Multiple Choice Question

As same as handout
Subject: Cs101
Tutor: Aiman Khan
Produced: Iman Kashif

Lecture # 01-10

Lesson # 01: Coordinates, graphs & lines

1. What is Calculus the study of?

a) Numbers and shapes

b) Continuous rates of change of quantities
c) Discrete mathematics
d) Imaginary numbers
Answer: b

2. Which of the following is an example of a natural number?

a) -2
b) 0
c) 5
d) 1/2
Answer: c

3. What is a set?

a) A single number
b) A collection of well-defined objects
c) A mathematical operation
d) A fraction
Answer: b

4. The set {1, 2, 3, 4} is a subset of which set?

a) {5, 6, 7, 8}
b) {1, 2, 3, 4, 5}
c) {0, 1, 2, 3}
d) {2, 4, 6, 8}
Answer: b

5. What symbol is used to denote an empty set?

a) ∞
b) {}
c) ∅
d) ⊆
Answer: c
6. What does the inequality a<ba < ba<b imply?

a) b−ab - ab−a is positive

b) b−ab - ab−a is negative
c) b−a=0b - a = 0b−a=0
d) b>0b > 0b>0
Answer: a

7. What are the endpoints of the closed interval [a, b]?

a) a only
b) b only
c) Neither a nor b
d) Both a and b
Answer: d

8. The solution set of x<5x < 5x<5 is represented as:

a) (5,∞)(5, ∞)(5,∞)
b) (−∞,5)(-∞, 5)(−∞,5)
c) (−∞,5](-∞, 5](−∞,5]
d) [5,∞)[5, ∞)[5,∞)
Answer: b

9. What is the geometric representation of an interval on the number line?

a) A dot
b) A line segment
c) A parabola
d) A curve
Answer: b

10. Which property of rational numbers allows them to be written as a ratio of integers?

a) Rational numbers are infinite

b) They can be expressed as terminating or repeating decimals
c) They have no decimal representation
d) They are irrational numbers
Answer: b

11. Why is division by zero not allowed?

a) It is undefined and leads to logical inconsistencies

b) It is too small to calculate
c) It results in infinity
d) It is a rule without a reason
Answer: a
12. Who demonstrated the existence of irrational numbers?

a) Pythagoras
b) Hippasus
c) Euclid
d) Archimedes
Answer: b

13. What kind of number is 2\sqrt{2}2?

a) Rational
b) Natural
c) Irrational
d) Imaginary
Answer: c

14. What defines the "order" of real numbers?

a) Their representation on the coordinate plane

b) Their size relative to other real numbers
c) Their divisibility properties
d) Their decimal expansion
Answer: b

15. Which symbol denotes that AAA is a subset of BBB?

a) ∅
b) ⊂
c) ∩
d) ∪
Answer: b

16. In set notation, what does A∩BA ∩ BA∩B represent?

a) Union of AAA and BBB

b) Difference between AAA and BBB
c) Elements common to AAA and BBB
d) Elements in BBB but not in AAA
Answer: c

17. What does the symbol ∪∪∪ mean in set theory?

a) Subset
b) Union
c) Intersection
d) Empty set
Answer: b

18. Which statement is true for open intervals?

a) The endpoints are included
b) The endpoints are excluded
c) Only one endpoint is included
d) It includes infinity
Answer: b

19. What type of interval is (a,b](a, b](a,b]?

a) Closed
b) Open
c) Half-open or half-closed
d) Infinite
Answer: c

20. Which mathematician is credited with analytic geometry?

a) Hippasus
b) Descartes
c) Newton
d) Leibniz
Answer: b

21. What does the coordinate line establish?

a) A way to divide numbers

b) A one-to-one correspondence between real numbers and points
c) A set of imaginary numbers
d) A geometric curve
Answer: b

22. The coordinate of the origin is always:

a) 1
b) -1
c) 0
d) Any positive number
Answer: c

23. Which property does not hold true for inequalities?

a) a<b⇒a+c<b+ca < b \Rightarrow a + c < b + ca<b⇒a+c<b+c

b) a<b⇒a−c>b−ca < b \Rightarrow a - c > b - ca<b⇒a−c>b−c
c) a<b and c>0⇒ac<bca < b \text{ and } c > 0 \Rightarrow ac < bca<b and c>0⇒ac<bc
d) a<b and c<d⇒a+c<b+da < b \text{ and } c < d \Rightarrow a + c < b + da<b and c<d⇒a+c<b+d
Answer: b

24. What does −∞-∞−∞ represent on a number line?

a) The smallest real number

b) A number smaller than all real numbers
c) Zero
d) Undefined
Answer: b

25. An interval with finite real endpoints is called:

a) Open interval
b) Infinite interval
c) Finite interval
d) Half-open interval
Answer: c

26. Which interval represents all real numbers?

a) (−∞,∞)(-∞, ∞)(−∞,∞)
b) [0,∞)[0, ∞)[0,∞)
c) (0,∞)(0, ∞)(0,∞)
d) (0,1)(0, 1)(0,1)
Answer: a

27. Which number is NOT a rational number?

a) 1.5
b) π\piπ
c) -2
d) 0.25
Answer: b

28. How do we denote the set of all x between 2 and 3, including 2 but not 3?

a) [2, 3]
b) (2, 3]
c) (2, 3)
d) [2, 3)
Answer: d

29. The inequality 7≤2−5x<97 ≤ 2 - 5x < 97≤2−5x<9 combines how many inequalities?

a) 1
b) 2
c) 3
d) 4
Answer: b

30. If x=−125x = -\frac{12}{5}x=−512 is the solution of an inequality, which interval represents the solution set?

a) [−125,∞)[\frac{-12}{5}, ∞)[5−12,∞)
b) (−∞,−125](-∞, \frac{-12}{5}](−∞,5−12]
c) (−125,∞)(\frac{-12}{5}, ∞)(5−12,∞)
d) (−∞,−125)(-∞, \frac{-12}{5})(−∞,5−12)
Answer: b

Lecture 2: Absolute Value:

1. What is the absolute value of a number aaa?

a) The negative of aaa

b) The square of aaa
c) The distance of aaa from zero on the number line
d) The reciprocal of aaa
Answer: c

2. How is the absolute value of aaa, denoted as ∣a∣|a|∣a∣, defined when a<0a < 0a<0?

a) aaa
b) −a-a−a
c) a2a^2a2
d) 000
Answer: b

3. What is the absolute value of 0?

a) -1
b) 0
c) 1
d) Undefined
Answer: b

4. If a≥0a \geq 0a≥0, what is ∣a∣|a|∣a∣?

a) aaa
b) −a-a−a
c) a2a^2a2
d) 000
Answer: a

5. What is the absolute value of −5-5−5?

a) -5
b) 0
c) 5
d) 25
Answer: c

6. Solve ∣x−3∣=4|x - 3| = 4∣x−3∣=4.

a) x=−1x = -1x=−1 or x=7x = 7x=7

b) x=7x = 7x=7 only
c) x=1x = 1x=1 or x=7x = 7x=7
d) x=−7x = -7x=−7 or x=1x = 1x=1
Answer: a

7. Which of the following is true about ∣a∣|a|∣a∣ for any real number aaa?

a) ∣a∣|a|∣a∣ is always negative

b) ∣a∣|a|∣a∣ is always non-negative
c) ∣a∣|a|∣a∣ is positive only if a≤0a \leq 0a≤0
d) ∣a∣|a|∣a∣ equals −a-a−a only for a≥0a \geq 0a≥0
Answer: b

8. Solve ∣x−2∣<5|x - 2| < 5∣x−2∣<5.

a) −3<x<7-3 < x < 7−3<x<7

b) x<−3x < -3x<−3 or x>7x > 7x>7
c) 3<x<53 < x < 53<x<5
d) −5<x<2-5 < x < 2−5<x<2
Answer: a

9. What is ∣a+b∣|a + b|∣a+b∣ always less than or equal to?

a) a+ba + ba+b
b) ∣a∣−∣b∣|a| - |b|∣a∣−∣b∣
c) ∣a∣+∣b∣|a| + |b|∣a∣+∣b∣
d) ∣a×b∣|a \times b|∣a×b∣
Answer: c

10. The formula for the distance ddd between two points AAA and BBB on a number line with coordinates aaa
and bbb is:

a) ∣a∣−∣b∣|a| - |b|∣a∣−∣b∣
b) ∣a+b∣|a + b|∣a+b∣
c) ∣b−a∣|b - a|∣b−a∣
d) ∣ab∣|ab|∣ab∣
Answer: c

11. Solve ∣x+4∣≥2|x + 4| \geq 2∣x+4∣≥2.

a) −6<x<−2-6 < x < -2−6<x<−2

b) x≤−6x \leq -6x≤−6 or x≥−2x \geq -2x≥−2
c) x≥−6x \geq -6x≥−6 or x≤−2x \leq -2x≤−2
d) x=−6x = -6x=−6 or x=−2x = -2x=−2
Answer: b

12. Which of the following is NOT a property of absolute value?

a) ∣a∣=∣−a∣|a| = |-a|∣a∣=∣−a∣
b) ∣a+b∣≤∣a∣+∣b∣|a + b| \leq |a| + |b|∣a+b∣≤∣a∣+∣b∣
c) ∣a×b∣=∣a∣×∣b∣|a \times b| = |a| \times |b|∣a×b∣=∣a∣×∣b∣
d) ∣a/b∣=∣b∣/∣a∣|a/b| = |b|/|a|∣a/b∣=∣b∣/∣a∣
Answer: d

13. What does the expression ∣x∣|x|∣x∣ geometrically represent?

a) The slope of the line passing through xxx

b) The distance between xxx and the origin
c) The midpoint of the interval containing xxx
d) The area under the curve at xxx
Answer: b

14. Solve ∣3x−2∣=5x+4|3x - 2| = 5x + 4∣3x−2∣=5x+4.

a) x=−3x = -3x=−3 or x=0x = 0x=0

b) x=0x = 0x=0 only
c) x=−3x = -3x=−3
d) x=3x = 3x=3
Answer: c

15. For a<0a < 0a<0, ∣a2∣|a^2|∣a2∣ equals:

a) a2a^2a2
b) −a2-a^2−a2
c) ∣a∣2|a|^2∣a∣2
d) aaa
Answer: a

16. What is ∣−4∣+∣5∣|-4| + |5|∣−4∣+∣5∣?

a) 1
b) 9
c) -9
d) -1
Answer: b

17. Which theorem explains that ∣a+b∣≤∣a∣+∣b∣|a + b| \leq |a| + |b|∣a+b∣≤∣a∣+∣b∣?

a) Triangle Inequality
b) Pythagorean Theorem
c) Distance Formula
d) Heisenberg Principle
Answer: a

18. Which is true for ∣a/b∣|a/b|∣a/b∣ when b≠0b \

a) ∣a/b∣=∣a∣×∣b∣|a/b| = |a| \times |b|∣a/b∣=∣a∣×∣b∣

b) ∣a/b∣=∣a∣+∣b∣|a/b| = |a| + |b|∣a/b∣=∣a∣+∣b∣
c) ∣a/b∣=∣a∣/∣b∣|a/b| = |a| / |b|∣a/b∣=∣a∣/∣b∣
d) ∣a/b∣=−∣a∣/∣b∣|a/b| = -|a| / |b|∣a/b∣=−∣a∣/∣b∣
Answer: c

19. What is the absolute value of −16-\sqrt{16}−16?

a) 4
b) -4
c) −4\sqrt{-4}−4
d) ±4\pm 4±4
Answer: a

20. Solve ∣x∣<3|x| < 3∣x∣<3.

a) x≥3x \geq 3x≥3

b) x≤−3x \leq -3x≤−3
c) −3<x<3-3 < x < 3−3<x<3
d) −3>x>3-3 > x > 3−3>x>3
Answer: c

21. What is ∣2a∣|2a|∣2a∣ if a=−3a = -3a=−3?

a) 6
b) -6
c) 9
d) -9
Answer: a

22. Solve ∣x∣>5|x| > 5∣x∣>5.

a) x≤5x \leq 5x≤5

b) x≥−5x \geq -5x≥−5
c) x<−5x < -5x<−5 or x>5x > 5x>5
d) −5<x<5-5 < x < 5−5<x<5
Answer: c

23. Which expression represents the distance between xxx and −a-a−a?

a) ∣x−a∣|x - a|∣x−a∣
b) ∣x+a∣|x + a|∣x+a∣
c) ∣a−x∣|a - x|∣a−x∣
d) ∣x×a∣|x \times a|∣x×a∣
Answer: b

24. If ∣x−2∣≤3|x - 2| \leq 3∣x−2∣≤3, what is the interval?

a) (−1,5](-1, 5](−1,5]
b) [0,5][0, 5][0,5]
c) [−1,5][-1, 5][−1,5]
d) [−1,2][-1, 2][−1,2]
Answer: c
25. If x≥0x \geq 0x≥0, then ∣x−3∣=3−x|x - 3| = 3 - x∣x−3∣=3−x is true when:

a) x≤3x \leq 3x≤3

b) x>3x > 3x>3
c) x=3x = 3x=3
d) x<0x < 0x<0
Answer: a

26. For any real aaa, ∣−a∣=∣a∣|-a| = |a|∣−a∣=∣a∣. This property states:

a) Absolute value is positive

b) Absolute value is commutative
c) Absolute value is invariant to sign
d) Absolute value is additive
Answer: c

27. Solve ∣x+2∣>6|x + 2| > 6∣x+2∣>6.

a) x>−8x > -8x>−8

b) x>4x > 4x>4 or x<−8x < -8x<−8
c) x>6x > 6x>6 or x<−6x < -6x<−6
d) −6<x<4-6 < x < 4−6<x<4
Answer: b

28. ∣a2∣|a^2|∣a2∣ is always equal to:

a) aaa
b) a2a^2a2
c) ∣a∣2|a|^2∣a∣2
d) −a2-a^2−a2
Answer: b

29. The triangle inequality is essential in:

a) Geometry only
b) Algebra only
c) Both geometry and algebra
d) Neither geometry nor algebra
Answer: c

30. a2=∣a∣\sqrt{a^2} = |a|a2=∣a∣ is true for:

a) All real aaa

b) Positive aaa only
c) Negative aaa only
d) Non-real aaa
Answer: a

Lecture 3: Graphs in the Coordinate Plane, Intercepts, and Symmetry:

1. What is the coordinate plane?
a) A one-dimensional line
b) The intersection of two lines at a 45-degree angle
c) The intersection of two perpendicular coordinate lines
d) A system for polar coordinates
Answer: c

2. What is an ordered pair?

a) Two real numbers in any order
b) Two real numbers with an assigned order
c) A pair of integers
d) A set of two equations
Answer: b

3. Which of the following correctly describes the quadrants in the rectangular coordinate system?
a) Divided into four regions labeled clockwise
b) Divided into three regions labeled counterclockwise
c) Divided into four regions labeled counterclockwise
d) Divided into six regions labeled randomly
Answer: c

4. In which quadrant does the point (-3, 2) lie?

a) Quadrant I
b) Quadrant II
c) Quadrant III
d) Quadrant IV
Answer: b

5. If a point lies on the x-axis, what is its y-coordinate?

a) 1
b) 0
c) -1
d) Undefined
Answer: b

Intercepts

6. What is the x-intercept of a graph?

a) A point where y=0y = 0y=0
b) A point where x=0x = 0x=0
c) A point where x>yx > yx>y
d) A point on the origin
Answer: a

7. The y-intercept of a graph has the form:

a) (a,0)(a, 0)(a,0)
b) (0,b)(0, b)(0,b)
c) (0,0)(0, 0)(0,0)
d) (x,y)(x, y)(x,y)
Answer: b

8. Find the x-intercept of y=3x−6y = 3x - 6y=3x−6.

a) (0, -6)
b) (-2, 0)
c) (2, 0)
d) (6, 0)
Answer: c

9. Find the y-intercept of 2x+y=42x + y = 42x+y=4.

a) (0, 4)
b) (0, -4)
c) (2, 0)
d) (4, 0)
Answer: a

10. What is the significance of intercepts in graphing equations?

a) They determine the slope of the line
b) They define the center of the plane
c) They identify where the graph crosses the axes
d) They determine symmetry of the graph
Answer: c

Symmetry

11. Which point is symmetric to (3, -5) about the x-axis?

a) (3, 5)
b) (-3, -5)
c) (-3, 5)
d) (5, -3)
Answer: a

12. If the graph of an equation is symmetric about the y-axis, what is true?
a) Replacing xxx with −x-x−x gives the same equation
b) Replacing yyy with −y-y−y gives the same equation
c) Replacing xxx with −y-y−y gives the same equation
d) Replacing yyy with xxx gives the same equation
Answer: a

13. Which symmetry does the graph of y=x2y = x^2y=x2 have?

a) Symmetric about the x-axis
b) Symmetric about the y-axis
c) Symmetric about the origin
d) No symmetry
Answer: b

14. The points (x, y) and (-x, -y) are symmetric about:
a) The x-axis
b) The y-axis
c) The origin
d) No axis
Answer: c

15. Determine whether the graph of y=x3y = x^3y=x3 is symmetric about the origin.
a) Yes
b) No
Answer: a

Graphs of Equations

16. What is the graph of y=x2y = x^2y=x2?

a) A straight line
b) A parabola opening upwards
c) A parabola opening downwards
d) A circle
Answer: b

17. Sketching the graph of x=y2x = y^2x=y2 gives:

a) A parabola opening upwards
b) A parabola opening downwards
c) A parabola opening to the left
d) A parabola opening to the right
Answer: d

18. Which type of graph does x2+y2=1x^2 + y^2 = 1x2+y2=1 represent?

a) A line
b) A circle
c) A parabola
d) A hyperbola
Answer: b

19. What is the solution of an equation in two variables?

a) A real number
b) An ordered pair of real numbers
c) A geometric point
d) A curve
Answer: b

20. Find if (2, 1) is a solution of y=x2−3y = x^2 - 3y=x2−3.

a) Yes
b) No
Answer: b

Using Symmetry to Graph

21. If a graph is symmetric about the x-axis, what is true of its points?
a) If (x,y)(x, y)(x,y) is on the graph, then (−x,y)(-x, y)(−x,y) is also on the graph
b) If (x,y)(x, y)(x,y) is on the graph, then (x,−y)(x, -y)(x,−y) is also on the graph
c) If (x,y)(x, y)(x,y) is on the graph, then (−x,−y)(-x, -y)(−x,−y) is also on the graph
d) If (x,y)(x, y)(x,y) is on the graph, then (−y,x)(-y, x)(−y,x) is also on the graph
Answer: b

22. If a graph is symmetric about the y-axis, what is true?

a) You only need to calculate for x≥0x \geq 0x≥0
b) You only need to calculate for y≤0y \leq 0y≤0
c) The graph does not depend on x
d) The graph passes through the origin
Answer: a

23. To graph y=1/xy = 1/xy=1/x, how can symmetry be used?

a) The graph is symmetric about the origin
b) The graph is symmetric about the x-axis only
c) The graph is symmetric about the y-axis only
d) The graph has no symmetry
Answer: a

24. Which is true for x=y2x = y^2x=y2?

a) Symmetric about x-axis
b) Symmetric about y-axis
c) Symmetric about the origin
d) Symmetric about both axes
Answer: a

25. Which equation’s graph is symmetric about the origin?

a) y=x2y = x^2y=x2
b) y=x3y = x^3y=x3
c) x=y2x = y^2x=y2
d) y=∣x∣y = |x|y=∣x∣
Answer: b

Additional Questions
26. If x=y2x = y^2x=y2, what is true about its graph?
a) It only exists in the first quadrant
b) It only exists in the second quadrant
c) It spans both quadrants of the x-axis
d) It spans all quadrants
Answer: c

27. Which symmetry reduces computation for y=x4+x2y = x^4 + x^2y=x4+x2?

a) Origin symmetry
b) y-axis symmetry
c) x-axis symmetry
d) No symmetry
Answer: b

28. The graph of y=xy = xy=x is:

a) A horizontal line
b) A vertical line
c) A line passing through the origin with slope 1
d) A parabola
Answer: c

29. What is the first step in graphing x2+y2=9x^2 + y^2 = 9x2+y2=9?

a) Plot the points for x≥0x \geq 0x≥0
b) Find the center and radius of the circle
c) Solve for y in terms of x
d) Reflect points over axes
Answer: b

30. Using symmetry, if (1, 2) lies on a symmetric graph, which point also lies on it for y-axis symmetry?
a) (1, -2)
b) (-1, 2)
c) (-1, -2)
d) None of these
Answer: b

Lecture # 4: Line and definition of slope

1. What is the definition of the slope of a line?

A) The ratio of the vertical change to the horizontal change

B) The distance between two points on the line
C) The distance from the origin to a point on the line
D) The length of the line segment

Answer: A) The ratio of the vertical change to the horizontal change

2. What is the slope of a vertical line?

A) 0
B) 1
C) Undefined
D) Infinite

Answer: C) Undefined

3. The slope of a line through the points (6,2) and (9,8) is:
A) 2
B) 3
C) 1/2
D) 2/3

Answer: A) 2

4. What does a positive slope indicate?

A) The line moves downward to the right

B) The line is horizontal
C) The line moves upward to the right
D) The line is vertical

Answer: C) The line moves upward to the right

5. What is the slope of the line through the points (2,9) and (4,3)?

A) 3
B) -3
C) 3/2
D) -3/2

Answer: D) -3/2

6. If the slope of a line is 1, the angle of inclination is:

A) 45 degrees
B) 30 degrees
C) 60 degrees
D) 90 degrees

Answer: A) 45 degrees

7. What does a zero slope represent?

A) A line inclined upward

B) A line inclined downward
C) A horizontal line
D) A vertical line

Answer: C) A horizontal line

8. If two lines are perpendicular, their slopes satisfy the condition:

A) m1 + m2 = 0
B) m1 * m2 = -1
C) m1 * m2 = 1
D) m1 - m2 = 1

Answer: B) m1 * m2 = -1
9. The angle of inclination for a line with slope -1 is:

A) 45 degrees
B) 135 degrees
C) 180 degrees
D) 0 degrees

Answer: B) 135 degrees

10. What is the equation of a line that is parallel to the y-axis and passes through (a, 0)?

A) y = a
B) x = a
C) y = 0
D) x = 0

Answer: B) x = a

11. The equation of a horizontal line passing through (0, b) is:

A) x = b
B) y = b
C) y = 0
D) x = 0

Answer: B) y = b

12. What form does the equation of a line in the point-slope form take?

A) y = mx + b
B) y - y1 = m(x - x1)
C) Ax + By = C
D) y = b

Answer: B) y - y1 = m(x - x1)

13. What is the slope-intercept form of a line?

A) y = mx + b
B) y - y1 = m(x - x1)
C) Ax + By = C
D) y = b

Answer: A) y = mx + b

14. The equation of a line passing through the point (3,4) with slope -3/2 is:

A) y = -3/2x + 5
B) y = -3/2x - 1
C) y = -3x + 5
D) y = -3x - 1

Answer: B) y = -3/2x - 1

15. A line passes through the points (-2,-1) and (3,4). What is the slope of the line?

A) 1
B) -1
C) -5/5
D) 5

Answer: A) 1

16. The equation of the line that passes through (0, -4) with slope -9 is:

A) y = -9x + 4
B) y = -9x - 4
C) y = 9x - 4
D) y = -9x + 32

Answer: B) y = -9x - 4

17. The general equation of a line is in the form:

A) Ax + By = C
B) y = mx + b
C) x = a
D) y = b

Answer: A) Ax + By = C

18. A line with equation 8x + 5y = 20 has slope:

A) 8/5
B) -8/5
C) 5/8
D) -5/8

Answer: B) -8/5

19. If a line has slope m = 2, what is the angle of inclination?

A) tan^(-1)(2)
B) 45 degrees
C) 90 degrees
D) 0 degrees

Answer: A) tan^(-1)(2)
20. Which of the following represents a first-degree equation in x and y?

A) x^2 + y^2 = 1
B) 4x + 6y - 5 = 0
C) y = x^2 + 3
D) 3x + 4y + 7 = 0

Answer: B) 4x + 6y - 5 = 0

21. The slope of a line that is parallel to the x-axis is:

A) 1
B) 0
C) Undefined
D) -1

Answer: B) 0

22. What is the formula for the slope of a line through the points (x1, y1) and (x2, y2)?

A) m = (x2 - x1) / (y2 - y1)

B) m = (y2 - y1) / (x2 - x1)
C) m = (x1 - x2) / (y1 - y2)
D) m = (y1 - y2) / (x1 - x2)

Answer: B) m = (y2 - y1) / (x2 - x1)

23. Which of the following is the equation of a line through the point (2,3) with slope -3/2?

A) y - 3 = -3/2(x - 2)
B) y = -3x + 5
C) y = 3x - 5
D) y = -3/2x + 6

Answer: A) y - 3 = -3/2(x - 2)

24. If two lines have slopes of 3 and -1/3, they are:

A) Parallel
B) Perpendicular
C) Horizontal
D) Vertical

Answer: B) Perpendicular

25. The slope of a line passing through the points (2, -5) and (6, 3) is:

A) 1/2
B) -2
C) 2
D) -1/2
Answer: D) -1/2

26. What is the slope of the line passing through the points (0, 0) and (4, 2)?

A) 1/2
B) 2
C) 1
D) 1/4

Answer: C) 1

27. What is the slope of the line passing through the points (1, 2) and (1, 4)?

A) 2
B) Undefined
C) 1
D) 0

Answer: B) Undefined

28. What is the slope of a line perpendicular to the line with slope 5?

A) 5
B) -1/5
C) -5
D) 1/5

Answer: B) -1/5

29. The point-slope form of a line is:

A) y = mx + b
B) y - y1 = m(x - x1)
C) Ax + By = C
D) x = a

Answer: B) y - y1 = m(x - x1)

30. The equation y = 3x + 2 represents a line with slope:

A) 2
B) 3
C) -3
D) 1/3

Answer: B) 3

31. The equation y = -2x + 4 represents a line with slope:

A) 2
B) -2
C) 4
D) -4

Answer: B) -2

32. The slope of a line that is horizontal is:

A) 0
B) Undefined
C) 1
D) -1

Answer: A) 0

33. A line through the points (4, 6) and (10, 3) has slope:

A) 3/4
B) -3/4
C) 3
D) -3

Answer: B) -3/4

34. What is the y-intercept of the line y = 4x + 7?

A) 4
B) 7
C) -7
D) 0

Answer: B) 7

35. The equation of a line is given as 2x - 3y = 6. What is the slope of the line?

A) 2/3
B) -2/3
C) 3/2
D) -3/2

Answer: B) -2/3

36. The slope of a line is -4. If the line passes through the point (1, 3), what is the equation of the line?

A) y - 3 = -4(x - 1)
B) y - 3 = 4(x - 1)
C) y - 3 = -4x + 3
D) y = -4x + 3

Answer: A) y - 3 = -4(x - 1)
37. A line with equation 5x - 2y = 10 has a slope of:

A) 5/2
B) -5/2
C) 2/5
D) -2/5

Answer: B) -5/2

38. A line through the points (0, -2) and (3, 4) has slope:

A) 2
B) -2
C) 6/3
D) 2/3

Answer: D) 2/3

39. What is the slope of the line passing through the points (5, -3) and (-2, 4)?

A) -7/7
B) 7/7
C) -7
D) 1

Answer: A) -7/7

40. The slope of a line through the points (-1, 2) and (2, -1) is:

A) -1
B) 1
C) -2/3
D) 3/2

Answer: C) -2/3

Lecture # 5: Distance, Circle & Equation:

1. The distance between two points P1(x1, y1) and P2(x2, y2) is given by:

A) (x2−x1)2+(y2−y1)2\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}(x2−x1)2+(y2−y1)2

B) (x2−x1)+(y2−y1)(x_2 - x_1) + (y_2 - y_1)(x2−x1)+(y2−y1)
C) (x1−x2)2+(y1−y2)2(x_1 - x_2)^2 + (y_1 - y_2)^2(x1−x2)2+(y1−y2)2
D) (x1−x2)2+(y1−y2)2\sqrt{(x_1 - x_2)^2 + (y_1 - y_2)^2}(x1−x2)2+(y1−y2)2

Answer: A) (x2−x1)2+(y2−y1)2\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}(x2−x1)2+(y2−y1)2

2. The distance between the points (-2, 3) and (1, 7) is:

A) 5
B) 4
C) 6
D) 7

Answer: A) 5

3. What is the midpoint of the line segment joining (3, -4) and (7, 2)?

A) (5, 1)
B) (4, 0)
C) (6, -1)
D) (2, 3)

Answer: A) (5, 1)

4. The equation of a circle with center (x0, y0) and radius r is:

A) (x−x0)2+(y−y0)2=r2(x - x_0)^2 + (y - y_0)^2 = r^2(x−x0)2+(y−y0)2=r2

B) (x−x0)2+(y−y0)=r(x - x_0)^2 + (y - y_0) = r(x−x0)2+(y−y0)=r
C) (x+x0)2+(y+y0)2=r2(x + x_0)^2 + (y + y_0)^2 = r^2(x+x0)2+(y+y0)2=r2
D) (x−x0)+(y−y0)=r2(x - x_0) + (y - y_0) = r^2(x−x0)+(y−y0)=r2

Answer: A) (x−x0)2+(y−y0)2=r2(x - x_0)^2 + (y - y_0)^2 = r^2(x−x0)2+(y−y0)2=r2

5. The equation (x+5)2+(y−3)2=16(x + 5)^2 + (y - 3)^2 = 16(x+5)2+(y−3)2=16 represents a circle with center at:

A) (5, -3)
B) (-5, 3)
C) (-5, -3)
D) (5, 3)

Answer: B) (-5, 3)

6. The equation (x−1)2+(y+2)2=25(x - 1)^2 + (y + 2)^2 = 25(x−1)2+(y+2)2=25 represents a circle with radius:

A) 5
B) 25
C) 10
D) 4

Answer: A) 5

7. What is the center of the circle with equation x2+y2−8x+2y+8=0x^2 + y^2 - 8x + 2y + 8 = 0x2+y2−8x+2y+8=0?

A) (4, -1)
B) (-4, 1)
C) (4, 1)
D) (-4, -1)

Answer: A) (4, -1)

8. The radius of the circle with equation x2+y2+12x−81/2=0x^2 + y^2 + 12x - 81/2 = 0x2+y2+12x−81/2=0 is:

A) 153/2\sqrt{153/2}153/2
B) 9\sqrt{9}9
C) 9
D) 3

Answer: A) 153/2\sqrt{153/2}153/2

9. The equation of a circle in the form Ax2+By2+Dx+Ey+F=0Ax^2 + By^2 + Dx + Ey + F = 0Ax2+By2+Dx+Ey+F=0

represents a circle if:

A) A=BA = BA=B
B) A=0A = 0A=0
C) A=B≠0A = B \
D) A≠0,B≠0A \neq 0, B \

Answer: C) A=B≠0A = B \

10. A circle equation that is degenerate (no graph) occurs when:

A) The radius is positive

B) k<0k < 0k<0
C) k=0k = 0k=0
D) k>0k > 0k>0

Answer: B) k<0k < 0k<0

11. The equation x2+y2=25x^2 + y^2 = 25x2+y2=25 represents a circle with center:

A) (0, 0) and radius 25

B) (0, 0) and radius 5
C) (5, 5) and radius 25
D) (5, 5) and radius 5

Answer: B) (0, 0) and radius 5

12. The x-coordinate of the vertex of a parabola given by y=ax2+bx+cy = ax^2 + bx + cy=ax2+bx+c is:

A) b2a\frac{b}{2a}2ab
B) −b2a\frac{-b}{2a}2a−b
C) −c2a\frac{-c}{2a}2a−c
D) −b2c\frac{-b}{2c}2c−b

Answer: B) −b2a\frac{-b}{2a}2a−b

13. The graph of y=x2−2x−2y = x^2 - 2x - 2y=x2−2x−2 has its vertex at:

A) (1, -3)
B) (2, -3)
C) (-1, -3)
D) (1, -2)
Answer: A) (1, -3)

14. The equation y=−x2+4x−5y = -x^2 + 4x - 5y=−x2+4x−5 represents a parabola that opens:

A) Upward
B) Downward
C) Left
D) Right

Answer: B) Downward

15. What is the vertex of the parabola given by y=−2x2+4x+6y = -2x^2 + 4x + 6y=−2x2+4x+6?

A) (1, 8)
B) (-1, 8)
C) (1, 6)
D) (-1, 6)

Answer: D) (-1, 6)

16. For the quadratic equation y=x2−2x−3y = x^2 - 2x - 3y=x2−2x−3, the x-intercepts are:

A) 1, -3
B) -1, 3
C) 2, -3
D) -2, 3

Answer: A) 1, -3

17. The equation y=4x2+2x−1y = 4x^2 + 2x - 1y=4x2+2x−1 represents a parabola that is:

A) Symmetric about the x-axis

B) Symmetric about the y-axis
C) Opening upward
D) Opening downward

Answer: C) Opening upward

18. The graph of the equation y=x2+2x−8y = x^2 + 2x - 8y=x2+2x−8 has:

A) One x-intercept
B) Two x-intercepts
C) No x-intercepts
D) One y-intercept

Answer: B) Two x-intercepts

19. The solution to the quadratic equation x2−2x−2=0x^2 - 2x - 2 = 0x2−2x−2=0 is:

A) x=1±3x = 1 \pm \sqrt{3}x=1±3
B) x=−1±3x = -1 \pm \sqrt{3}x=−1±3
C) x=2±2x = 2 \pm \sqrt{2}x=2±2
D) x=1±2x = 1 \pm 2x=1±2

Answer: A) x=1±3x = 1 \pm \sqrt{3}x=1±3

20. The maximum value of the function s=−4.9t2+24.5ts = -4.9t^2 + 24.5ts=−4.9t2+24.5t occurs at:

A) t=2.5t = 2.5t=2.5 sec

B) t=5t = 5t=5 sec
C) t=10t = 10t=10 sec
D) t=0t = 0t=0 sec

Answer: A) t=2.5t = 2.5t=2.5 sec

21. The height of the ball in the equation s=−4.9t2+24.5ts = -4.9t^2 + 24.5ts=−4.9t2+24.5t is 30.625 meters when t=t =t=:

A) 0 sec
B) 2.5 sec
C) 5 sec
D) 10 sec

Answer: B) 2.5 sec

22. The graph of the equation x=ay2+by+cx = ay^2 + by + cx=ay2+by+c is:

A) A parabola opening left or right

B) A parabola opening up or down
C) A straight line
D) A circle

Answer: A) A parabola opening left or right

23. For the equation y=3x2+6x−1y = 3x^2 + 6x - 1y=3x2+6x−1, the vertex is:

A) (-1, -4)
B) (1, -1)
C) (-1, -3)
D) (1, 3)

Answer: C) (-1, -3)

3. The length of the latus rectum of a parabola y2=4axy^2 = 4axy2=4ax is:

A) aaa
B) 4a4a4a
C) 2a2a2a
D) None of these

Answer: B) 4a4a4a
24. The eccentricity of a circle is:

A) 0
B) 1
C) Greater than 1
D) None of these

Answer: A) 0

25. The general equation of a parabola is given by:

A) x2+y2+Ax+By+C=0x^2 + y^2 + Ax + By + C = 0x2+y2+Ax+By+C=0

B) y2=4axy^2 = 4axy2=4ax
C) x2=4ayx^2 = 4ayx2=4ay
D) Both B and C

Answer: D) Both B and C

26. The center of the ellipse x29+y216=1\frac{x^2}{9} + \frac{y^2}{16} = 19x2+16y2=1 is:

A) (0, 0)
B) (3, 4)
C) (9, 16)
D) None of these

Answer: A) (0, 0)

27. The equation of the hyperbola with foci at (±c,0)(\pm c, 0)(±c,0) and vertices at (±a,0)(\pm a, 0)(±a,0) is:

A) x2a2−y2b2=1\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1a2x2−b2y2=1

B) y2a2−x2b2=1\frac{y^2}{a^2} - \frac{x^2}{b^2} = 1a2y2−b2x2=1
C) x2b2+y2a2=1\frac{x^2}{b^2} + \frac{y^2}{a^2} = 1b2x2+a2y2=1
D) None of these

Answer: A) x2a2−y2b2=1\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1a2x2−b2y2=1

28. The distance between the foci of the ellipse x29+y216=1\frac{x^2}{9} + \frac{y^2}{16} = 19x2+16y2=1 is:

A) 5
B) 4
C) 2
D) None of these

Answer: C) 2

29. The directrix of the parabola x2=−4ayx^2 = -4ayx2=−4ay is:

A) y=−ay = -ay=−a
B) y=ay = ay=a
C) x=ax = ax=a
D) x=−ax = -ax=−a
Answer: A) y=−ay = -ay=−a

30. The standard equation of a circle is given by:

A) (x−h)2+(y−k)2=r2(x - h)^2 + (y - k)^2 = r^2(x−h)2+(y−k)2=r2

B) x2+y2+Ax+By+C=0x^2 + y^2 + Ax + By + C = 0x2+y2+Ax+By+C=0
C) Both A and B
D) None of these

Answer: A) (x−h)2+(y−k)2=r2(x - h)^2 + (y - k)^2 = r^2(x−h)2+(y−k)2=r2

31. The slope of the line passing through (2,3)(2, 3)(2,3) and (5,−1)(5, -1)(5,−1) is:

A) 4/3
B) -4/3
C) 3/4
D) -3/4

Answer: B) -4/3

32. The distance between two points (x1,y1)(x_1, y_1)(x1,y1) and (x2,y2)(x_2, y_2)(x2,y2) is given by:

A) (x2−x1)2+(y2−y1)2\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}(x2−x1)2+(y2−y1)2

B) (x2−x1)+(y2−y1)(x_2 - x_1) + (y_2 - y_1)(x2−x1)+(y2−y1)
C) (x2−x1)2+(y2−y1)2(x_2 - x_1)^2 + (y_2 - y_1)^2(x2−x1)2+(y2−y1)2
D) None of these

Answer: A) (x2−x1)2+(y2−y1)2\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}(x2−x1)2+(y2−y1)2

33. If two lines are parallel, their slopes are:

A) Equal
B) Negative reciprocals
C) Zero
D) Undefined

Answer: A) Equal

34. The area of a triangle with vertices at (x1,y1)(x_1, y_1)(x1,y1), (x2,y2)(x_2, y_2)(x2,y2), and (x3,y3)(x_3, y_3)(x3,y3) is
given by:

A) 12∣x1(y2−y3)+x2(y3−y1)+x3(y1−y2)∣\frac{1}{2} | x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2) |21∣x1(y2−y3)+x2(y3

−y1)+x3(y1−y2)∣
B) 12[x1(y2−y3)+x2(y3−y1)+x3(y1−y2)]\frac{1}{2} [ x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2) ]21[x1(y2−y3)+x2(y3
−y1)+x3(y1−y2)]
C) ∣x1(y2−y3)+x2(y3−y1)+x3(y1−y2)∣| x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2) |∣x1(y2−y3)+x2(y3−y1)+x3(y1−y2)∣
D) None of these

Answer: A) 12∣x1(y2−y3)+x2(y3−y1)+x3(y1−y2)∣\frac{1}{2} | x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2) |21∣x1(y2−y3

)+x2(y3−y1)+x3(y1−y2)∣
35. The midpoint of the segment joining (x1,y1)(x_1, y_1)(x1,y1) and (x2,y2)(x_2, y_2)(x2,y2) is:

A) (x2−x1,y2−y1)(x_2 - x_1, y_2 - y_1)(x2−x1,y2−y1)

B) (x1+x22,y1+y22)(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2})(2x1+x2,2y1+y2)
C) (x1−x22,y1−y22)(\frac{x_1 - x_2}{2}, \frac{y_1 - y_2}{2})(2x1−x2,2y1−y2)
D) None of these

Answer: B) (x1+x22,y1+y22)(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2})(2x1+x2,2y1+y2)

36. A circle is defined as the set of all points that are:

A) Equidistant from a fixed line

B) Equidistant from a fixed point
C) At varying distances from a fixed point
D) None of these

Answer: B) Equidistant from a fixed point

37. The equation of a line passing through the point (3,−2)(3, -2)(3,−2) with a slope of 4 is:

A) y+2=4(x−3)y + 2 = 4(x - 3)y+2=4(x−3)

B) y−2=4(x−3)y - 2 = 4(x - 3)y−2=4(x−3)
C) y−2=3(x−4)y - 2 = 3(x - 4)y−2=3(x−4)
D) None of these

Answer: A) y+2=4(x−3)y + 2 = 4(x - 3)y+2=4(x−3)

38. The equation of a line in the slope-intercept form is:

A) y=mx+cy = mx + cy=mx+c
B) ax+by+c=0ax + by + c = 0ax+by+c=0
C) Both A and B
D) None of these

Answer: A) y=mx+cy = mx + cy=mx+c

39. The equation of the line perpendicular to y=2x+3y = 2x + 3y=2x+3 is:

A) y=−12x+cy = -\frac{1}{2}x + cy=−21x+c

B) y=2x+cy = 2x + cy=2x+c
C) y=−2x+cy = -2x + cy=−2x+c
D) y=12x+cy = \frac{1}{2}x + cy=21x+c

Answer: A) y=−12x+cy = -\frac{1}{2}x + cy=−21x+c

40. The slope of a vertical line is:

A) 0
B) Undefined
C) 1
D) Infinite
Answer: B) Undefined

Lecture # 6: Functions:
1. Who first introduced the term "function"?
A) Euler
B) Leibniz
C) Newton
D) Gauss

Answer: B) Leibniz

2. In the equation A=πr2A = \pi r^2A=πr2, the area AAA is a function of:
A) π\piπ
B) rrr
C) r2r^2r2
D) None of these

Answer: B) rrr

3. A function relates one variable to another such that:

A) One value of xxx gives multiple values of yyy
B) One value of xxx gives exactly one value of yyy
C) xxx and yyy are always equal
D) None of these

Answer: B) One value of xxx gives exactly one value of yyy

4. Which of the following equations does NOT represent a function?

A) y=4x+1y = 4x + 1y=4x+1
B) y=±xy = \pm xy=±x
C) y=x2y = x^2y=x2
D) y=2x−3y = 2x - 3y=2x−3

Answer: B) y=±xy = \pm xy=±x

5. The notation y=f(x)y = f(x)y=f(x) was introduced by:

A) Leibniz
B) Euler
C) Newton
D) Pythagoras

Answer: B) Euler

6. In y=f(x)y = f(x)y=f(x), xxx is typically:

A) Dependent
B) Independent
C) Constant
D) Coefficient

Answer: B) Independent
7. The expression f(3)=9f(3) = 9f(3)=9 means:
A) fff is multiplied by 333
B) The value of f(x)f(x)f(x) when x=3x = 3x=3 is 9
C) fff is divided by 333
D) None of these

Answer: B) The value of f(x)f(x)f(x) when x=3x = 3x=3 is 9

8. In the equation s=f(t)s = f(t)s=f(t), sss is:

A) A function of ttt
B) Independent variable
C) Always constant
D) None of these

Answer: A) A function of ttt

9. Two functions are considered the same if:

A) They have different formulas but the same variables
B) Their formulas are identical regardless of variable names
C) They have the same output for different inputs
D) None of these

Answer: B) Their formulas are identical regardless of variable names

10. The domain of a function is:

A) The set of dependent variable values
B) The set of independent variable values
C) The range of the function
D) None of these

Answer: B) The set of independent variable values

11. For the function h(x)=1x−2h(x) = \frac{1}{x - 2}h(x)=x−21, the domain excludes:
A) x=0x = 0x=0
B) x=1x = 1x=1
C) x=2x = 2x=2
D) None of these

Answer: C) x=2x = 2x=2

12. If no domain is explicitly stated, the domain is:

A) All real numbers
B) All values for which the function is undefined
C) The natural domain
D) None of these

Answer: C) The natural domain

13. The range of a function is:
A) The set of possible yyy-values
B) The set of possible xxx-values
C) Always positive
D) Always constant

Answer: A) The set of possible yyy-values

14. The natural domain of h(x)=1(x−1)(x−3)h(x) = \frac{1}{(x-1)(x-3)}h(x)=(x−1)(x−3)1 is:

A) (−∞,∞)(-\infty, \infty)(−∞,∞)
B) (−∞,1)∪(1,3)∪(3,∞)(-\infty, 1) \cup (1, 3) \cup (3, \infty)(−∞,1)∪(1,3)∪(3,∞)
C) (1,3)(1, 3)(1,3)
D) None of these

Answer: B) (−∞,1)∪(1,3)∪(3,∞)(-\infty, 1) \cup (1, 3) \cup (3, \infty)(−∞,1)∪(1,3)∪(3,∞)

15. Simplifying h(x)=x2−4x−2h(x) = \frac{x^2 - 4}{x - 2}h(x)=x−2x2−4 without altering its domain gives:
A) h(x)=x+2, x≠2h(x) = x + 2, \, x \
B) h(x)=x+2h(x) = x + 2h(x)=x+2
C) h(x)=x−2h(x) = x - 2h(x)=x−2
D) None of these

Answer: A) h(x)=x+2, x≠2h(x) = x + 2, \, x \neq 2h

16. The function y=x2y = x^2y=x2 has a range of:

A) All real numbers
B) Non-negative real numbers
C) Positive integers
D) None of these

Answer: B) Non-negative real numbers

17. The range of g(x)=1x−1g(x) = \frac{1}{x - 1}g(x)=x−11 is:

A) All real numbers except y=1y = 1y=1
B) All real numbers
C) All positive real numbers
D) None of these

Answer: A) All real numbers except y=1y = 1y=1

18. Piecewise functions are:

A) Always continuous
B) Defined by different formulas over different intervals
C) Only defined for integers
D) None of these

Answer: B) Defined by different formulas over different intervals

19. If f(x)=2x2−1f(x) = 2x^2 - 1f(x)=2x2−1, then f(t−1)=f(t - 1) =f(t−1)=:

A) 2t2−4t+12t^2 - 4t + 12t2−4t+1
B) t2−4t+1t^2 - 4t + 1t2−4t+1
C) 2t2−t+12t^2 - t + 12t2−t+1
D) None of these

Answer: A) 2t2−4t+12t^2 - 4t + 12t2−4t+1

20. Reversing the roles of xxx and yyy in 3x+2y=63x + 2y = 63x+2y=6 gives:
A) y=−32x+3y = -\frac{3}{2}x + 3y=−23x+3
B) x=−23y+2x = -\frac{2}{3}y + 2x=−32y+2
C) Both A and B
D) None of these

Answer: C) Both A and B

21. For g(x)=x−1g(x) = \sqrt{x - 1}g(x)=x−1, the domain is:

A) [1,∞)[1, \infty)[1,∞)
B) (−∞,1)(-\infty, 1)(−∞,1)
C) [0,∞)[0, \infty)[0,∞)
D) None of these

Answer: A) [1,∞)[1, \infty)[1,∞)

(Continued in the next message for questions 22-40...)

MCQs (Continued)

22. The range of g(x)=x−1g(x) = \sqrt{x - 1}g(x)=x−1 is:

A) [0,∞)[0, \infty)[0,∞)
B) (−∞,1)(-\infty, 1)(−∞,1)
C) [1,∞)[1, \infty)[1,∞)
D) None of these

Answer: A) [0,∞)[0, \infty)[0,∞)

23. The function f(x)=1xf(x) = \frac{1}{x}f(x)=x1 is defined for:

A) All real numbers except x=0x = 0x=0
B) All positive integers
C) All real numbers
D) None of these

Answer: A) All real numbers except x=0x = 0x=0

24. Which is an example of a piecewise function?

A) f(x)=3x+1f(x) = 3x + 1f(x)=3x+1
B) f(x)={1.75if 0<x≤11.75+0.5(x−1)if x>1f(x) = \begin{cases} 1.75 & \text{if } 0 < x \leq 1 \\ 1.75 + 0.5(x - 1) & \text{if } x > 1
\end{cases}f(x)={1.751.75+0.5(x−1)if 0<x≤1if x>1
C) f(x)=x2+1f(x) = x^2 + 1f(x)=x2+1
D) None of these
Answer: B) f(x)={1.75if 0<x≤11.75+0.5(x−1)if x>1f(x) = \begin{cases} 1.75 & \text{if } 0 < x \leq 1 \\ 1.75 + 0.5(x - 1) &
\text{if } x > 1 \end{cases}f(x)={1.751.75+0.5(x−1)if 0<x≤1if x>1

25. The natural domain of a square root function f(x)=x−2f(x) = \sqrt{x - 2}f(x)=x−2 is:
A) x≥2x \geq 2x≥2
B) x>2x > 2x>2
C) x≤2x \leq 2x≤2
D) None of these

Answer: A) x≥2x \geq 2x≥2

26. The function y=x+1y = x + 1y=x+1 has a range of:

A) All real numbers
B) Positive integers
C) Negative integers
D) None of these

Answer: A) All real numbers

27. Which equation shows xxx as a function of yyy?

A) y=3x2y = 3x^2y=3x2
B) x=2y+3x = 2y + 3x=2y+3
C) x2+y2=1x^2 + y^2 = 1x2+y2=1
D) None of these

Answer: B) x=2y+3x = 2y + 3x=2y+3

28. A restricted domain is often used to:

A) Simplify functions
B) Solve quadratic equations
C) Remove constants
D) None of these

Answer: A) Simplify functions

29. The range of f(x)=x2−3f(x) = x^2 - 3f(x)=x2−3 is:

A) [−3,∞)[-3, \infty)[−3,∞)
B) (−∞,−3](-\infty, -3](−∞,−3]
C) [0,∞)[0, \infty)[0,∞)
D) None of these

Answer: A) [−3,∞)[-3, \infty)[−3,∞)

30. What happens if you cancel factors in the function h(x)=x2−4x−2h(x) = \frac{x^2 - 4}{x - 2}h(x)=x−2x2−4 without
altering the domain?
A) You must write x≠2x \
B) The function becomes invalid
C) The range is reduced
D) None of these

Answer: A) You must write x≠2x \

31. For the function y=x2+3x+2y = x^2 + 3x + 2y=x2+3x+2, the domain is:
A) All real numbers
B) Non-negative real numbers
C) Positive integers
D) None of these

Answer: A) All real numbers

32. If f(x)=1x2−4f(x) = \frac{1}{x^2 - 4}f(x)=x2−41, the domain excludes:

A) x=±2x = \pm 2x=±2
B) x=0x = 0x=0
C) x=2x = 2x=2
D) None of these

Answer: A) x=±2x = \pm 2x=±2

33. The equation y=x2+3x+2y = x^2 + 3x + 2y=x2+3x+2 represents:

A) A linear function
B) A quadratic function
C) A constant function
D) None of these

Answer: B) A quadratic function

34. The cost of a taxicab ride defined by two formulas is an example of:
A) A continuous function
B) A piecewise function
C) A quadratic function
D) None of these

Answer: B) A piecewise function

35. The equation y=2+1x−1y = 2 + \frac{1}{x - 1}y=2+x−11 has a range of:

A) (−∞,2)∪(2,∞)( -\infty, 2 ) \cup ( 2, \infty )(−∞,2)∪(2,∞)
B) (1,∞)( 1, \infty )(1,∞)
C) [2,∞)[2, \infty)[2,∞)
D) None of these

Answer: A) (−∞,2)∪(2,∞)( -\infty, 2 ) \cup ( 2, \infty )(−∞,2)∪(2,∞)

36. In reversing roles, 3x+2y=63x + 2y = 63x+2y=6 can be expressed as:

A) y=−32x+3y = -\frac{3}{2}x + 3y=−23x+3
B) x=−23y+2x = -\frac{2}{3}y + 2x=−32y+2
C) Both A and B
D) None of these

Answer: C) Both A and B

37. The function y=1x2−4y = \frac{1}{x^2 - 4}y=x2−41 is undefined for:
A) x=±2x = \pm 2x=±2
B) x=0x = 0x=0
C) x=4x = 4x=4
D) None of these

Answer: A) x=±2x = \pm 2x=±2

38. The cost function f(x)={1.75if 0<x≤11.75+0.5(x−1)if x>1f(x) = \begin{cases} 1.75 & \text{if } 0 < x \leq 1 \\ 1.75 + 0.5(x -
1) & \text{if } x > 1 \end{cases}f(x)={1.751.75+0.5(x−1)if 0<x≤1if x>1 changes:
A) After 1 mile
B) At 1.5 miles
C) At 2 miles
D) None of these

Answer: A) After 1 mile

39. The domain of f(x)=x−3f(x) = \sqrt{x - 3}f(x)=x−3 is:

A) [3,∞)[3, \infty)[3,∞)
B) (−∞,3)(-\infty, 3)(−∞,3)
C) [0,∞)[0, \infty)[0,∞)
D) None of these

Answer: A) [3,∞)[3, \infty)[3,∞)

40. The range of g(x)=x+5g(x) = x + 5g(x)=x+5 is:

A) All real numbers
B) [5,∞)[5, \infty)[5,∞)
C) [0,∞)[0, \infty)[0,∞)
D) None of these

Answer: A) All real numbers

Lecture # 7: Operations on Functions:

1. What does the sum of two functions f(x)f(x)f(x) and g(x)g(x)g(x) represent?
A) A new function defined as f(x)−g(x)f(x) - g(x)f(x)−g(x)
B) A new function defined as f(x)⋅g(x)f(x) \cdot g(x)f(x)⋅g(x)
C) A new function defined as f(x)+g(x)f(x) + g(x)f(x)+g(x)
D) None of these

Answer: C) A new function defined as f(x)+g(x)f(x) + g(x)f(x)+g(x)

2. For the functions f(x)=x2f(x) = x^2f(x)=x2 and g(x)=xg(x) = xg(x)=x, what is (f+g)(x)(f + g)(x)(f+g)(x)?
A) x2+xx^2 + xx2+x
B) x2−xx^2 - xx2−x
C) x2⋅xx^2 \cdot xx2⋅x
D) None of these

Answer: A) x2+xx^2 + xx2+x

3. The difference of two functions f(x)f(x)f(x) and g(x)g(x)g(x) is defined as:
A) (f−g)(x)=f(x)−g(x)(f - g)(x) = f(x) - g(x)(f−g)(x)=f(x)−g(x)
B) (f−g)(x)=f(x)+g(x)(f - g)(x) = f(x) + g(x)(f−g)(x)=f(x)+g(x)
C) (f−g)(x)=f(x)⋅g(x)(f - g)(x) = f(x) \cdot g(x)(f−g)(x)=f(x)⋅g(x)
D) None of these

Answer: A) (f−g)(x)=f(x)−g(x)(f - g)(x) = f(x) - g(x)(f−g)(x)=f(x)−g(x)

4. If f(x)=x2f(x) = x^2f(x)=x2 and g(x)=x+2g(x) = x + 2g(x)=x+2, what is (f⋅g)(x)(f \cdot g)(x)(f⋅g)(x)?

A) x2+(x+2)x^2 + (x + 2)x2+(x+2)
B) x2⋅(x+2)x^2 \cdot (x + 2)x2⋅(x+2)
C) x2−(x+2)x^2 - (x + 2)x2−(x+2)
D) None of these

Answer: B) x2⋅(x+2)x^2 \cdot (x + 2)x2⋅(x+2)

5. What is the domain of the function (f/g)(x)(f / g)(x)(f/g)(x) where g(x)≠0g(x) \

A) The domain of f(x)f(x)f(x) only
B) The domain of g(x)g(x)g(x) only
C) The intersection of the domains of f(x)f(x)f(x) and g(x)g(x)g(x), excluding points where g(x)=0g(x) = 0g(x)=0
D) None of these

Answer: C) The intersection of the domains of f(x)f(x)f(x) and g(x)g(x)g(x), excluding points where g(x)=0g(x) = 0g(x)=0

6. If f(x)=x2−4f(x) = x^2 - 4f(x)=x2−4 and g(x)=x−2g(x) = x - 2g(x)=x−2, the domain of (f/g)(x)(f / g)(x)(f/g)(x) excludes:
A) x=2x = 2x=2
B) x=−2x = -2x=−2
C) Both x=2x = 2x=2 and x=−2x = -2x=−2
D) None of these

Answer: A) x=2x = 2x=2

7. If f(x)=1+x2f(x) = 1 + x^2f(x)=1+x2 and g(x)=x−1g(x) = x - 1g(x)=x−1, what is (f+g)(x)(f + g)(x)(f+g)(x)?

A) 1+x2−x+11 + x^2 - x + 11+x2−x+1
B) x2+xx^2 + xx2+x
C) 1+x2+x−11 + x^2 + x - 11+x2+x−1
D) None of these

Answer: C) 1+x2+x−11 + x^2 + x - 11+x2+x−1

8. If f(x)=x2+1f(x) = x^2 + 1f(x)=x2+1 and g(x)=3xg(x) = 3xg(x)=3x, what is (f⋅g)(x)(f \cdot g)(x)(f⋅g)(x)?
A) 3x⋅(x2+1)3x \cdot (x^2 + 1)3x⋅(x2+1)
B) x2+3x+1x^2 + 3x + 1x2+3x+1
C) 3x+13x + 13x+1
D) None of these

Answer: A) 3x⋅(x2+1)3x \cdot (x^2 + 1)3x⋅(x2+1)

9. Which of the following represents the composition of f(x)f(x)f(x) and g(x)g(x)g(x), written as (f∘g)(x)(f \circ g)(x)(f∘g)(x)?
A) f(x)+g(x)f(x) + g(x)f(x)+g(x)
B) f(g(x))f(g(x))f(g(x))
C) g(f(x))g(f(x))g(f(x))
D) None of these

Answer: B) f(g(x))f(g(x))f(g(x))

10. If f(x)=x2f(x) = x^2f(x)=x2 and g(x)=x+1g(x) = x + 1g(x)=x+1, what is (f∘g)(x)(f \circ g)(x)(f∘g)(x)?
A) (x+1)2(x + 1)^2(x+1)2
B) x2+1x^2 + 1x2+1
C) (x2+1)(x^2 + 1)(x2+1)
D) None of these

Answer: A) (x+1)2(x + 1)^2(x+1)2

11. If g(x)=x+1g(x) = x + 1g(x)=x+1 and f(x)=x3f(x) = x^3f(x)=x3, then (f∘g)(x)(f \circ g)(x)(f∘g)(x) is:
A) (x+1)3(x + 1)^3(x+1)3
B) x3+1x^3 + 1x3+1
C) (x3+1)3(x^3 + 1)^3(x3+1)3
D) None of these

Answer: A) (x+1)3(x + 1)^3(x+1)3

12. The domain of (f∘g)(x)(f \circ g)(x)(f∘g)(x) is:

A) The domain of f(x)f(x)f(x) only
B) The domain of g(x)g(x)g(x) only
C) The domain of g(x)g(x)g(x) where g(x)g(x)g(x) lies in the domain of f(x)f(x)f(x)
D) None of these

Answer: C) The domain of g(x)g(x)g(x) where g(x)g(x)g(x) lies in the domain of f(x)f(x)f(x)

13. If f(x)=x+2f(x) = x + 2f(x)=x+2 and g(x)=x2−1g(x) = x^2 - 1g(x)=x2−1, then (g∘f)(x)(g \circ f)(x)(g∘f)(x) is:
A) (x+2)2−1(x + 2)^2 - 1(x+2)2−1
B) (x2+2)2−1(x^2 + 2)^2 - 1(x2+2)2−1
C) (x+1)2−2(x + 1)^2 - 2(x+1)2−2
D) None of these

Answer: A) (x+2)2−1(x + 2)^2 - 1(x+2)2−1

14. If f(x)=xf(x) = \sqrt{x}f(x)=x and g(x)=x+3g(x) = x + 3g(x)=x+3, what is (f∘g)(x)(f \circ g)(x)(f∘g)(x)?
A) x+3\sqrt{x + 3}x+3
B) x+3x + \sqrt{3}x+3
C) x+3\sqrt{x} + 3x+3
D) None of these

Answer: A) x+3\sqrt{x + 3}x+3

15. The composition of two functions, fff and ggg, is generally:

A) Always commutative
B) Always associative
C) Neither commutative nor associative
D) None of these

Answer: B) Always associative

16. If f(x)=x2+1f(x) = x^2 + 1f(x)=x2+1 and g(x)=x−1g(x) = x - 1g(x)=x−1, what is (f−g)(x)(f - g)(x)(f−g)(x)?
A) x2+1−(x−1)x^2 + 1 - (x - 1)x2+1−(x−1)
B) x2−x+2x^2 - x + 2x2−x+2
C) x2−x−2x^2 - x - 2x2−x−2
D) Both A and B

Answer: D) Both A and B

17. If f(x)=x2f(x) = x^2f(x)=x2 and g(x)=1xg(x) = \frac{1}{x}g(x)=x1, the domain of (f⋅g)(x)(f \cdot g)(x)(f⋅g)(x) excludes:
A) x=0x = 0x=0
B) x=1x = 1x=1
C) x=−1x = -1x=−1
D) None of these

Answer: A) x=0x = 0x=0

18. Which of the following functions can be written as a composition?

A) h(x)=(x+1)2h(x) = (x + 1)^2h(x)=(x+1)2
B) h(x)=x2+1h(x) = \sqrt{x^2 + 1}h(x)=x2+1
C) Both A and B
D) None of these

Answer: C) Both A and B

19. For the function f(x)=x2+1f(x) = x^2 + 1f(x)=x2+1, what is f2(x)f^2(x)f2(x)?

A) (x2+1)2(x^2 + 1)^2(x2+1)2
B) x4+1x^4 + 1x4+1
C) x2+2x+1x^2 + 2x + 1x2+2x+1
D) None of these

Answer: A) (x2+1)2(x^2 + 1)^2(x2+1)2

20. Which of the following is true for a constant function f(x)=cf(x) = cf(x)=c?
A) It assigns the same value to every xxx in the domain
B) It assigns a different value to each xxx in the domain
C) It is undefined for all xxx
D) None of these

Answer: A) It assigns the same value to every xxx in the domain

21. If f(x)=x2+3x+2f(x) = x^2 + 3x + 2f(x)=x2+3x+2, which two functions can be composed to form f(x)f(x)f(x)?
A) g(x)=x2,h(x)=3x+2g(x) = x^2, h(x) = 3x + 2g(x)=x2,h(x)=3x+2
B) g(x)=x+2,h(x)=x2+3g(x) = x + 2, h(x) = x^2 + 3g(x)=x+2,h(x)=x2+3
C) g(x)=x+1,h(x)=x2+2g(x) = x + 1, h(x) = x^2 + 2g(x)=x+1,h(x)=x2+2
D) None of these
Answer: D) None of these

22. If f(x)=x3f(x) = x^3f(x)=x3 and g(x)=x2+1g(x) = x^2 + 1g(x)=x2+1, what is (f∘g)(x)(f \circ g)(x)(f∘g)(x)?
A) (x2+1)3(x^2 + 1)^3(x2+1)3
B) x6+1x^6 + 1x6+1
C) (x2)3+1(x^2)^3 + 1(x2)3+1
D) None of these

Answer: A) (x2+1)3(x^2 + 1)^3(x2+1)3

23. A monomial in xxx must:

A) Have a positive integer power of xxx
B) Have a constant coefficient
C) Both A and B
D) None of these

Answer: C) Both A and B

24. If f(x)=x2−4f(x) = x^2 - 4f(x)=x2−4 and g(x)=x+2g(x) = x + 2g(x)=x+2, what is the domain of (f/g)(x)(f / g)(x)(f/g)(x)?
A) All real numbers except x=−2x = -2x=−2
B) All real numbers except x=2x = 2x=2
C) All real numbers except x=±2x = \pm 2x=±2
D) None of these

Answer: A) All real numbers except x=−2x = -2x=−2

25. If f(x)=x2+1f(x) = x^2 + 1f(x)=x2+1 and g(x)=xg(x) = \sqrt{x}g(x)=x, the domain of (f∘g)(x)(f \circ g)(x)(f∘g)(x) is:
A) [0,∞)[0, \infty)[0,∞)
B) (−∞,∞)(-\infty, \infty)(−∞,∞)
C) (−∞,0)(-\infty, 0)(−∞,0)
D) None of these

Answer: A) [0,∞)[0, \infty)[0,∞)

26. If f(x)=x2f(x) = x^2f(x)=x2 and g(x)=ln⁡(x)g(x) = \ln(x)g(x)=ln(x), the domain of (f∘g)(x)(f \circ g)(x)(f∘g)(x) is:
A) All real numbers
B) x>0x > 0x>0
C) x≥0x \geq 0x≥0
D) None of these

Answer: B) x>0x > 0x>0

27. Which of the following is an example of a polynomial?

A) f(x)=3x2−2x+1f(x) = 3x^2 - 2x + 1f(x)=3x2−2x+1
B) f(x)=x−1+x2f(x) = x^{-1} + x^2f(x)=x−1+x2
C) f(x)=2x+5f(x) = 2\sqrt{x} + 5f(x)=2x+5
D) None of these

Answer: A) f(x)=3x2−2x+1f(x) = 3x^2 - 2x + 1f(x)=3x2−2x+1

28. If f(x)=2x+1f(x) = 2x + 1f(x)=2x+1 and g(x)=3x−1g(x) = 3x - 1g(x)=3x−1, find (f+g)(x)(f + g)(x)(f+g)(x):
A) 5x5x5x
B) 5x5x5x + 000
C) 5x5x5x + 000
D) (sum eq-null division

Answer Proper Layer fix

Here’s the corrected Question 28 and its options:

28. If f(x)=2x+1f(x) = 2x + 1f(x)=2x+1 and g(x)=3x−1g(x) = 3x - 1g(x)=3x−1, find (f+g)(x)(f + g)(x)(f+g)(x):

A) 5x+05x + 05x+0
B) 5x+15x + 15x+1
C) 5x−25x - 25x−2
D) 5x5x5x

Answer: C) 5x−25x - 25x−2

29. For f(x)=3xf(x) = 3xf(x)=3x and g(x)=x2g(x) = x^2g(x)=x2, find (g∘f)(x)(g \circ f)(x)(g∘f)(x):
A) 3x23x^23x2
B) (3x)2(3x)^2(3x)2
C) 9x29x^29x2
D) None of these

Answer: C) 9x29x^29x2

30. Which of the following is NOT a polynomial?

A) f(x)=4x3+x2−1f(x) = 4x^3 + x^2 - 1f(x)=4x3+x2−1
B) f(x)=2x−2+x+1f(x) = 2x^{-2} + x + 1f(x)=2x−2+x+1
C) f(x)=5x2+x+2f(x) = 5x^2 + x + 2f(x)=5x2+x+2
D) None of these

Answer: B) f(x)=2x−2+x+1f(x) = 2x^{-2} + x + 1f(x)=2x−2+x+1

31. What is the degree of the polynomial f(x)=4x3+2x2+7f(x) = 4x^3 + 2x^2 + 7f(x)=4x3+2x2+7?
A) 2
B) 3
C) 4
D) None of these

Answer: B) 3

32. If f(x)=sin⁡(x)f(x) = \sin(x)f(x)=sin(x), then f2(x)f^2(x)f2(x) is:

A) (sin⁡(x))2(\sin(x))^2(sin(x))2
B) 2sin⁡(x)2\sin(x)2sin(x)
C) (sin⁡(x))(\sin(x))(sin(x))
D) None of these

Answer: A) (sin⁡(x))2(\sin(x))^2(sin(x))2
33. Which of the following is a constant function?
A) f(x)=5f(x) = 5f(x)=5
B) f(x)=x2f(x) = x^2f(x)=x2
C) f(x)=ln⁡(x)f(x) = \ln(x)f(x)=ln(x)
D) None of these

Answer: A) f(x)=5f(x) = 5f(x)=5

34. For f(x)=x3+x+1f(x) = x^3 + x + 1f(x)=x3+x+1, and g(x)=2xg(x) = 2xg(x)=2x, find (f∘g)(x)(f \circ g)(x)(f∘g)(x):
A) (2x)3+2x+1(2x)^3 + 2x + 1(2x)3+2x+1
B) (2x3)+2x(2x^3) + 2x(2x3)+2x
C) x3+1x^3 + 1x3+1
D) None of these

Answer: A) (2x)3+2x+1(2x)^3 + 2x + 1(2x)3+2x+1

35. A function h(x)h(x)h(x) is decomposed into f(g(x))f(g(x))f(g(x)) where g(x)=x+1g(x) = x + 1g(x)=x+1 and f(x)=x2f(x) =
x^2f(x)=x2. What is h(x)h(x)h(x)?
A) (x+1)2(x + 1)^2(x+1)2
B) x2+1x^2 + 1x2+1
C) x+1x + 1x+1
D) None of these

Answer: A) (x+1)2(x + 1)^2(x+1)2

36. What is the domain of f(x)=1x2+1f(x) = \frac{1}{x^2 + 1}f(x)=x2+11?

A) All real numbers
B) x≠0x \
C) x>0x > 0x>0
D) None of these

Answer: A) All real numbers

37. If f(x)=x3+1f(x) = x^3 + 1f(x)=x3+1 and g(x)=x−2g(x) = x - 2g(x)=x−2, find (f−g)(x)(f - g)(x)(f−g)(x):
A) x3+3x^3 + 3x3+3
B) x3+1−(x−2)x^3 + 1 - (x - 2)x3+1−(x−2)
C) x3−x+3x^3 - x + 3x3−x+3
D) Both B and C

Answer: D) Both B and C

38. Which of the following cannot be decomposed as a composition?

A) f(x)=x2+1f(x) = x^2 + 1f(x)=x2+1
B) f(x)=x2+1f(x) = \sqrt{x^2 + 1}f(x)=x2+1
C) f(x)=x+1f(x) = x + 1f(x)=x+1
D) None of these

Answer: C) f(x)=x+1f(x) = x + 1f(x)=x+1

39. If f(x)=x2+4x+4f(x) = x^2 + 4x + 4f(x)=x2+4x+4, which pair of functions can decompose it?
A) g(x)=x+2,h(x)=x2g(x) = x + 2, h(x) = x^2g(x)=x+2,h(x)=x2
B) g(x)=x2,h(x)=4x+4g(x) = x^2, h(x) = 4x + 4g(x)=x2,h(x)=4x+4
C) g(x)=x+2,h(x)=x+4g(x) = x + 2, h(x) = x + 4g(x)=x+2,h(x)=x+4
D) None of these

Answer: A) g(x)=x+2,h(x)=x2g(x) = x + 2, h(x) = x^2g(x)=x+2,h(x)=x2

40. The function f(x)=2x2−3f(x) = 2x^2 - 3f(x)=2x2−3 is a:

A) Polynomial of degree 2
B) Monomial
C) Polynomial of degree 1
D) None of these

Answer: A) Polynomial of degree 2

Lesson # 08: Graphs of functions:

1. The graph of f(x)f(x)f(x) is the graph of the equation:
A) x=f(y)x = f(y)x=f(y)
B) y=f(x)y = f(x)y=f(x)
C) y=f(y)y = f(y)y=f(y)
D) None of these

Answer: B) y=f(x)y = f(x)y=f(x)

2. The graph of f(x)=x+2f(x) = x + 2f(x)=x+2 is a:

A) Parabola
B) Straight line with slope 1
C) Straight line with slope 2
D) None of these

Answer: B) Straight line with slope 1

3. The absolute value function f(x)=∣x∣f(x) = |x|f(x)=∣x∣:

A) Is a parabola
B) Is piecewise defined
C) Has a discontinuity
D) None of these

Answer: B) Is piecewise defined

4. The absolute value function f(x)=∣x∣f(x) = |x|f(x)=∣x∣ consists of two parts:

A) xxx if x>0x > 0x>0 and −x-x−x if x≤0x \leq 0x≤0
B) xxx if x≥0x \geq 0x≥0 and −x-x−x if x<0x < 0x<0
C) xxx if x≤0x \leq 0x≤0 and −x-x−x if x>0x > 0x>0
D) None of these

Answer: B) xxx if x≥0x \geq 0x≥0 and −x-x−x if x<0x < 0x<0

5. What happens at x=2x = 2x=2 in the function t(x)=x2−4x−2t(x) = \frac{x^2 - 4}{x - 2}t(x)=x−2x2−4?
A) There is a jump discontinuity
B) There is a hole in the graph
C) The function is undefined everywhere
D) None of these

Answer: B) There is a hole in the graph

6. A graph is the graph of a function if and only if:

A) It passes the horizontal line test
B) It passes the vertical line test
C) It passes both horizontal and vertical line tests
D) None of these

Answer: B) It passes the vertical line test

7. The function f(x)=x2f(x) = x^2f(x)=x2:

A) Is symmetric about the x-axis
B) Is symmetric about the y-axis
C) Passes the horizontal line test
D) None of these

Answer: B) Is symmetric about the y-axis

8. The graph of y=f(x)+cy = f(x) + cy=f(x)+c is obtained by:

A) Shifting the graph of y=f(x)y = f(x)y=f(x) up by ccc units
B) Shifting the graph of y=f(x)y = f(x)y=f(x) down by ccc units
C) Shifting the graph of y=f(x)y = f(x)y=f(x) right by ccc units
D) Shifting the graph of y=f(x)y = f(x)y=f(x) left by ccc units

Answer: A) Shifting the graph of y=f(x)y = f(x)y=f(x) up by ccc units

9. The graph of y=f(x−c)y = f(x - c)y=f(x−c) is obtained by:

A) Shifting the graph of y=f(x)y = f(x)y=f(x) up by ccc units
B) Shifting the graph of y=f(x)y = f(x)y=f(x) down by ccc units
C) Shifting the graph of y=f(x)y = f(x)y=f(x) right by ccc units
D) Shifting the graph of y=f(x)y = f(x)y=f(x) left by ccc units

Answer: C) Shifting the graph of y=f(x)y = f(x)y=f(x) right by ccc units

10. The graph of f(x)=x2−4x+5f(x) = x^2 - 4x + 5f(x)=x2−4x+5 can be rewritten as:

A) (x+2)2−1(x + 2)^2 - 1(x+2)2−1
B) (x−2)2+1(x - 2)^2 + 1(x−2)2+1
C) (x+2)2+1(x + 2)^2 + 1(x+2)2+1
D) (x−2)2−1(x - 2)^2 - 1(x−2)2−1

Answer: B) (x−2)2+1(x - 2)^2 + 1(x−2)2+1

11. If f(x)=x2f(x) = x^2f(x)=x2, the graph of y=−f(x)y = -f(x)y=−f(x) is:

A) A reflection about the x-axis
B) A reflection about the y-axis
C) Stretched vertically
D) Compressed vertically
Answer: A) A reflection about the x-axis

12. The graph of y=f(−x)y = f(-x)y=f(−x) is:

A) A reflection of y=f(x)y = f(x)y=f(x) about the y-axis
B) A reflection of y=f(x)y = f(x)y=f(x) about the x-axis
C) Stretched vertically
D) Compressed vertically

Answer: A) A reflection of y=f(x)y = f(x)y=f(x) about the y-axis

13. The graph of y=2f(x)y = 2f(x)y=2f(x) is obtained by:

A) Stretching the graph of y=f(x)y = f(x)y=f(x) horizontally by a factor of 2
B) Compressing the graph of y=f(x)y = f(x)y=f(x) horizontally by a factor of 2
C) Stretching the graph of y=f(x)y = f(x)y=f(x) vertically by a factor of 2
D) Compressing the graph of y=f(x)y = f(x)y=f(x) vertically by a factor of 2

Answer: C) Stretching the graph of y=f(x)y = f(x)y=f(x) vertically by a factor of 2

14. If f(x)=x3f(x) = x^3f(x)=x3, then y=f(−x)y = f(-x)y=f(−x) is:

A) A reflection of y=x3y = x^3y=x3 about the y-axis
B) A reflection of y=x3y = x^3y=x3 about the x-axis
C) Stretched vertically
D) Compressed vertically

Answer: A) A reflection of y=x3y = x^3y=x3 about the y-axis

15. The function y=x2+4x+5y = x^2 + 4x + 5y=x2+4x+5 is shifted how to obtain y=x2+4x+3y = x^2 + 4x + 3y=x2+4x+3?
A) Up by 2 units
B) Down by 2 units
C) Left by 2 units
D) Right by 2 units

Answer: B) Down by 2 units

16. The vertical line test fails if:

A) A vertical line intersects the graph more than once
B) A horizontal line intersects the graph more than once
C) The graph has no slope
D) None of these

Answer: A) A vertical line intersects the graph more than once

17. Which of the following is NOT a graph of a function?

A) y=x2y = x^2y=x2
B) x2+y2=25x^2 + y^2 = 25x2+y2=25
C) y=∣x∣y = |x|y=∣x∣
D) y=x+2y = x + 2y=x+2

Answer: B) x2+y2=25x^2 + y^2 = 25x2+y2=25

18. The horizontal line test determines:
A) If a function is one-to-one
B) If a graph is symmetric
C) If a graph represents a function
D) None of these

Answer: A) If a function is one-to-one

19. Which of the following represents a reflection about the x-axis?

A) y=−f(x)y = -f(x)y=−f(x)
B) y=f(−x)y = f(-x)y=f(−x)
C) y=f(x)+cy = f(x) + cy=f(x)+c
D) y=f(x−c)y = f(x - c)y=f(x−c)

Answer: A) y=−f(x)y = -f(x)y=−f(x)

20. Which of the following represents a reflection about the y-axis?

A) y=−f(x)y = -f(x)y=−f(x)
B) y=f(−x)y = f(-x)y=f(−x)
C) y=f(x)+cy = f(x) + cy=f(x)+c
D) y=f(x−c)y = f(x - c)y=f(x−c)

Answer: B) y=f(−x)y = f(-x)y=f(−x)

21. The graph of y=f(x)+cy = f(x) + cy=f(x)+c:

A) Shifts upward
B) Shifts downward
C) Reflects about the x-axis
D) Reflects about the y-axis

Answer: A) Shifts upward

22. The graph of y=f(x−c)y = f(x - c)y=f(x−c):

A) Shifts upward
B) Shifts downward
C) Shifts to the right
D) Shifts to the left

Answer: C) Shifts to the right

23. The graph of y=2f(x)y = 2f(x)y=2f(x):

A) Is stretched vertically by a factor of 2
B) Is compressed vertically by a factor of 2
C) Is shifted up by 2 units
D) Is shifted down by 2 units

Answer: A) Is stretched vertically by a factor of 2

24. The graph of y=12f(x)y = \frac{1}{2}f(x)y=21f(x):

A) Is stretched vertically by a factor of 2
B) Is compressed vertically by a factor of 12\frac{1}{2}21
C) Is shifted up by 12\frac{1}{2}21 units
D) Is shifted down by 12\frac{1}{2}21 units

Answer: B) Is compressed vertically by a factor of 12\frac{1}{2}21

25. Which equation is a parabola?

A) y=x2−4x+5y = x^2 - 4x + 5y=x2−4x+5
B) x2+y2=25x^2 + y^2 = 25x2+y2=25
C) y=xy = \sqrt{x}y=x
D) y=∣x∣y = |x|y=∣x∣

Answer: A) y=x2−4x+5y = x^2 - 4x + 5y=x2−4x+5

26. The function y=f(x+c)y = f(x + c)y=f(x+c):

A) Shifts to the left
B) Shifts to the right
C) Stretches vertically
D) Reflects about the x-axis

Answer: A) Shifts to the left

27. A hole in the graph of a function occurs when:

A) The function is discontinuous at a point
B) The function is undefined at a point
C) The graph intersects the x-axis
D) The graph intersects the y-axis

Answer: B) The function is undefined at a point

28. Which of the following graphs is NOT a function?

A) y=x3y = x^3y=x3
B) x2+y2=16x^2 + y^2 = 16x2+y2=16
C) y=xy = \sqrt{x}y=x
D) y=∣x∣y = |x|y=∣x∣

Answer: B) x2+y2=16x^2 + y^2 = 16x2+y2=16

29. For y=x2y = x^2y=x2, the domain is:

A) [0,∞)[0, \infty)[0,∞)
B) (−∞,∞)(-\infty, \infty)(−∞,∞)
C) (−∞,0](-\infty, 0](−∞,0]
D) None of these

Answer: B) (−∞,∞)(-\infty, \infty)(−∞,∞)

30. For y=xy = \sqrt{x}y=x, the range is:

A) [0,∞)[0, \infty)[0,∞)
B) (−∞,∞)(-\infty, \infty)(−∞,∞)
C) (−∞,0](-\infty, 0](−∞,0]
D) None of these
Answer: A) [0,∞)[0, \infty)[0,∞)

31. The function y=x2+2x+1y = x^2 + 2x + 1y=x2+2x+1 is equivalent to:

A) (x+1)2(x + 1)^2(x+1)2
B) (x−1)2(x - 1)^2(x−1)2
C) (x+2)2−1(x + 2)^2 - 1(x+2)2−1
D) (x−2)2+1(x - 2)^2 + 1(x−2)2+1

Answer: A) (x+1)2(x + 1)^2(x+1)2

32. A vertical line test checks:

A) If a graph is symmetric
B) If a graph is a function
C) If a function is one-to-one
D) None of these

Answer: B) If a graph is a function

33. The graph of y=x3y = x^3y=x3 is symmetric about:

A) The x-axis
B) The y-axis
C) The origin
D) None of these

Answer: C) The origin

34. The graph of y=−x2y = -x^2y=−x2:

A) Opens upward
B) Opens downward
C) Is a straight line
D) Is a reflection about the y-axis

Answer: B) Opens downward

35. The graph of y=∣x∣y = |x|y=∣x∣:

A) Is a parabola
B) Has a V-shape
C) Is a straight line
D) None of these

Answer: B) Has a V-shape

36. The reflection of y=f(x)y = f(x)y=f(x) about the y-axis is given by:
A) y=−f(x)y = -f(x)y=−f(x)
B) y=f(−x)y = f(-x)y=f(−x)
C) y=−f(−x)y = -f(-x)y=−f(−x)
D) None of these

Answer: B) y=f(−x)y = f(-x)y=f(−x)

37. The graph of y=(x−2)2y = (x - 2)^2y=(x−2)2:
A) Is shifted 2 units left
B) Is shifted 2 units right
C) Is shifted 2 units up
D) Is shifted 2 units down

Answer: B) Is shifted 2 units right

38. The graph of y=1xy = \frac{1}{x}y=x1:

A) Has a hole
B) Has a vertical asymptote at x=0x = 0x=0
C) Has a horizontal asymptote at y=0y = 0y=0
D) Both B and C

Answer: D) Both B and C

39. For the function y=x2y = x^2y=x2, the range is:

A) [0,∞)[0, \infty)[0,∞)
B) (−∞,∞)(-\infty, \infty)(−∞,∞)
C) (−∞,0](-\infty, 0](−∞,0]
D) None of these

Answer: A) [0,∞)[0, \infty)[0,∞)

40. The horizontal line test is used to check if a function is:

A) Even
B) Odd
C) One-to-one
D) Continuous

Answer: C) One-to-one

LECTURE # 9: LIMITS
1. Calculus that arises from the tangent problem is called:

A) Integral Calculus
B) Differential Calculus
C) Analytical Geometry
D) Coordinate Geometry

Answer: B) Differential Calculus

2. Calculus that arises from the area problem is called:

A) Integral Calculus
B) Differential Calculus
C) Analytical Geometry
D) Trigonometry

Answer: A) Integral Calculus

3. The precise definition of tangent depends on:

A) Geometry
B) Derivatives
C) Limits
D) Algebra

Answer: C) Limits

4. A secant line is defined as a line that passes through:

A) One point on the curve

B) Two points on the curve
C) The origin
D) None of these

Answer: B) Two points on the curve

5. The tangent line is defined as:

A) A line perpendicular to the curve

B) A limiting position of the secant line
C) A line parallel to the curve
D) None of these

Answer: B) A limiting position of the secant line

6. The process of finding an area under a curve is closely related to:

A) Derivatives
B) Tangents
C) Limits
D) Integration

Answer: D) Integration

7. What is used to approximate areas under a curve?

A) Circles
B) Rectangles
C) Triangles
D) Trapezoids

Answer: B) Rectangles

8. The better approximation of area under a curve is obtained by:

A) Increasing the number of rectangles

B) Decreasing the number of rectangles
C) Using larger rectangles
D) None of these

Answer: A) Increasing the number of rectangles

9. A limit is a way to study the behavior of a function's y-values as:

A) y approaches infinity
B) x approaches a specific value
C) y decreases without bound
D) None of these

Answer: B) x approaches a specific value

10. The right-hand limit is represented as:

A) lim⁡x→0−f(x)\lim_{x \to 0^-} f(x)limx→0−f(x)

B) lim⁡x→0+f(x)\lim_{x \to 0^+} f(x)limx→0+f(x)
C) lim⁡x→∞f(x)\lim_{x \to \infty} f(x)limx→∞f(x)
D) lim⁡x→0f(x)\lim_{x \to 0} f(x)limx→0f(x)

Answer: B) lim⁡x→0+f(x)\lim_{x \to 0^+} f(x)limx→0+f(x)

11. The left-hand limit is represented as:

A) lim⁡x→0−f(x)\lim_{x \to 0^-} f(x)limx→0−f(x)

B) lim⁡x→0+f(x)\lim_{x \to 0^+} f(x)limx→0+f(x)
C) lim⁡x→∞f(x)\lim_{x \to \infty} f(x)limx→∞f(x)
D) lim⁡x→0f(x)\lim_{x \to 0} f(x)limx→0f(x)

Answer: A) lim⁡x→0−f(x)\lim_{x \to 0^-} f(x)limx→0−f(x)

12. When both the left-hand and right-hand limits are equal, we say:

A) The limit does not exist

B) The limit exists
C) The function is undefined
D) None of these

Answer: B) The limit exists

13. What is the behavior of sin⁡(x)\sin(x)sin(x) as xxx approaches 0?

A) sin⁡(x)\sin(x)sin(x) approaches −1-1−1

B) sin⁡(x)\sin(x)sin(x) approaches 000
C) sin⁡(x)\sin(x)sin(x) approaches 111
D) None of these

Answer: C) sin⁡(x)\sin(x)sin(x) approaches 111

14. The limit of sin⁡(x)\sin(x)sin(x) as xxx approaches π\piπ does not exist because:

A) sin⁡(x)\sin(x)sin(x) oscillates
B) sin⁡(x)\sin(x)sin(x) is undefined
C) sin⁡(x)\sin(x)sin(x) approaches infinity
D) None of these

Answer: A) sin⁡(x)\sin(x)sin(x) oscillates

15. A limit fails to exist due to oscillations because:

A) The function diverges

B) The function settles on a single value
C) The function does not approach a single value
D) None of these

Answer: C) The function does not approach a single value

16. The usual culprits for a limit not existing are:

A) Oscillations and boundedness

B) Oscillations and unboundedness
C) Symmetry and periodicity
D) None of these

Answer: B) Oscillations and unboundedness

17. If a function increases without bound as x→x0x \to x_0x→x0, the limit is classified as:

A) Zero
B) Infinity
C) Negative infinity
D) Undefined

Answer: B) Infinity

18. When a function decreases without bound, the limit is classified as:

A) Zero
B) Infinity
C) Negative infinity
D) Undefined

Answer: C) Negative infinity

19. For a limit to exist, the left-hand and right-hand limits must:

A) Differ by 1
B) Be equal
C) Be undefined
D) Oscillate
Answer: B) Be equal

20. A limit that does not exist due to unboundedness is classified as:

A) Oscillating
B) Infinity or negative infinity
C) Continuous
D) None of these

Answer: B) Infinity or negative infinity

21. A graph oscillating between 1 and -1 is an example of:

A) A limit that exists

B) A limit that does not exist due to oscillations
C) A limit that diverges
D) None of these

Answer: B) A limit that does not exist due to oscillations

22. For the function f(x)=1/xf(x) = 1/xf(x)=1/x, the limit as x→0+x \to 0^+x→0+:

A) −∞-\infty−∞
B) +∞+\infty+∞
C) 000
D) Undefined

Answer: B) +∞+\infty+∞

23. For the function f(x)=1/xf(x) = 1/xf(x)=1/x, the limit as x→0−x \to 0^-x→0−:

A) −∞-\infty−∞
B) +∞+\infty+∞
C) 000
D) Undefined

Answer: A) −∞-\infty−∞

24. If a limit exists at x→x0x \to x_0x→x0, the graph of the function will:

A) Always have a discontinuity at x0x_0x0

B) Be undefined at x0x_0x0
C) Approach the same value from both sides of x0x_0x0
D) Oscillate at x0x_0x0

Answer: C) Approach the same value from both sides of x0x_0x0

25. A limit at x→∞x \to \inftyx→∞ describes the behavior of the function as:

A) xxx approaches 0
B) xxx becomes very large
C) xxx approaches −∞-\infty−∞
D) None of these

Answer: B) xxx becomes very large

26. When a function oscillates indefinitely as x→∞x \to \inftyx→∞:

A) The limit exists

B) The limit does not exist
C) The function is undefined
D) None of these

Answer: B) The limit does not exist

27. A function oscillating with decreasing amplitude as x→∞x \to \inftyx→∞:

A) Settles at a specific value

B) Does not have a limit
C) Approaches a limiting value
D) Is undefined

Answer: C) Approaches a limiting value

28. The limit of a function as x→∞x \to \inftyx→∞ depends on the:

A) Behavior of the function at x=0x = 0x=0

B) Trend of the function as xxx increases
C) Oscillations at x=0x = 0x=0
D) None of these

Answer: B) Trend of the function as xxx increases

29. For the function f(x)=1/xf(x) = 1/xf(x)=1/x, the limit as x→∞x \to \inftyx→∞ is:

A) 0
B) 1
C) ∞\infty∞
D) Undefined

Answer: A) 0

30. For the function f(x)=1/xf(x) = 1/xf(x)=1/x, the limit as x→−∞x \to -\inftyx→−∞ is:

A) 0
B) 1
C) −∞-\infty−∞
D) Undefined

Answer: A) 0
31. The limit of f(x)f(x)f(x) as x→∞x \to \inftyx→∞ is written as:

A) lim⁡x→0f(x)\lim_{x \to 0} f(x)limx→0f(x)

B) lim⁡x→∞f(x)\lim_{x \to \infty} f(x)limx→∞f(x)
C) lim⁡x→−∞f(x)\lim_{x \to -\infty} f(x)limx→−∞f(x)
D) lim⁡x→x0f(x)\lim_{x \to x_0} f(x)limx→x0f(x)

Answer: B) lim⁡x→∞f(x)\lim_{x \to \infty} f(x)limx→∞f(x)

32. If lim⁡x→∞f(x)=4\lim_{x \to \infty} f(x) = 4limx→∞f(x)=4, it means:

A) f(x)f(x)f(x) becomes equal to 4 for large xxx

B) f(x)f(x)f(x) approaches 4 as xxx becomes large
C) f(x)f(x)f(x) oscillates around 4
D) None of these

Answer: B) f(x)f(x)f(x) approaches 4 as xxx becomes large

33. A limit does not exist at x→x0x \to x_0x→x0 if:

A) Left-hand and right-hand limits are equal

B) Left-hand and right-hand limits are not equal
C) The function is continuous
D) The function has no oscillations

Answer: B) Left-hand and right-hand limits are not equal

34. The notation lim⁡x→x0+f(x)\lim_{x \to x_0^+} f(x)limx→x0+f(x) means:

A) xxx approaches x0x_0x0 from the left

B) xxx approaches x0x_0x0 from the right
C) x0x_0x0 is undefined
D) x0x_0x0 is a finite value

Answer: B) xxx approaches x0x_0x0 from the right

35. If a function is unbounded as x→∞x \to \inftyx→∞, the limit is:

A) Infinity or negative infinity

B) Zero
C) Undefined
D) A finite number

Answer: A) Infinity or negative infinity

36. For the graph of y=f(x)y = f(x)y=f(x) oscillating between 1 and -1, the limit at infinity is:

A) 1
B) -1
C) Undefined
D) None of these
Answer: C) Undefined

37. A function's limit at x→x0x \to x_0x→x0 does not exist due to:

A) Undefined values only

B) Oscillations or unboundedness
C) Continuity of the function
D) None of these

Answer: B) Oscillations or unboundedness

38. A limit can exist at x→∞x \to \inftyx→∞ even if:

A) The function oscillates but stabilizes

B) The function increases indefinitely
C) The function decreases indefinitely
D) None of these

Answer: A) The function oscillates but stabilizes

39. When the limit of a function is −∞-\infty−∞ as x→x0x \to x_0x→x0, it means:

A) The function is increasing without bound

B) The function is decreasing without bound
C) The function is constant
D) The function oscillates

Answer: B) The function is decreasing without bound

40. If lim⁡x→∞f(x)=−2\lim_{x \to \infty} f(x) = -2limx→∞f(x)=−2, the function will eventually approach:

A) -2
B) Infinity
C) 0
D) Undefined

Answer: A) -2

Lecture # 10: LIMITS AND COMPUTATIONAL TECHNIQUES:

1. Which of the following functions is a polynomial?

A) f(x)=2x3+4x2−5x+1f(x) = 2x^3 + 4x^2 - 5x + 1f(x)=2x3+4x2−5x+1

B) f(x)=2x3x+1f(x) = \frac{2x^3}{x+1}f(x)=x+12x3
C) f(x)=sin⁡(x)f(x) = \sin(x)f(x)=sin(x)
D) f(x)=1xf(x) = \frac{1}{x}f(x)=x1

Answer: A) f(x)=2x3+4x2−5x+1f(x) = 2x^3 + 4x^2 - 5x + 1f(x)=2x3+4x2−5x+1

2. What is the limit of a constant function f(x)=kf(x) = kf(x)=k as x→ax \to ax→a?
A) kkk
B) 000
C) Undefined
D) aaa

Answer: A) kkk

3. The limit of g(x)=xg(x) = xg(x)=x as x→ax \to ax→a is:

A) 000
B) aaa
C) a2a^2a2
D) Undefined

Answer: B) aaa

4. According to Theorem 2.5.1, what is the limit of f(x)+g(x)f(x) + g(x)f(x)+g(x) as x→ax \to ax→a?

A) lim⁡f(x)+lim⁡g(x)\lim f(x) + \lim g(x)limf(x)+limg(x)

B) lim⁡f(x)−lim⁡g(x)\lim f(x) - \lim g(x)limf(x)−limg(x)
C) lim⁡f(x)×lim⁡g(x)\lim f(x) \times \lim g(x)limf(x)×limg(x)
D) Undefined

Answer: A) lim⁡f(x)+lim⁡g(x)\lim f(x) + \lim g(x)limf(x)+limg(x)

5. If the limits of f(x)f(x)f(x) and g(x)g(x)g(x) exist as x→ax \to ax→a, what is the limit of f(x)×g(x)f(x) \times
g(x)f(x)×g(x)?

A) lim⁡f(x)×lim⁡g(x)\lim f(x) \times \lim g(x)limf(x)×limg(x)

B) lim⁡f(x)−lim⁡g(x)\lim f(x) - \lim g(x)limf(x)−limg(x)
C) lim⁡f(x)+lim⁡g(x)\lim f(x) + \lim g(x)limf(x)+limg(x)
D) Undefined

Answer: A) lim⁡f(x)×lim⁡g(x)\lim f(x) \times \lim g(x)limf(x)×limg(x)

6. What is the limit of the constant factor kkk multiplied by a function g(x)g(x)g(x)?

A) k×lim⁡g(x)k \times \lim g(x)k×limg(x)

B) k+lim⁡g(x)k + \lim g(x)k+limg(x)
C) k÷lim⁡g(x)k \div \lim g(x)k÷limg(x)
D) Undefined

Answer: A) k×lim⁡g(x)k \times \lim g(x)k×limg(x)

7. What is the limit of xnx^nxn as x→∞x \to \inftyx→∞ when nnn is a positive integer?

A) 0
B) +∞+\infty+∞
C) Undefined
D) −∞-\infty−∞
Answer: B) +∞+\infty+∞

8. What is the limit of xnx^nxn as x→−∞x \to -\inftyx→−∞ when nnn is an even integer?

A) 000
B) +∞+\infty+∞
C) −∞-\infty−∞
D) Undefined

Answer: B) +∞+\infty+∞

9. The limit of a rational function as x→ax \to ax→a can be evaluated using:

A) Polynomial division
B) Algebraic manipulation
C) Theorem 2.5.1
D) All of the above

Answer: D) All of the above

10. What is the limit of 5x3+4x−3\frac{5x^3 + 4}{x - 3}x−35x3+4 as x→2x \to 2x→2?

A) −44-44−44
B) 444444
C) 101010
D) Undefined

Answer: A) −44-44−44

11. If both the numerator and denominator of a rational function approach 0 as x→ax \to ax→a, the limit is:

A) ∞\infty∞
B) Undefined
C) 0
D) Can be determined by simplifying the expression

Answer: D) Can be determined by simplifying the expression

12. The limit of x2−4x−2\frac{x^2 - 4}{x - 2}x−2x2−4 as x→2x \to 2x→2 is:

A) 0
B) 2
C) 4
D) 8

Answer: B) 2

13. What happens to the limit of 2x−3x−4\frac{2x - 3}{x - 4}x−42x−3 as x→4x \to 4x→4?
A) The limit is +∞+\infty+∞
B) The limit is −∞-\infty−∞
C) The limit is 0
D) The limit does not exist

Answer: A) The limit is +∞+\infty+∞

14. If the denominator of a rational function tends to 0 while the numerator does not as x→ax \to ax→a, the
limit is:

A) 000
B) +∞+\infty+∞ or −∞-\infty−∞ depending on the sign
C) Undefined
D) The function is continuous at that point

Answer: B) +∞+\infty+∞ or −∞-\infty−∞ depending on the sign

15. The limit of x2−xx\frac{x^2 - x}{x}xx2−x as x→∞x \to \inftyx→∞ is:

A) 1
B) 0
C) ∞\infty∞
D) Undefined

Answer: A) 1

16. What is the behavior of 1xn\frac{1}{x^n}xn1 as x→∞x \to \inftyx→∞ for n>0n > 0n>0?

A) The limit is 0
B) The limit is ∞\infty∞
C) The limit is −∞-\infty−∞
D) The function oscillates

Answer: A) The limit is 0

17. What is the limit of x2+4x+3x^2 + 4x + 3x2+4x+3 as x→5x \to 5x→5?

A) 33
B) 53
C) 43
D) 55

Answer: A) 33

18. What happens when you divide a polynomial by another polynomial as x→∞x \to \inftyx→∞?

A) The highest degree term dominates the limit

B) The degree of the denominator determines the limit
C) The degree of the numerator determines the limit
D) The result is always 0
Answer: A) The highest degree term dominates the limit

19. What happens to the limit of 2x3+xx3−3x\frac{2x^3 + x}{x^3 - 3x}x3−3x2x3+x as x→∞x \to \inftyx→∞?

A) 2
B) 0
C) ∞\infty∞
D) 1

Answer: A) 2

20. What is the limit of x2+3x+2x2−4\frac{x^2 + 3x + 2}{x^2 - 4}x2−4x2+3x+2 as x→2x \to 2x→2?

A) 70\frac{7}{0}07
B) Undefined
C) 1
D) 74\frac{7}{4}47

Answer: B) Undefined

21. The limit of x2+4x+4x+2\frac{x^2 + 4x + 4}{x + 2}x+2x2+4x+4 as x→−2x \to -2x→−2 is:

A) 0
B) 4
C) 1
D) Undefined

Answer: B) 4

22. The limit of 2x2−5xx2−4x\frac{2x^2 - 5x}{x^2 - 4x}x2−4x2x2−5x as x→4x \to 4x→4 is:

A) ∞\infty∞
B) 0
C) −∞-\infty−∞
D) 1

Answer: D) 1

23. What is the limit of 5x+4x2+x\frac{5x + 4}{x^2 + x}x2+x5x+4 as x→∞x \to \inftyx→∞?

A) 0
B) ∞\infty∞
C) 1
D) 4

Answer: A) 0

24. What is the limit of a rational function as x→∞x \to \inftyx→∞ if the degree of the numerator is greater
than the degree of the denominator?
A) ∞\infty∞
B) 0
C) 1
D) Undefined

Answer: A) ∞\infty∞

25. If f(x)=x2−1x−1f(x) = \frac{x^2 - 1}{x - 1}f(x)=x−1x2−1, what is the limit as x→1x \to 1x→1?

A) 2
B) 0
C) 1
D) Undefined

Answer: A) 2

26. The limit of 2x3−3xx2−2x\frac{2x^3 - 3x}{x^2 - 2x}x2−2x2x3−3x as x→2x \to 2x→2 is:

A) Undefined
B) 0
C) 1
D) 3

Answer: A) Undefined

27. What is the limit of 1x\frac{1}{x}x1 as x→0+x \to 0^+x→0+?

A) +∞+\infty+∞
B) −∞-\infty−∞
C) 0
D) Undefined

Answer: A) +∞+\infty+∞

28. What happens to the limit of xxx as x→∞x \to \inftyx→∞?

A) It approaches 1
B) It approaches 0
C) It approaches ∞\infty∞
D) It approaches −∞-\infty−∞

Answer: C) It approaches ∞\infty∞

29. What is the limit of x2x2+1\frac{x^2}{x^2 + 1}x2+1x2 as x→∞x \to \inftyx→∞?

A) 1
B) 0
C) ∞\infty∞
D) Undefined

Answer: A) 1
30. The limit of 1x2\frac{1}{x^2}x21 as x→∞x \to \inftyx→∞ is:

A) 1
B) 0
C) ∞\infty∞
D) Undefined

Answer: B) 0

31. The limit of x+3x−2\frac{x + 3}{x - 2}x−2x+3 as x→2x \to 2x→2 is:

A) ∞\infty∞
B) Undefined
C) 5
D) 1

Answer: A) ∞\infty∞

32. What is the limit of x3+2xx2+1\frac{x^3 + 2x}{x^2 + 1}x2+1x3+2x as x→∞x \to \inftyx→∞?

A) 2
B) 0
C) ∞\infty∞
D) 1

Answer: C) ∞\infty∞

33. What is the limit of 1xn\frac{1}{x^n}xn1 as x→−∞x \to -\inftyx→−∞ for any nnn?

A) 0
B) ∞\infty∞
C) −∞-\infty−∞
D) Undefined

Answer: A) 0

34. What is the limit of x+1x2\frac{x + 1}{x^2}x2x+1 as x→∞x \to \inftyx→∞?

A) 1
B) 0
C) ∞\infty∞
D) Undefined

Answer: B) 0

35. What happens to x2+4x+4x2−4\frac{x^2 + 4x + 4}{x^2 - 4}x2−4x2+4x+4 as x→2x \to 2x→2?

A) Undefined
B) 1
C) 4
D) 0
Answer: A) Undefined

36. What is the limit of x3x2\frac{x^3}{x^2}x2x3 as x→∞x \to \inftyx→∞?

A) 1
B) 0
C) ∞\infty∞
D) 3

Answer: A) 1

37. If lim⁡x→01x\lim_{x \to 0} \frac{1}{x}limx→0x1, the limit is:

A) 0
B) ∞\infty∞
C) Undefined
D) 1

Answer: C) Undefined

38. What happens to the limit of xx2+1\frac{x}{x^2 + 1}x2+1x as x→∞x \to \inftyx→∞?

A) 0
B) 1
C) ∞\infty∞
D) Undefined

Answer: A) 0

39. What happens to 3x2x3+4x2\frac{3x^2}{x^3 + 4x^2}x3+4x23x2 as x→∞x \to \inftyx→∞?

A) 0
B) ∞\infty∞
C) 1
D) Undefined

Answer: A) 0

40. The limit of x2+1x2−1\frac{x^2 + 1}{x^2 - 1}x2−1x2+1 as x→∞x \to \inftyx→∞ is:

A) 1
B) 0
C) Undefined
D) 2

Answer: A) 1