MATH 3339 1 ACTUAL EXAM PAPER 2026
QUESTIONS WITH ANSWERS GRADED A+
◍ The symbols and have different meanings. Can you tell the difference?
Ineach case, determine if the given statement is true or false.(a) a {a, b,
c}(b) a {a, b, c}(c) {a} {a, b, c}(d) {a} {a, b, c}(e) Ø {a, b, c}(f)
Ø {a, b, c}(g) {d, e} {d, e, {d}, {e}, {d, e}}(h) {d, e} {d, e, {d},
{e}, {d, e}}(i) {d, {d}} {d, e, {d}, {e}, {d, e}}(j) {d, {d}} {d, e, {d},
{e}, {d, e}}.
Answer: (a) True(b) False(c) False(d) True(e) False(f) True(g) True(h)
True(i) False(j) True
◍ y-intercepts.
Answer: Y intercept: (0. #)let x = 0, then solve for y
◍ Perpendicular lines & slope.
Answer: Have opposite reciprocal slopes (flipped and sign changed) Ex.
3/2--> -2/3
◍ Express each of the following sets using set-builder notation.a) The set of
integers greater than 100.b) The set of real numbers belonging to the closed
interval [1, 4].c) The set of rational numbers whose square is less than 7.d)
The set of odd integers.e) {. . . , -7, -4, -1, 2, 5, 8, . . . }.
Answer: a) {x Z | x > 100}b) {x R | 1 ≤ x ≤ 4}c) {x Q | x² ≤ 7}d) {x
Z | x = 2k + 1 where k Z}e) {x Z | x = 3k − 1 where k Z}
◍ Consider the universal set U = R, and let Bx = [x, ∞) for each x R+.(a)
Bx^c(b) Union of Bx^c(c) Intersection of Bx^c(d) Complement of the union
of Bx^c(e) Complement of the intersection of Bx^c.
Answer: (a) (−∞, x)(b) (−∞,∞)(c) (−∞, 0](d) ((0,∞))^c = (−∞, 0](e) (Ø)^c =
(−∞,∞)
◍ Prove the given statement:If x B and A B, then x A.
, Answer: Assume x B and A B, and assume for the sake of
contradiction that x A. Then, x B, since A B. Therefore, x B and x
B. This is a contradiction. Therefore, x A.The proof is complete.
◍ Prove that if A B, then A ∩ C B ∩ C.
Answer: Proof: Assume A B. We want to show that A ∩ C B ∩ C.Let x
A ∩ C. Then, x A and x C. Since x A and A B, it follows that x
B. Therefore, x B and x C, which means x B ∩ C. This completes
the proof.
◍ A = {An | n N} where An = {. . . , −3n, −2n, −n, 0, n, 2n, 3n, . . .}(a)
Union of A over all natural numbers(b) Intersection of A over all natural
numbers.
Answer: (a) Z(b) {0}
◍ Q3.
Answer: Middle number between median and largest number in Data set
◍ Quadratic equation.
Answer: *Highest power is 2a, b, c are numbers
◍ Add/Subtract Polynomials.
Answer: Combine Like Terms**Like Terms have the same letters and
exponents
◍ Prove the given statement:If A B and B C, then A C.
Answer: Assume A B and B C. We want to show A C. Assume x
A. Then, x B, since A B. Therefore, x C, since B C. This proves
that if x A, then x C. That is, A C.
◍ What would you do to simplify (x+y)^2.
Answer: Write out the binomial factors and FOIL
◍ Prove that if A B, then A − B = Ø.
Answer: Proof: Assume A B, and assume for the sake of contradiction
that A−B 6= Ø.Then there exists an element x A − B. Therefore, x A
and x / B. Thiscontradicts the assumption that A B. Therefore A − B =
, Ø.The proof is complete
◍ Exponential Growth or Decay Formula for Word Problems.
Answer: % increasing or decreasingb is the % KEPTAdd or Subtract from
100 and change to a decimal
◍ Exponent rule for dividing monomials.
Answer: Subtract (big-small) and the answer goes where the bigger # is.
◍ A = {An | n N} where An = [1/n, 3 −1/n](a) Union of A over all natural
numbers(b) Intersection of A over all natural numbers.
Answer: (a) (0,3)(b) [1,2]
◍ Form of Exponential Equation.
Answer: a= starting number b= rate x= timey=ending amount
◍ When the problem says "Consecutive integers"....
Answer: Use X, X+1, X+2, X+3..
◍ "Golden Rule" when solving Inequalities.
Answer: If you multiply or divide both sides on inequality by a NEGATIVE
number, FLIP the inequality sign around!!
◍ In each case, find the power set P(S) of the given set S(a) S = {a}(b) S = {b,
c}(c) S = {1, 2, 3}(d) S = {4, {5, 6}}(e) S = {Ø, R}.
Answer: (a) (S) = {Ø, {a}}(b) P(S) = {Ø, {b}, {c}, {b, c}}(c) P(S) = {Ø,
{1}, {2}, {3}, {1, 2}, {1, 3}, {2, 3}, {1, 2, 3}}(d) P(S) = {Ø, {4}, {{5, 6}},
{4, {5, 6}}}(e) P(S) = {Ø, {Ø}, {R}, {Ø, R}}
◍ What is the slope-intercept form of a line?.
Answer: y=mx+b
◍ Line of best fit.
Answer: Also called "regression line", "linear model", "prediction equation",
or "trend line".Use calculator to findSTAT, edit, input L1 and L2, Stat, ->
CALC,#4 LinReg
◍ How do you tell if a parabola opens up/down?.
Answer: If "a" is positive: opens up. If "a" is negative: opens down.
QUESTIONS WITH ANSWERS GRADED A+
◍ The symbols and have different meanings. Can you tell the difference?
Ineach case, determine if the given statement is true or false.(a) a {a, b,
c}(b) a {a, b, c}(c) {a} {a, b, c}(d) {a} {a, b, c}(e) Ø {a, b, c}(f)
Ø {a, b, c}(g) {d, e} {d, e, {d}, {e}, {d, e}}(h) {d, e} {d, e, {d},
{e}, {d, e}}(i) {d, {d}} {d, e, {d}, {e}, {d, e}}(j) {d, {d}} {d, e, {d},
{e}, {d, e}}.
Answer: (a) True(b) False(c) False(d) True(e) False(f) True(g) True(h)
True(i) False(j) True
◍ y-intercepts.
Answer: Y intercept: (0. #)let x = 0, then solve for y
◍ Perpendicular lines & slope.
Answer: Have opposite reciprocal slopes (flipped and sign changed) Ex.
3/2--> -2/3
◍ Express each of the following sets using set-builder notation.a) The set of
integers greater than 100.b) The set of real numbers belonging to the closed
interval [1, 4].c) The set of rational numbers whose square is less than 7.d)
The set of odd integers.e) {. . . , -7, -4, -1, 2, 5, 8, . . . }.
Answer: a) {x Z | x > 100}b) {x R | 1 ≤ x ≤ 4}c) {x Q | x² ≤ 7}d) {x
Z | x = 2k + 1 where k Z}e) {x Z | x = 3k − 1 where k Z}
◍ Consider the universal set U = R, and let Bx = [x, ∞) for each x R+.(a)
Bx^c(b) Union of Bx^c(c) Intersection of Bx^c(d) Complement of the union
of Bx^c(e) Complement of the intersection of Bx^c.
Answer: (a) (−∞, x)(b) (−∞,∞)(c) (−∞, 0](d) ((0,∞))^c = (−∞, 0](e) (Ø)^c =
(−∞,∞)
◍ Prove the given statement:If x B and A B, then x A.
, Answer: Assume x B and A B, and assume for the sake of
contradiction that x A. Then, x B, since A B. Therefore, x B and x
B. This is a contradiction. Therefore, x A.The proof is complete.
◍ Prove that if A B, then A ∩ C B ∩ C.
Answer: Proof: Assume A B. We want to show that A ∩ C B ∩ C.Let x
A ∩ C. Then, x A and x C. Since x A and A B, it follows that x
B. Therefore, x B and x C, which means x B ∩ C. This completes
the proof.
◍ A = {An | n N} where An = {. . . , −3n, −2n, −n, 0, n, 2n, 3n, . . .}(a)
Union of A over all natural numbers(b) Intersection of A over all natural
numbers.
Answer: (a) Z(b) {0}
◍ Q3.
Answer: Middle number between median and largest number in Data set
◍ Quadratic equation.
Answer: *Highest power is 2a, b, c are numbers
◍ Add/Subtract Polynomials.
Answer: Combine Like Terms**Like Terms have the same letters and
exponents
◍ Prove the given statement:If A B and B C, then A C.
Answer: Assume A B and B C. We want to show A C. Assume x
A. Then, x B, since A B. Therefore, x C, since B C. This proves
that if x A, then x C. That is, A C.
◍ What would you do to simplify (x+y)^2.
Answer: Write out the binomial factors and FOIL
◍ Prove that if A B, then A − B = Ø.
Answer: Proof: Assume A B, and assume for the sake of contradiction
that A−B 6= Ø.Then there exists an element x A − B. Therefore, x A
and x / B. Thiscontradicts the assumption that A B. Therefore A − B =
, Ø.The proof is complete
◍ Exponential Growth or Decay Formula for Word Problems.
Answer: % increasing or decreasingb is the % KEPTAdd or Subtract from
100 and change to a decimal
◍ Exponent rule for dividing monomials.
Answer: Subtract (big-small) and the answer goes where the bigger # is.
◍ A = {An | n N} where An = [1/n, 3 −1/n](a) Union of A over all natural
numbers(b) Intersection of A over all natural numbers.
Answer: (a) (0,3)(b) [1,2]
◍ Form of Exponential Equation.
Answer: a= starting number b= rate x= timey=ending amount
◍ When the problem says "Consecutive integers"....
Answer: Use X, X+1, X+2, X+3..
◍ "Golden Rule" when solving Inequalities.
Answer: If you multiply or divide both sides on inequality by a NEGATIVE
number, FLIP the inequality sign around!!
◍ In each case, find the power set P(S) of the given set S(a) S = {a}(b) S = {b,
c}(c) S = {1, 2, 3}(d) S = {4, {5, 6}}(e) S = {Ø, R}.
Answer: (a) (S) = {Ø, {a}}(b) P(S) = {Ø, {b}, {c}, {b, c}}(c) P(S) = {Ø,
{1}, {2}, {3}, {1, 2}, {1, 3}, {2, 3}, {1, 2, 3}}(d) P(S) = {Ø, {4}, {{5, 6}},
{4, {5, 6}}}(e) P(S) = {Ø, {Ø}, {R}, {Ø, R}}
◍ What is the slope-intercept form of a line?.
Answer: y=mx+b
◍ Line of best fit.
Answer: Also called "regression line", "linear model", "prediction equation",
or "trend line".Use calculator to findSTAT, edit, input L1 and L2, Stat, ->
CALC,#4 LinReg
◍ How do you tell if a parabola opens up/down?.
Answer: If "a" is positive: opens up. If "a" is negative: opens down.
