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ISEE Upper Level Math : Equations

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Example Questions

Example Question #832 : Isee Upper Level (Grades 9 12) Mathematics Achievement

Define an operation $\blacktriangledown$ on two real numbers as follows:

For any real numbers a,b , $a \blacktriangledown b = a - b^{2}$

Solve for : $x \blacktriangledown 5 = 96$

Possible Answers:

x = 121

x = 71

x= 101

x = 91

Correct answer:

x = 121

Explanation:

$a \blacktriangledown b = a - b^{2}$ , so

$x \blacktriangledown 5 = x - 5^{2} = x - 25$

Now replace:

$x \blacktriangledown 5 = 96$

x - 25 = 96

x - 25+ 25 = 96+ 25

x = 121

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Example Question #831 : Isee Upper Level (Grades 9 12) Mathematics Achievement

Define an operation $\blacktriangle$ on a real number as follows:

For any real nonnegative number ,

$\blacktriangle A = A^{2} - \sqrt{A}$

Evaluate $\blacktriangle 100$ .

Possible Answers:

9,990

900

9,900

9 90

Correct answer:

9,990

Explanation:

Substitute A = 100 :

$\blacktriangle A = A^{2} - \sqrt{A}$

$\blacktriangle 100 = 100^{2} - \sqrt{100} = 10,000 - 10 = 9,990$

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Example Question #501 : Problem Solving

Greg, a barista at a coffeehouse, has to mix two types of coffeebeans together - Strawberry Dream and Vanilla Madness - to produce eighty pounds of a new flavor.

The Strawberry Dream beans cost $9 per pound; the Vanilla Madness beans cost $14 per pound. The new blend will cost $12 per pound, with the beans costing the same as if they were still separate.

Let be the number of pounds of Strawberry Dream beans that will go into the mixture. Which of the following equations can Greg set up to answer the question of how many pounds of each bean he will need?

Possible Answers:

9(80-S) + 14 S= 12

9(80-S) + 14 S= 960

9S + 14 (80-S)= 960

9S + 14 (80-S)= 12

Correct answer:

9S + 14 (80-S)= 960

Explanation:

The total cost of beans is the price per pound times the number of pounds. The total cost of the mixture is equal to that of the Strawberry Dream beans and that of the Vanilla Madness beans. Each can be found by multiplying the cost per pound by the number of pounds:

Strawberry Dream: $\$ 9 \cdot S$

Vanilla Madness: $\$14 \cdot (80-S)$ , since the total number of pounds is eighty.

Total mixture: $\$12 \cdot 80 = \$960$

Add to get the equation:

9S + 14 (80-S)= 960

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Example Question #834 : Isee Upper Level (Grades 9 12) Mathematics Achievement

In appreciation for a job well done, Loretta is being awarded a bonus equal to 7.5% of the salary she earned over the last two years. Her monthly salary over the last twelve months was $3,500 per month; she received a raise of $500 per month twelve months ago. What is the amount of the bonus?

Possible Answers:

$\$6,750$

$\$5,400$

$\$6,300$

$\$5,850$

Correct answer:

$\$5,850$

Explanation:

Loretta earned $3,000 per month for twelve months, and $3,500 per month for the following twelve months. This is a total of

$\$ 3,000 \times 12 + \$3,500 \times 12 = \$36,000 + \$42,000 =\$ 78,000$

earned over two years. 7.5% of this is

$\$ 78,000 \times 0.075 = \$5,850$ ,

the amount of Loretta's bonus.

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Example Question #131 : Equations

A line on the coordinate plane includes the points (5,a ), (9, a + 3) . What is the slope of the line?

Possible Answers:

$\frac{3}{4}$

$\frac{4} {3}$

$\frac{4} {3-a}$

$\frac{3-a} {4}$

Correct answer:

$\frac{3}{4}$

Explanation:

Use the slope formula, substituting x_1 = 5, y_1 = a , x_2 = 9, y_2 = a + 3 :

$m = \frac{y_2-y_1}{x_2-x_1} = \frac{\left ( a+ 3 \right )- a}{9-5} = \frac{3}{4}$

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Example Question #132 : Equations

Define

$f (x) = \left\{\begin{matrix} x+7\textrm{ if }x<0\\ x-2\textrm{ if }x \geq 0 \end{matrix}\right.$

and

$g(x) = x^{2}$ .

Evaluate:

$\left ( f\circ g\right ) (-4)$

Possible Answers:

Correct answer:

Explanation:

$\left ( f\circ g\right ) (-4) = f(g(-4))$

First, evaluate g (-4) :

$g(x) = x^{2}$

$g(-4) = (-4)^{2}= 16$

Therefore,

$\left ( f\circ g\right ) (-4) = f(g(-4 )) = f(16)$

which can be evaluated using the definition of for nonnegative values of :

f (x) = x-2

f (16) = 16-2 = 14

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Example Question #133 : Equations

For what value of does this quadratic equation have exactly one solution?

$x ^{2} + 9x + C = 0$

Possible Answers:

$20 \frac{1}{4}$

$40\frac{1}{2}$

Correct answer:

$20 \frac{1}{4}$

Explanation:

For $x ^{2} + 9x + C = 0$ to have exactly one solution, the discriminant of the expression $x ^{2} + 9x + C$ must be 0.

The discriminant of $Ax ^{2} + Bx + C$ is $B ^{2} - 4AC$ , so we substitute A = 1, B = 9 , set the discriminant equal to 0, and solve for :

$B ^{2} - 4AC = 0$

$9 ^{2} - 4 \cdot 1 \cdot C = 0$

81 - 4C = 0

81 - 4C+4C = 0+4C

4C = 81

$4C \div 4= 81 \div 4$

$C = 20 \frac{1}{4}$

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Example Question #838 : Isee Upper Level (Grades 9 12) Mathematics Achievement

For what value of does this quadratic equation have exactly one solution?

$x ^{2} + Bx + 36 = 0$

Possible Answers:

$B = -4 \textrm{ or } B = 4$

$B = -12 \textrm{ or } B = 12$

$B = -6 \textrm{ or } B =6$

$B = -9\textrm{ or } B = 9$

Correct answer:

$B = -12 \textrm{ or } B = 12$

Explanation:

For $x ^{2} + Bx + 36 = 0$ to have exactly one solution, the discriminant of the expression $x ^{2} + Bx + 36$ must be 0.

The discriminant of $Ax ^{2} + Bx + C$ is $B ^{2} - 4AC$ , so we substitute A = 1, C= 36 , set the discriminant equal to 0, and solve for :

$B ^{2} - 4AC = 0$

$B ^{2} - 4 \cdot 1 \cdot 36 = 0$

$B ^{2} - 144 = 0$

$B ^{2} =144$

$B = -12 \textrm{ or } B = 12$

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Example Question #833 : Isee Upper Level (Grades 9 12) Mathematics Achievement

A line with slope N - 10 includes point (3, 5) . What is the -intercept of this line?

Possible Answers:

(0,-3 N +35)

(0,3 N -25)

(0,-5 N +53)

(0,5 N -47)

Correct answer:

(0,-3 N +35)

Explanation:

The -intercept will be point (0,b) for some real .

We can substitute m = N - 10, x_1 = 3, y_1 = 5, x_2 = 0,y_2 =b in the slope formula and solve for :

$\frac{y_2-y_1}{x_2-x_1} = m$

$\frac{b-5}{0-3} = N - 10$

$\frac{b-5}{-3} = N - 10$

$\frac{b-5}{-3} \cdot (-3) =\left ( N - 10 \right ) \cdot (-3)$

b-5 =-3 N +30

b-5+5 =-3 N +30 +5

b =-3 N +35

The -intercept is (0,-3 N +35)

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Example Question #134 : Equations

Solve for :
x^2-4=5

Possible Answers:

$x=\pm3$

x=3

x=1

$x=\pm1$

Correct answer:

$x=\pm3$

Explanation:

x^2-4=5

x^2-4+4=5+4

x^2=9

$\sqrt{x^2}=\sqrt9$

$x=\pm3$

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All ISEE Upper Level Math Resources

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ISEE Upper Level Math : Equations

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

All ISEE Upper Level Math Resources

Example Questions

Example Question #832 : Isee Upper Level (Grades 9 12) Mathematics Achievement

Example Question #831 : Isee Upper Level (Grades 9 12) Mathematics Achievement

Example Question #501 : Problem Solving

Example Question #834 : Isee Upper Level (Grades 9 12) Mathematics Achievement

Example Question #131 : Equations

Example Question #132 : Equations

Example Question #133 : Equations

Example Question #838 : Isee Upper Level (Grades 9 12) Mathematics Achievement

Example Question #833 : Isee Upper Level (Grades 9 12) Mathematics Achievement

Example Question #134 : Equations

All ISEE Upper Level Math Resources

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