .

That is, (B) is greater.

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Example Question #103 : How To Find The Solution To An Equation

Two lines have -intercept (0, 9) . Line A has -intercept (-5,0) ; Line B has -intercept (-7,0) . Which is the greater quantity?

(A) The slope of Line A

(B) The slope of Line B

Possible Answers:

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

(B) is greater

(A) and (B) are equal

(A) is greater

Correct answer:

(A) is greater

Explanation:

To get the slope of each line, use the slope formula

$m = \frac{y_{2}-y_{1}}{x_{2}-x_{1 }}$

For Line A, $x_{1} =-5, y_{1} =0, x_{2} = 0, y_{2} = 9$ . Substitute in the slope formula.

The slope is

$m = \frac{y_{2}-y_{1}}{x_{2}-x_{1}} = \frac{9-0}{0-(-5)} = \frac{9}{5}$

For Line B, $x_{1} =-7, y_{1} =0, x_{2} = 0, y_{2} = 9$ . Substitute in the slope formula.

The slope is

$m = \frac{y_{2}-y_{1}}{x_{2}-x_{1}} = \frac{9-0}{0-(-7)} = \frac{9}{7}$

Since

$\frac{9}{5} = \frac{63}{35}> \frac{45}{35} = \frac{9}{7}$ ,

Line A has the greater slope, and (A) is greater.

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Example Question #103 : How To Find The Solution To An Equation

Line A has -intercept (5, 0) and -intercept (0, 7) . Line C has -intercept (14, 0) and -intercept (0, -10) ; Line B is perpendicular to Line C. Which is the greater quantity?

(A) The slope of Line A

(B) The slope of Line B

Possible Answers:

(B) is greater

(A) is greater

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

(A) and (B) are equal

Correct answer:

(A) and (B) are equal

Explanation:

To get the slope of Line A and Line C, use the slope formula

$m = \frac{y_{2}-y_{1}}{x_{2}-x_{1 }}$

For Line A, $x_{1} =5, y_{1} =0, x_{2} = 0, y_{2} = 7$ . Substitute in the slope formula.

The slope is

$m = \frac{y_{2}-y_{1}}{x_{2}-x_{1 }}= \frac{7-0}{0-5} = \frac{7}{-5} = - \frac{7}{5}$

For Line C, $x_{1} =14, y_{1} =0, x_{2} = 0, y_{2} = -10$ . Substitute in the slope formula.

The slope is

$m = \frac{y_{2}-y_{1}}{x_{2}-x_{1 }}= \frac{-10-0}{0-14} = \frac{-10}{-14} = \frac{5}{7}$

Since Line B is perpendicular to Line C, its slope is the opposite of the reciprocal of that of Line C; this is $- \frac{5}{7}$ , which is equal to the slope of Line A.

The two quantities are equal.

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All ISEE Upper Level Quantitative Resources

8 Diagnostic Tests 234 Practice Tests Question of the Day Flashcards Learn by Concept

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ISEE Upper Level Quantitative : ISEE Upper Level (grades 9-12) Quantitative Reasoning

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

All ISEE Upper Level Quantitative Resources

Example Questions

Example Question #91 : Equations

Example Question #93 : How To Find The Solution To An Equation

Example Question #92 : How To Find The Solution To An Equation

Example Question #95 : How To Find The Solution To An Equation

Example Question #91 : How To Find The Solution To An Equation

Example Question #91 : Equations

Example Question #101 : How To Find The Solution To An Equation

Example Question #102 : How To Find The Solution To An Equation

Example Question #103 : How To Find The Solution To An Equation

Example Question #103 : How To Find The Solution To An Equation

All ISEE Upper Level Quantitative Resources

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