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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 #125 : How To Find The Solution To An Equation

3t = 7u

4u = 7v

9v = 2w

Which of the following is a true statement?

Possible Answers:

189t = 56w

27t = 8w

6t = w

147t = 8w

54t = 49 w

Correct answer:

54t = 49 w

Explanation:

3t = 7u

$\frac{3t}{3} =\frac{ 7u}{3}$

$t =\frac{ 7}{3}u$

Similarly,

4u = 7v

$u = \frac{7}{4}v$

9v = 2w

$v = \frac{2}{9}w$

By substitution:

$t =\frac{ 7}{3}u =\frac{ 7}{3} \cdot \frac{7}{4}v =\frac{ 7}{3} \cdot \frac{7}{4} \cdot \frac{2}{9}w$

$t = \frac{49}{54}w$

$54 \cdot t =54 \cdot \frac{49}{54}w$

54t = 49 w

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Example Question #126 : How To Find The Solution To An Equation

$8 ^{3x- 8} = 4 ^{y}$

Express in terms of .

Possible Answers:

$y = 2x-\frac{16}{3}$

$y = \frac{3}{2}x- 4$

y = 6x- 16

$y = \frac{9}{2}x- 12$

y = 3x-8

Correct answer:

$y = \frac{9}{2}x- 12$

Explanation:

$8 ^{3x- 8} = 4 ^{y}$

$\left (2^{3} \right )^{3x- 8} = (2^{2}) ^{y}$

$2^{3(3x- 8)} = 2^{2y}$

2y = 3(3x-8)

2y = 9x-24

$\frac{2y }{2}= \frac{9x-24}{2}$

$y = \frac{9}{2}x- 12$

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Example Question #127 : How To Find The Solution To An Equation

$6 ^{x} \cdot 36 ^{y} = 216 ^{t}$

Which of the following is true of ?

Possible Answers:

$t = \frac{1}{3}x+ \frac{2}{3}y$

t = 6x+ 3y

t = x+2y

$t = \frac{2}{3}x+ \frac{1}{3}y$

t = 3x+6y

Correct answer:

$t = \frac{1}{3}x+ \frac{2}{3}y$

Explanation:

$6 ^{x} \cdot 36 ^{y} = 216 ^{t}$

$6 ^{x} \cdot\left ( 6 ^{2} \right )^{y} = \left (6^{3} \right )^{t}$

$6 ^{x} \cdot 6 ^{2 y} = 6^{3 t}$

$6 ^{x + 2 y} = 6^{3 t}$

3t = x+2y

$\frac{3t }{3}= \frac{x+2y}{3}$

$t = \frac{1}{3}x+ \frac{2}{3}y$

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Example Question #128 : How To Find The Solution To An Equation

$\frac{4 ^{x} }{8 ^{y} } = 16^{t}$

Which of the following is true of ?

Possible Answers:

$t = 2x-\frac{4}{3} y$

None of the other responses gives a correct answer.

$t = \frac{1}{2}x- \frac{3}{4}y$

t = 8x-12y

t = 4x-2y

Correct answer:

$t = \frac{1}{2}x- \frac{3}{4}y$

Explanation:

$\frac{4 ^{x} }{8 ^{y} } = 16^{t}$

$\frac{\left (2^{2} \right )^{x} }{\left (2^{3} \right ) ^{y} } =\left ( 2^{4} \right ) ^{t}$

$\frac{ 2^{2x} }{ 2^{3y} } = 2^{4t}$

$2^{2x -3y} = 2^{4t}$

4t = 2x-3y

$\frac{4t }{4}= \frac{2x-3y}{4}$

$t = \frac{1}{2}x- \frac{3}{4}y$

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Example Question #129 : How To Find The Solution To An Equation

Function 4

Define to be the function graphed in the figure above, and g(x) = 4x+8 .

Evaluate $(g \circ f) (-3)$

Possible Answers:

$(g \circ f) (-3) = -4$

is outside the domain of $g \circ f$

$(g \circ f) (-3) = 16$

$(g \circ f) (-3) = 4$

$(g \circ f) (-3) = 8$

Correct answer:

$(g \circ f) (-3) = 16$

Explanation:

$(g \circ f) (-3) = g [f(-3)]$

From the diagram below, it can be seen that f(-3) = 2 .

Function 4a

$(g \circ f) (-3) = g [f(-3)] = g(2)$

g(x) = 4x+8 , so

g(x) = 4 (2)+8 = 8 + 8 = 16

Therefore,

$(g \circ f) (-3) = 16$ .

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Example Question #130 : How To Find The Solution To An Equation

Function 4

Define to be the function graphed in the figure above, and g(x) = 4x+8 .

Evaluate $(f \circ g) (-3)$

Possible Answers:

is outside the domain of $f \circ g$

$(f \circ g) (-3) = 8$

$(f \circ g) (-3) = -4$

$(f \circ g) (-3) = 16$

$(f \circ g) (-3) = 4$

Correct answer:

$(f \circ g) (-3) = 4$

Explanation:

$\left (f \circ g \right ) (-3) = f \left [ g (-3)\right ]$

g(x) = 4x+8

g(-3) = 4 (-3)+8 = -12 + 8 = -4

$f \left [ g (-3)\right ] = f(-4)$ .

Examine the diagram below.

Function 4a

As can be seen, f(-4)= 4 . Therefore, $(f \circ g) (-3) = 4$ .

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Example Question #131 : How To Find The Solution To An Equation

Function h

Let be the function whose graph is shown in the above figure. is defined by the equation

g(x) = f(-x) .

Give the -intercept of the graph of .

Possible Answers:

(0, -5)

(0, 5)

(0, 2)

(0, -2)

The graph of has no -intercept.

Correct answer:

(0, 5)

Explanation:

The -intercept of a function is the point at which x = 0 , so we can find this by evaluating g(0) .

g(x) = f(-x)

g(0) = f(-0) = f(0)

As seen in the diagram below, f(0) = 5 .

Function h1

Therefore, g(0) = f(0) = 5 , and the -intercept of the graph of is (0, 5) .

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Example Question #132 : How To Find The Solution To An Equation

Function h

Let be the function whose graph is shown in the above figure. is defined by the equation

g(x) = -f(x) .

Give the -intercept of the graph of .

Possible Answers:

The graph of has no -intercept.

(0, 2)

(0, 5)

(0, -2)

(0, -5)

Correct answer:

(0, -5)

Explanation:

The -intercept of a function is the point at which x = 0 , so we can find this by evaluating g(0) .

g(x) = -f(x)

g(0) = -f(0)

As seen in the diagram below, f(0) = 5 .

Function h1

Therefore, g(0) = -f(0) = -5 , and the -intercept of the graph of is (0, -5)

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Example Question #131 : How To Find The Solution To An Equation

Function h

Let be the function whose graph is shown in the above figure. is defined by the equation

$g(x) =- f^{-1}(x)$ .

Give the -intercept of the graph of .

Possible Answers:

The graph of has no -intercept.

(0, 5)

(0, -5)

(0, -2)

(0, 2)

Correct answer:

(0, 2)

Explanation:

The -intercept of a function is the point at which x = 0 , so we can find this by evaluating g(0) .

$g(x) =- f^{-1}(x)$

$g(0) = - f^{-1}(0)$

The graph of includes the point (-2,0) , as can be seen in the diagram below:

Function h1

Therefore, f(-2) = 0 and $g(0) =- f^{-1}(0)=- \left (-2 \right )= 2$ . The -intercept of the graph of is (0, 2) .

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Example Question #134 : How To Find The Solution To An Equation

Give the solution set of the equation

|70-t| - 70 = 70 .

Possible Answers:

$\left \{ -210 , - 70 \right \}$

$\left \{ -210 , 70 \right \}$

$\left \{ 70, 210\right \}$

The equation has no solution.

$\left \{ -70, 210\right \}$

Correct answer:

$\left \{ -70, 210\right \}$

Explanation:

|70-t| - 70 = 70

|70-t| - 70 + 70 = 70 + 70

|70-t| = 140

Either

70- t = 140 or 70- t = -140

so we solve each separately.

70- t = 140

70- t + t - 140 = 140 + t - 140

-70 = t

70- t+ t + 140 = -140+ t + 140

210 = t

The solution set is $\left \{ -70, 210\right \}$ .

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All ISEE Upper Level Quantitative Resources

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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 #125 : How To Find The Solution To An Equation

Example Question #126 : How To Find The Solution To An Equation

Example Question #127 : How To Find The Solution To An Equation

Example Question #128 : How To Find The Solution To An Equation

Example Question #129 : How To Find The Solution To An Equation

Example Question #130 : How To Find The Solution To An Equation

Example Question #131 : How To Find The Solution To An Equation

Example Question #132 : How To Find The Solution To An Equation

Example Question #131 : How To Find The Solution To An Equation

Example Question #134 : How To Find The Solution To An Equation

All ISEE Upper Level Quantitative Resources

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