MA110 Mathematical Methods Quiz

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Questions and Answers

Which of the following statements are true? (Select all that apply.)

The operation * on the set of real numbers, a*b = b^a, is commutative.
The operation * on the set of real numbers, a * b = ba, is commutative. (correct)
The operation * on the set of real numbers, a* b = ba, is a binary operation on the set of real numbers. (correct)
The operation * on the set of real numbers, a*b = ba, is associative.

What is the remainder when `g(x) = x² + ax² + 3x + 6` is divided by `3x - 2` given that `g(-1) = 2`?

-4/3
2/3
2
4/3 (correct)

Given the equation `px² - 2(p + 3)x + p - 1 = 0` has real roots, what is the range of values of `p`?

The range of values of p is p >1 or p < -3.

Given that the function `f(x) = ax² + bx + c has a maximum value of 4 where x = - 1`, find the value of `a` and `b`.

The value of <code>a</code> is -2 and the value of <code>b</code> is 4. Signup and view all the answers

What are the dimensions of the largest rectangular field that can be enclosed using 1200 m of fencing?

The dimensions of the largest rectangular field are 300m by 300m. Signup and view all the answers

Which of the following properties of union and intersection of sets are correct?

AU (A'∩B) = AUB (A), AUB = (A∩B) บ (A∩B') U (A'∩B) (B), (A∩B) U (A∩B') = A (C) Signup and view all the answers

If CCD, then simplify ______

C' U D' Signup and view all the answers

Given that X, Y, and Z are sets, simplify the following if possible: [X' U (YnZ)]'

X∩(Y'UZ) Signup and view all the answers

The expression (XY) U (XOY') simplifies to X

True (A) Signup and view all the answers

Given that X and Y are subsets of some universal set U, simplify the following: [(XOX)(XUF)]' = ______

(X'UY') Signup and view all the answers

Let A = {x ∈ R: - 4 ≤ x < 2} and B = {x ∈ R: x ≥ −1} . Find A∩B.

{-1 ≤ x < 2} Signup and view all the answers

Let A = (-9,9) be the universal set and X = (−1,5], Y = [−5,3] and Z = [−1,7). Find X'.

(-9, -1] U (5, 9) Signup and view all the answers

Let R, the set of real numbers, be the universal set. If A = [−7,8)[11,∞) and B = [0, 20], find A' and display it on the number line.

(-∞, -7]U [8, 11) Signup and view all the answers

Let X = (-10, 10) be the universal set and A= (-2, 6], B = [-5, 3] and C = [-1,8). Find (B-A) ∩ C.

[-5, -2) Signup and view all the answers

Express the following in the form of a/b , where a and b are integers, b≠0: 0.33

33/100 Signup and view all the answers

Prove that √3 is an irrational number

Proof by contradiction: Assume √3 is rational. Therefore √3 = a/b, where a and b are integers with no common factors. Squaring both sides, we find that 3 = a²/b². This implies that a² = 3b² which means that a² is a multiple of 3, so a is also a multiple of 3. This means that a can be written as 3k where k is an integer. Substituting into the equation, we find 9k² = 3b². This simplifies to 3k² = b². This means that b² is a multiple of 3, so b is also a multiple of 3. This contradicts the initial assumption that a and b have no common factors. Therefore, the only possibility is that √3 is irrational. Signup and view all the answers

Express 3.1212 in the form a/b where a and b are integers and b≠0.

3119/990 Signup and view all the answers

Evaluate the following using the definition of Absolute value: |x - 2| = 6

x = 8 or x = -4 Signup and view all the answers

Evaluate the following using the definition of Absolute value: |2n + 1| = 11

n =5 or n = -6 Signup and view all the answers

Evaluate the following using the definition of Absolute value: |2x - 3| ≤ 5

-1 ≤ x ≤ 4 Signup and view all the answers

Solve |x - 1| > |x + 1|

x < 0 Signup and view all the answers

Solve the inequality: |x + 2| > 3.

x < -5 or x > 1 Signup and view all the answers

Rationalize the denominator of each of the following: `2√3-√2/4√3`

(√6 - 1)/6 Signup and view all the answers

Rationalize the denominator of each of the following: x/1

x/√y Signup and view all the answers

Rationalize the denominator of each of the following: `√x²-9/x+√x²-9`

<code>√x²-9/x</code> Signup and view all the answers

Rationalize the numerator of each of the following: `√2+1)√3-1)/√3-1`

<code>√2+1/3</code> Signup and view all the answers

Rationalize the numerator of each of the following: `3-2√3/√5+h-3`

<code>3√5-2√15 +h√3 -2h/h² - 2h√5 + 5</code> Signup and view all the answers

Write each of the following in terms of i, perform the indicated operations, and simplify if possible: `√-4√-16`

8i Signup and view all the answers

Write each of the following in terms of i, perform the indicated operations, and simplify if possible: `√-36/-4`

3i Signup and view all the answers

Let z₁ = 2+i, z₂ = 1-i√3 and z₃ = 3 + 4i. Verify the following identities: ______

z₁z₂z₃ = z₃z₁z₂ Signup and view all the answers

Solve for x and y given that: (x + iy)(4i) = 8

x = 2, y = -2 Signup and view all the answers

Flashcards

De Morgan's Law for Intersection

A rule stating that the complement of the intersection of two sets is equal to the union of their complements.

De Morgan's Law for Union

A rule stating that the complement of the union of two sets is equal to the intersection of their complements.