Algebra Class 10 Quiz
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Questions and Answers

What value should be added to both sides of the equation $x^2 + 16x = 2$ to complete the square?

  • -16
  • 16
  • 8
  • 64 (correct)

What is the solution of the equation $(-8x + 6) + 8 = -7(x + 4)$?

  • {-10} (correct)
  • {-26}
  • {-42}
  • {42}

What is the result of the operation $(4 + 4i) - (-9 + i)$ expressed in standard form?

  • 13 - 3i
  • -13 - 3i
  • 13 + 3i (correct)
  • -5 + 5i

Using the square root property, what are the solutions to the equation $(2x - 1)^2 = 9$?

<p>{-1, 2} (C)</p> Signup and view all the answers

What is the result when using the quadratic formula on $16x^2 + 1 = 3x$?

<p>$\frac{3 \pm 55}{32}$ (C)</p> Signup and view all the answers

What is the profit the vendor makes from selling 600 pretzels?

<p>$511 (A)</p> Signup and view all the answers

What is the domain and range of the relation {(9, 4), (-6, -4), (-7, -1), (-7, -9)}?

<p>domain = {-6, -7, 9}; range = {-4, -1, 4, -9} (D)</p> Signup and view all the answers

Which set-builder notation correctly represents the interval [-4, 1)?

<p>{x | -4 ≤ x &lt; 1} (A)</p> Signup and view all the answers

Which equation represents the line in point-slope form with a slope of 9, passing through the point (4, 3)?

<p>y - 3 = 9(x - 4) (C)</p> Signup and view all the answers

What does the function $P(x) = 0.65x - 89$ model?

<p>The relationship between the number of pretzels sold and the profit made (B)</p> Signup and view all the answers

What is the slope-intercept form of the line passing through (5, 2) and perpendicular to the line whose equation is y = 9x + 7?

<p>y = -9x - 47 (C)</p> Signup and view all the answers

What is the domain of the function f(x) = 1/(x + 6)?

<p>(-∞, -6) ∪ (-6, ∞) (B)</p> Signup and view all the answers

What is the midpoint of the line segment between the points (4, 5) and (1, 8)?

<p>(3, 13/2) (D)</p> Signup and view all the answers

For a quadratic function with a vertex at (-1, -6) that opens upwards, what can be inferred about its range?

<p>[-6, ∞) (B)</p> Signup and view all the answers

If the slope of a line is 9, what would be the slope of a line perpendicular to it?

<p>-1/9 (D)</p> Signup and view all the answers

What is the equation of a line that has a slope of 9 and passes through the point (0, 0) in slope-intercept form?

<p>y = 9x (C)</p> Signup and view all the answers

What does the inequality $x^2 + y^2 \leq 36$ represent in a Cartesian plane?

<p>A circle with a radius of 6 centered at the origin (C)</p> Signup and view all the answers

Which inequality describes a line that intercepts the x-axis at 2 and the y-axis at 8?

<p>4x + y \leq 8 (A)</p> Signup and view all the answers

What is the maximum value of y that satisfies both inequalities when represented graphically?

<p>5 (D)</p> Signup and view all the answers

Which graph represents the solution to the inequalities $x^2 + y^2 \leq 36$ and $8x + 2y \leq 16$ correctly?

<p>Graph C (B)</p> Signup and view all the answers

In the inequality $8x + 2y \leq 16$, what is the slope of the boundary line?

<p>-2 (A)</p> Signup and view all the answers

At which point do the inequalities $x^2 + y^2 \leq 36$ and $8x + 2y \leq 16$ intersect?

<p>(2, 0) (D)</p> Signup and view all the answers

Which of these points lies outside the feasible region defined by the inequalities?

<p>(5, 0) (A)</p> Signup and view all the answers

If we increase the right-hand side of the inequality $8x + 2y \leq 16$ to 20, what effect does this have?

<p>The feasible region expands (C)</p> Signup and view all the answers

What is the solution set for the inequality $(x - 3)(x + 2) > 0$?

<p>(-∞, -2) ∪ (3, ∞) (C)</p> Signup and view all the answers

If y varies directly as x, and y = 6 when x = 5, what is y when x = 10?

<p>12 (D)</p> Signup and view all the answers

What is the expected rabbit population next April given the growth formula $y = 190(2.7)^{0.09t}$ and starting with 190 rabbits?

<p>555 (C)</p> Signup and view all the answers

What is the approximate value of $e^{1.6}$ rounded to three decimal places?

<p>4.349 (D)</p> Signup and view all the answers

What is the exact solution to the logarithmic equation log 3 (x + 4) = 2?

<p>{13} (B)</p> Signup and view all the answers

Which choice correctly represents the cost function C(x) for a business selling vases at $30 each with supplies costing $17 per vase and an initial investment of $26,000?

<p>C(x) = 17x + 26,000 (B)</p> Signup and view all the answers

Which equation represents the equivalent exponential form of $log_2 32 = x$?

<p>$2^x = 32$ (A)</p> Signup and view all the answers

What is the equivalent logarithmic form of $2^{-3} = 8$?

<p>$log_{1/2} 2 = -3$ (B)</p> Signup and view all the answers

Using the substitution method, what is the solution to the system of equations 5x - 4y = 147 and x = 5y?

<p>{(7, 35)} (C)</p> Signup and view all the answers

From the addition method, what is the solution for the system of equations 6x + 7y = 30 and 6x + 2y = 60?

<p>{(-12, 6)} (C)</p> Signup and view all the answers

What does the expression $log_2 (xy)$ expand to using properties of logarithms?

<p>$log_2 x + log_2 y$ (C)</p> Signup and view all the answers

What is the solution for x in the equation $(x - 3)(x + 2) = 0$?

<p>x = -2 (C), x = 3 (D)</p> Signup and view all the answers

If you invested $8800 to sell vases costing $12 each and selling at $23, how many units must be sold to break even?

<p>801 units (D)</p> Signup and view all the answers

What is the solution set for the system of inequalities -4x - y = -24 and y = x^2 - 8?

<p>{(4, 8), (-8, 56)} (B)</p> Signup and view all the answers

Which of the following is the revenue function for selling vases at $30 each, based on the previous investment?

<p>R(x) = 30x (A)</p> Signup and view all the answers

What values must be rejected from the logarithmic expression log 3 (x + 4) = 2?

<p>x &lt; -4 (B)</p> Signup and view all the answers

What is the domain of the polynomial function f(x) = 2(x - 3)(x + 6)4?

<p>(- ∞, ∞) (C)</p> Signup and view all the answers

What are the zeros of the function f(x) = 2(x - 3)(x + 6)4, and how do they behave at the x-axis?

<p>3, multiplicity 1, crosses x-axis; -6, multiplicity 4, touches x-axis and turns around (C)</p> Signup and view all the answers

Using the Rational Zero Theorem, which of the following is a complete list of all possible rational zeros for f(x) = 5x^4 - x^2 + 2?

<p>±1, ±2, ±5, ±1/2 (B)</p> Signup and view all the answers

What are the vertical asymptotes for the rational function f(x) = (x + 4)/(x(x + 5))?

<p>x = 0 and x = -5 (C)</p> Signup and view all the answers

Which of the following describes the multiplicity of the zeros found in f(x) = 2(x - 3)(x + 6)4?

<p>3 has multiplicity 1; -6 has multiplicity 4 (B)</p> Signup and view all the answers

What is the range of the function f(x) = 2(x - 3)(x + 6)4?

<p>[-6, ∞) (B)</p> Signup and view all the answers

What is the behavior of the graph at x = 3 for the polynomial function f(x) = 2(x - 3)(x + 6)4?

<p>The graph crosses the x-axis (D)</p> Signup and view all the answers

What can be concluded about the vertical asymptote of f(x) = (x + 4)/(x(x + 5))?

<p>The function approaches infinity as it nears the vertical asymptote (B)</p> Signup and view all the answers

Flashcards

Completing the Square

A method to rewrite a quadratic equation in the form (x + a)² = b to solve for x.

Solving Linear Equations

Finding the value of the variable that makes the equation true.

Complex Numbers Addition

Adding or subtracting complex numbers, combining real and imaginary parts separately.

Square Root Property

If x² = a, then x = ±√a.

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Quadratic Formula

A formula used to solve quadratic equations in the form ax² + bx + c = 0.

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Domain of a Relation

The set of all possible input values (x-coordinates).

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Range of a Relation

The set of all possible output values (y-coordinates).

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Interval Notation

A way to describe a set of numbers using intervals (ranges).

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Profit from selling 600 pretzels

The amount of money a vendor makes from selling 600 pretzels, given a certain price per pretzel and expenses.

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Point-slope form of a line

A linear equation representing a line in the form y – y₁ = m(x – x₁), where (x₁, y₁) is a point on the line and m is the slope.

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Perpendicular lines

Two lines are perpendicular if their slopes are negative reciprocals of each other.

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Domain of a function

The set of all possible input values (x-values) for a function.

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Midpoint of a line segment

The point that divides a line segment into two equal parts.

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Domain of a quadratic function

Set of all x-values for which the quadratic function is defined.

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Range of a quadratic function

Set of all y-values for which the quadratic function is defined, in context of the given graph.

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Vertex of a parabola

The highest or lowest point on the graph of a quadratic function.

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Polynomial Function Zeros (Multiplicity)

Zeros of a polynomial function are the values of x where the function equals zero. Multiplicity indicates how many times a zero appears as a factor in the function.

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Graph Crossing/Touching X-axis

If a zero has an odd multiplicity, the graph crosses the x-axis at that zero. If a zero has an even multiplicity, the graph touches the x-axis but turns around at that zero.

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Rational Zero Theorem

A theorem to find potential rational zeros of a polynomial. Possible zeros are created by factors of the constant term divided by factors of the leading coefficient.

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Possible Rational Zeros

A list of possible rational numbers that could be zeros of a polynomial function.

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Vertical Asymptote (Rational Function)

A vertical asymptote is a vertical line that the graph of a rational function approaches but never touches, where the denominator equals zero and the corresponding factor from the numerator isn't cancelled.

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Range of a function

The set of all possible output values (y-values) for a function.

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Function

A relationship between inputs (x-values) and outputs (y-values) where each input has exactly one output.

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Direct Variation

A relationship between two variables where one variable is a constant multiple of the other. In the equation y = kx, k is the constant of variation.

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Exponential Growth

A pattern of increase where the rate of growth is proportional to the current value. It is often represented by the equation y = a(1 + r)^t, where a is the initial value, r is the growth rate, and t is time.

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Logarithm

The inverse operation of exponentiation. It answers the question 'what power do I need to raise the base to, to get this number?'

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Logarithm Base

The base of the logarithm, which determines the exponential relationship being used. For example, log2 8 asks 'what power do I need to raise 2 to, to get 8?'

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Solving Inequalities

Finding the set of values that satisfy an inequality, typically by isolating the variable using the same rules as for equations, with the exception that multiplying or dividing by a negative number flips the inequality sign.

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Set Notation

A way to represent a set of numbers using braces {} and listing the elements or describing them using a rule.

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Union of Sets

Combining two sets into a single set that includes all elements from both original sets. It is represented by the symbol U.

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Logarithmic Equation

An equation that involves logarithms. The goal is to find the value of the unknown variable that makes the equation true.

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Solving Logarithmic Equations

The process of finding the solution(s) to a logarithmic equation, which are the values of the variable that satisfy the equation.

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Substitution Method

A technique used to solve a system of equations by solving one equation for one variable and then substituting that expression into the other equation.

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Addition Method

A technique to solve a system of equations by adding or subtracting the equations together to eliminate one variable.

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Cost Function

A mathematical expression that represents the total cost incurred in producing a certain number of items.

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Revenue Function

A mathematical expression that represents the total income earned from selling a certain number of items.

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Break-Even Point

The point at which total revenue equals total cost, meaning the business is neither making a profit nor a loss.

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System of Equations

A set of two or more equations that share the same variables. The goal is to find the values of the variables that satisfy all equations simultaneously.

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System of Inequalities

A set of two or more inequalities that are considered together. The solution to a system of inequalities is the set of all points that satisfy all of the inequalities.

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Graphing Inequalities

Graphing a system of inequalities involves shading the areas that represent the solutions to each inequality. The solution to the whole system is where the shaded areas overlap.

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Boundary Lines

The lines that define the boundaries of the solutions to an inequality. These lines are usually drawn dashed if the inequality is strict (not including the line itself) or solid if the inequality is not strict (including the line itself).

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Testing Points

A method used to determine which side of a boundary line to shade. Choose a point not on the line and substitute its coordinates into the inequality. If the inequality is true, shade the side of the line containing the point. If the inequality is false, shade the other side.

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x² + y² ≤ 36

Represents a circle centered at the origin with a radius of 6. The inequality sign (≤) means that the shaded area includes the circle itself.

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8x + 2y ≤ 16

Represents a line. The inequality sign (≤) means that the shaded area includes the line itself. To graph this line, find its x and y intercepts.

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Finding the solution to the system

The solution area to the system of inequalities is the overlapping shaded areas from the original inequalities. This area represents all the points that fulfill both inequalities.

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Matching the graph

By carefully graphing both inequalities and finding the overlap, you can find the correct graph that matches the given system.

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Study Notes

Multiple Choice Questions

  • Question 1: To solve x² + 16x = 2 by completing the square, add 64 to both sides of the equation.
  • Question 2: The solution to the linear equation (-8x + 6) + 8 = -7(x + 4) is {42}.
  • Question 3: (4 + 4i) - (-9 + i) = 13 + 3i.
  • Question 4: The solution to the equation (2x – 1)² = 9 is {-1, 2}.
  • Question 5: The solution to the quadratic equation 16x² + 1 = 3x using the quadratic formula is (3 + √55)/32.
  • Question 6: The domain of the relation {(9, 4), (-6, −4), (−7, −1), (-7, −9)} is {-6, -7, 9} and the range is {-4, -1, 4, −9}.

Intervals

  • Question 7: The interval [-4, 1) in set-builder notation is {x | -4 ≤ x < 1}.

Functions

  • Question 8: P(600) = $390 (profit for selling 600 pretzels).
  • Question 9: The equation for the line with slope 8/8 passing through (4, 3) is y - 3= (x - 4).
  • Question 10: The equation of the line passing through (5, 2) and perpendicular to y = 1/9x + 7 in slope-intercept form is y = -9x + 47.
  • Question 11: The domain of the function f(x) = 1/(x - 6) is all real numbers except 6.
  • Question 12: The midpoint of the line segment with endpoints (4, 5) and (1, 8) is (5/2, 13/2).

Quadratic Functions

  • Question 13: A quadratic function with vertex (-1, -6) and opening upwards has a domain of (-∞, ∞) and a range of [-6, ∞).

Polynomials

  • Question 14: For f(x) = 2(x − 3)(x + 6)⁴, the zeros are x = 3 (multiplicity 1) and x = -6 (multiplicity 4). The graph touches the x-axis and turns around at x = 3 and crosses the x-axis at x = −6.

Rational Functions

  • Question 15: The possible rational zeros of f(x) = 5x⁴ - x² + 2 are ±1/5, ±1/2, ±1 and ±2.
  • Question 16: The vertical asymptote for f(x) = (x + 4)/(x(x + 5)) is x = 0.

Polynomial Inequalities

  • Question 17: The solution to the inequality (x − 3)(x + 2) > 0 is (-2, 3).

Exponential and Logarithmic Functions

  • Question 18: If y varies directly as x, and y = 6 when x = 5, then y = 12 when x = 10.
  • Question 19: The approximate rabbit population in the next April, using the given function, is expected to be 554.
  • Question 20: e¹·⁶ ≈ 4.953 (rounded to three decimal places).
  • Question 21: log₂32= x is equivalent to 2x = 32.
  • Question 22: The equivalent logarithmic form of 2⁻³ = ¹⁄₈ is log₂1/8 = –3.
  • Question 23: log₃(y³/x⁷) = 3log₃x - 7log₃y
  • Question 24: The solution to log₃(x + 4) = 2 is x = 10.

Systems of Equations

  • Question 25: The solution to the system of equations 5x - 4y = 147 and x = 5y is (35, 7).
  • Question 26: The solution to the system of equations 6x + 7y = 30 and 6x + 2y = 60 is (12, -6)
  • Question 27: The cost function C(x) = 17x + 26,000 and the revenue function R(x) = 30x
  • Question 28: The break-even point in this business scenario is 800 units.
  • Question 29: The solution to the system -4x - y = -24 and y = x² - 8 is (4, 8) and (-8, 56).

Systems of Inequalities

  • Question 30 The solution graph of the system of inequalities x² + y² ≤ 36 and 8x + 2y ≤ 16 is shown in image C.

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Test your knowledge of algebraic concepts with this quiz designed for 10th-grade students. It covers topics such as completing the square, solving linear and quadratic equations, and understanding functions and intervals. Each question challenges you to apply your algebra skills effectively.

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