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Chapter 1-4

MTH001 Elementary Maths

1 / 47

Which logical connective is represented by the symbol “∨”?

 

2 / 47

What does a negation (~) do to the truth value of a proposition p?

 

3 / 47

What is the result of a conjunction (p ∧ q) in a truth table when both p and q are false?

 

4 / 47

What is the truth value of the compound statement (h ∧ w) ∧ (~s) if h, w, and s represent “Zia is healthy”, “Zia is wealthy”, and “Zia is wise”, respectively, and Zia is healthy, wealthy but not wise?

 

5 / 47

How is the disjunction of p and q symbolically represented, if p states “Islamabad is the capital of Pakistan” and q states “17 is divisible by 3”?

 

6 / 47

What symbol is used in logical operations to represent “and”?

 

7 / 47

Which of the following is NOT a proposition?

 

8 / 47

What is the main focus of Discrete Mathematics?

 

9 / 47

Using logical laws, what is the simplified form of the expression 𝑝∨[∼(∼𝑝∧𝑞)]p∨[∼(∼p∧q)]?

 

10 / 47

What is the symbolic representation for the statement “It is not hot but sunny”?

 

11 / 47

 

In the statement translation, if m = “Ali is good in Mathematics” and c = “Ali is a Computer Science student”, how would you represent “Ali is a Computer Science student or good in Maths” in symbolic form?

 

12 / 47

What outcome is specified in a truth table for the disjunction of p and q when both p and q are false?

 

13 / 47

 

Which of the following statements best describes the construction of a truth table for a compound statement?

 

14 / 47

What is the result of the truth table for the expression ~(p ∧ q)?

 

15 / 47

Which of the following statements correctly applies De Morgan’s Laws to the statement p ∨ q?

 

16 / 47

Which of the following expressions is tautology?

 

17 / 47

How would you write the negation of the inequality -1 < x ≤ 4 using De Morgan’s Laws?

 

18 / 47

Which of the following expressions accurately represents the logical equivalence of ∼(𝑝∧𝑞)∼(p∧q)?

 

19 / 47

Evaluate the logical equivalence for the expression (𝑝∧𝑞)∨𝑟(p∧q)∨r compared to 𝑝∧(𝑞∨𝑟)p∧(q∨r).

 

20 / 47

What is the truth value of the expression (𝑝∨𝑞)∧∼(𝑝∧𝑞)(p∨q)∧∼(p∧q) when 𝑝p is true and 𝑞q is false?

 

21 / 47

Using the double negative property, what does the expression ∼(∼𝑝)∼(∼p) simplify to?

 

22 / 47

What is the correct negation for the statement “The fan is slow or it is very hot” using De Morgan’s Laws?

 

23 / 47

Identify the statement that is always false, known as a contradiction.

 

24 / 47

Which logical equivalence is demonstrated by the expression ∼(∼𝑝∧𝑞)∧(𝑝∨𝑞)≡𝑝∼(∼p∧q)∧(p∨q)≡p?

 

25 / 47

What is the simplified English version of the logical expression (𝑝∧𝑞)∨(𝑝∧𝑟)(p∧q)∨(p∧r)?

 

26 / 47

What is the simplified English version of the logical expression (𝑝∧𝑞)∨(𝑝∧𝑟)(p∧q)∨(p∧r)?

 

27 / 47

What is the correct contrapositive of the statement “If you earn an A in Math, then I’ll buy you a computer”?

 

28 / 47

Using truth tables, which statement proves 𝑝→𝑞≡∼𝑞→∼𝑝p→q≡∼q→∼p?

 

29 / 47

What is the logical equivalence used to simplify the negation of the conditional statement 𝑝→𝑞p→q?

 

30 / 47

Given the statement “If it is raining, I have forgotten my umbrella or I have forgotten my hat,” how can it be simplified using logical equivalence?

 

31 / 47

Which of the following is the correct inverse of the conditional statement “If P is a square, then P is a rectangle”?

 

32 / 47

What is the correct translation of the symbolic proposition ∼𝑞→𝑟∼q→r?

 

33 / 47

Determine the truth value of the statement “If today is not Friday, then 2 + 3 ≠ 5.”

 

34 / 47

Which expression accurately represents the contrapositive of the statement “If my car is in the repair shop, then I cannot get to class”?

 

35 / 47

What is the truth value of the biconditional statement “1+1 = 3 if and only if the Earth is flat”?

 

36 / 47

Which logical operation best describes the expression (𝑝↔𝑞)≡(𝑝→𝑞)∧(𝑞→𝑝)(p↔q)≡(p→q)∧(q→p)?

 

37 / 47

Translate the following logical form into English using the concept of biconditional: 𝑝∧∼𝑟↔𝑞∨𝑟p∧∼r↔q∨r

 

38 / 47

Which statement shows that ∼(𝑝⊕𝑞)∼(p⊕q) and 𝑝↔𝑞p↔q are logically equivalent?

 

39 / 47

Rewrite the statement form ∼𝑝∨𝑞→𝑟∨∼𝑞∼p∨q→r∨∼q to a logically equivalent form using only ∼∼ and ∧∧.

 

40 / 47

Suppose that 𝑝→𝑞p→q is false. What is the truth value of 𝑝∨𝑞p∨q?

 

41 / 47

What is the truth value of the biconditional statement “Sky is blue if and only if 1 = 0”?

 

42 / 47

How can you express the logical form 𝑝∧∼𝑟↔𝑞∨𝑟p∧∼r↔q∨r using the concept of implication and equivalence?

 

43 / 47

Identify the correct logical equivalence for the biconditional operation expressed as 𝑝↔𝑞p↔q.

 

44 / 47

What is the contrapositive of the statement 𝑝→𝑞p→q?

 

45 / 47

Which of the following statements is always false, known as a contradiction?

 

46 / 47

Rewrite the statement form 𝑝∧∼𝑞→𝑟p∧∼q→r using only logical equivalences without the implication operator.

 

47 / 47

Determine the truth value of the biconditional 𝑝↔𝑞p↔q when both 𝑝p and 𝑞q are false.

 

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Lecture 7 MCQs

MTH-001 Lecture 7

1 / 20

If 𝐴∪𝐵=𝐴, what can be inferred about sets 𝐴 and 𝐵?

2 / 20

Which of the following is true about the difference of sets 𝐴 and 𝐵?

3 / 20

Which of the following sets are disjoint?

4 / 20

Which of the following is true about the union of sets 𝐴 and 𝐵?

5 / 20

If 𝐴∩𝐵=𝐴, which of the following must be true?

6 / 20

Which of the following is true about the complement of set 𝐴?

7 / 20

Which Venn diagram correctly represents 𝐴∩𝐵?

8 / 20

Which of the following is true about the intersection of sets 𝐴 and 𝐵?

9 / 20

If 𝐴={1,2,3} and 𝐵={4,5,6}, what is 𝐴∪𝐵?

10 / 20

If 𝐴∩𝐵=𝐵, what can be inferred about sets 𝐴 and 𝐵?

11 / 20

If 𝐴∪𝐵=𝑈 and 𝐴∩𝐵=𝜙, what can be said about sets 𝐴 and 𝐵?

12 / 20

If 𝐴∪𝐵=𝐴, what can be inferred about sets 𝐴 and 𝐵?

13 / 20

Which membership table corresponds to the intersection of sets 𝐴 and 𝐵?

14 / 20

Which membership table corresponds to the union of sets 𝐴 and 𝐵?

15 / 20

If 𝑈={𝑎,𝑏,𝑐,𝑑,𝑒,𝑓,𝑔} and 𝐴={𝑎,𝑐,𝑒,𝑔}, what is 𝐴𝑐?

16 / 20

If 𝑈={𝑎,𝑏,𝑐,𝑑,𝑒,𝑓,𝑔}, 𝐴={𝑎,𝑐,𝑒,𝑔}, and 𝐵={𝑑,𝑒,𝑓,𝑔}, what is 𝐴∩𝐵?

17 / 20

Which Venn diagram correctly represents 𝐴∪𝐵?

18 / 20

If 𝑈={𝑎,𝑏,𝑐,𝑑,𝑒,𝑓,𝑔}, 𝐴={𝑎,𝑐,𝑒,𝑔}, and 𝐵={𝑑,𝑒,𝑓,𝑔}, what is 𝐴−𝐵?

19 / 20

Given 𝑈={1,2,3,4,5,6,7,8,9,10} and 𝐴={2,3,5,7}, what is 𝐴∪𝐴𝑐?

20 / 20

Given 𝑈={1,2,3,4,5,6,7,8,9,10} and 𝐴={2,4,6,8,10}, what is 𝐴𝑐?

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Lecture 8 MCQs

MTH-001 Lecture 8

1 / 23

Which of the following is not an ordered pair?

2 / 23

Given sets A = {1, 2} and B = {a, b, c}, which of the following is a binary relation from A to B?

3 / 23

If A = {1, 2} and B = {a, b, c}, what is A × B?

4 / 23

Which of the following is an ordered triple?

5 / 23

If 𝐴={1,2,3} and 𝐵={4,5,6}, what is 𝐴∪𝐵?

6 / 23

Given sets A = {1, 2} and B = {a, b, c}, which of the following is a binary relation from A to B?

7 / 23

The matrix representation of a relation R from A = {1, 2, 3} to B = {x, y} is given as 100111101011. What is the relation R?

8 / 23

Find the values of x and y if (2x, x + y) = (6, 2).

9 / 23

What is the domain of the relation R = {(1, 2), (2, 3), (3, 4)}?

10 / 23

What is the matrix representation of the relation R = {(1, y), (2, x), (2, y), (3, x)} from A = {1, 2, 3} to B = {x, y}?

11 / 23

The range of the relation R = {(2, 4), (3, 6), (4, 8)} is:

12 / 23

Given the ordered pairs (a, b) and (c, d), when are they equal?

13 / 23

If A = {0, 1} and B = {1}, what is the total number of binary relations from A to B?

14 / 23

The matrix representation of a relation R from A = {1, 2, 3} to B = {x, y} is given as 100111101​011​. What is the relation R?

15 / 23

Let R be a binary relation on set A = {1, 2, 3} defined by R = {(1, 2), (2, 3)}. What is the range of R?

16 / 23

If A = {1, 2, 3} and R = {(1, 2), (2, 1), (2, 3), (3, 2)}, what type of relation is R?

17 / 23

Let R be a binary relation on set A = {1, 2, 3} defined by R = {(1, 2), (2, 3)}. What is the domain of R?

18 / 23

Which of the following statements is true about Cartesian products?

19 / 23

The Cartesian product of sets A and B is denoted by:

20 / 23

How many elements are there in the Cartesian product of two sets A and B, where |A| = 2 and |B| = 3?

21 / 23

Which of the following is the Cartesian product A × A for A = {1, 2}?

22 / 23

Which of the following is the universal relation on set A = {1, 2, 3}?

23 / 23

The range of the relation R = {(1, 2), (2, 3), (3, 4)} is:

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Lecture 9 MCQs

MTH-001 Lecture-9

1 / 20

If 𝑅 is a transitive relation on set 𝐴 and (𝑎,𝑏)∈𝑅 and (𝑏,𝑐)∈𝑅, which of the following must be true?

2 / 20

For the set 𝐴={1,2,3}, the relation 𝑅={(1,2),(2,1),(2,3),(3,2),(3,3)} is:

3 / 20

For set A = {1, 2, 3}, which of the following relations is not reflexive?

4 / 20

The matrix representation of a transitive relation on set 𝐴={1,2,3} must satisfy which condition?

5 / 20

If R is a relation on set A and R is not reflexive, then:

6 / 20

Which of the following relations on set 𝐴={1,2,3} is transitive?

7 / 20

A relation R on set A is symmetric if:

8 / 20

Which matrix representation corresponds to a reflexive relation on set A = {1, 2, 3}?

9 / 20

A directed graph representing a reflexive relation always includes:

10 / 20

Which of the following is true for the relation 𝑅={(1,2),(2,1)} on set 𝐴={1,2,3}?

11 / 20

Which of the following statements is true about a transitive relation?

12 / 20

Which of the following is not a characteristic of a symmetric relation?

13 / 20

Which of the following is a transitive relation on set 𝐴={1,2,3}?

14 / 20

A relation 𝑅 on set 𝐴 is transitive if:

15 / 20

Which of the following relations on set A = {1, 2, 3} is reflexive?

16 / 20

Which of the following relations on set 𝐴={1,2,3,4} is neither reflexive, symmetric, nor transitive?

17 / 20

Which statement is true for a symmetric relation on set 𝐴?

18 / 20

For the set 𝐴={1,2,3,4}, which relation is transitive?

19 / 20

Which of the following relations on set A = {1, 2, 3} is symmetric?

20 / 20

For the set 𝐴={1,2,3}, the relation 𝑅={(1,2),(2,3),(1,3)} is:

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Lecture 10 MCQs

MTH-001 Lecture-10

1 / 20

A partial order relation is:

2 / 20

The matrix representation of an antisymmetric relation R on a set A has:

3 / 20

Which of the following is true for a symmetric relation?

4 / 20

A relation R on set A is reflexive if:

5 / 20

The subset relation ⊆ on the power set P(A) is:

6 / 20

Which of the following is an example of a partial order relation?

7 / 20

For the relation 𝑅 defined on set 𝐴 as 𝑅={(𝑎,𝑏)∣𝑎>𝑏}, 𝑅 is:

8 / 20

The matrix representation of an irreflexive relation has:

9 / 20

Which of the following is a property of an antisymmetric relation 𝑅 on set 𝐴?

10 / 20

A relation 𝑅 on set 𝐴 is irreflexive if:

11 / 20

Which of the following relations is neither reflexive nor irreflexive?

12 / 20

A relation 𝑅R on a set 𝐴A is symmetric if:

13 / 20

A relation 𝑅R on set 𝐴A is transitive if:

14 / 20

The relation 𝑅={(1,1),(2,2),(3,3)}R={(1,1),(2,2),(3,3)} on the set 𝐴={1,2,3}A={1,2,3} is:

15 / 20

Which of the following statements is true for an antisymmetric relation 𝑅R?

16 / 20

If a relation 𝑅R is both symmetric and irreflexive, which of the following is true?

17 / 20

A relation 𝑅R on a set 𝐴A is called irreflexive if for all 𝑎∈𝐴a∈A:

18 / 20

If 𝑅R and 𝑆S are transitive, is 𝑅∩𝑆R∩S transitive?

19 / 20

If 𝑅R and 𝑆S are symmetric, is 𝑅∩𝑆R∩S symmetric?

20 / 20

If 𝑅R and 𝑆S are reflexive, is 𝑅∩𝑆R∩S reflexive?

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Lecture 11 MCQs

MTH001-Lecture-11

1 / 20

Which of the following relations is not a function?

2 / 20

If a binary operation * on a set A is defined as a * b = a + b for all a, b ∈ A, then this operation is:

3 / 20

How many functions are there from a set with three elements to a set with four elements?

4 / 20

If A = {1, 2} and B = {2, 3}, which of the following is the image of A under f where f: A → B is defined by f(x) = x + 1?

5 / 20

The graph of y = x^2 is a:

6 / 20

Which of the following is true about the inverse image of a function f: X → Y?

7 / 20

For the function f: X → Y, what is the co-domain?

8 / 20

Which of the following is a function from X = {2, 4, 5} to Y = {1, 2, 4, 6}?

9 / 20

Which of the following is not a binary operation?

10 / 20

If f: X → Y, and f(x) = y, where X = {a, b, c} and Y = {1, 2, 3, 4}, what is the range of f if f(a) = 2, f(b) = 4, and f(c) = 2?

11 / 20

In the vertical line test, if any vertical line intersects the graph of a relation more than once, then the relation:

12 / 20

For the function f(x) = x^2 + 1, what is the range?

13 / 20

The range of the function f(x) = x^2 + 3 for x in R is:

14 / 20

Which of the following sets is the domain of the function f: X → Y defined by f(x) = x^2 + 1, where X and Y are real numbers?

15 / 20

What is the domain of the function f(x) = sqrt(x)?

16 / 20

Let A = {4, 5, 6} and B = {5, 6}. Which of the following relations R from A to B is a function?

17 / 20

If the set X = {a, b, c} and the set Y = {1, 2, 3}, which of the following represents a one-to-one function from X to Y?

18 / 20

The composition of two functions f and g, denoted by g ∘ f, is defined as:

19 / 20

The inverse image of a set C under a function f: X → Y is defined as:

20 / 20

In an arrow diagram of a function, which of the following must be true?

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Math Lesson 12

1 / 20

Which of the following sequences converges to 0?

2 / 20

Explain the difference between an arithmetic and a geometric sequence:

3 / 20

The function ℎ:𝑍→𝑍 defined by ℎ(𝑛)=2𝑛+1 is:

4 / 20

What is the image of the function 𝑔:𝑍→𝑍 defined by 𝑔(𝑛)=𝑛^2?

5 / 20

Is the function 𝑓:𝑅→𝑅 defined by 𝑓(𝑥)=2𝑥−5 surjective?

6 / 20

Which of the following is true for an injective function?

7 / 20

If a function 𝑓:𝑋→𝑌 is both injective and surjective, it is called:

8 / 20

Given the sequence an=3n-2, what are the first 5 terms?

9 / 20

Given 𝑓:𝑋→𝑌 where 𝑓(𝑥)=3𝑥+2, which of the following statements is true?

10 / 20

What is the domain of the function 𝑓(𝑥)=𝑥−1?

11 / 20

Which of the following functions is bijective?

12 / 20

Given the sequence an=2n+3, find the first 5 terms and determine if it is arithmetic or geometric.

13 / 20

If a function 𝑓:𝑋→𝑌 is not one-to-one, which of the following is true?

14 / 20

Which arrow diagram represents a one-to-one function?

15 / 20

Calculate the 13th term of the Fibonacci sequence.

16 / 20

How many one-to-one functions are there from a set with 3 elements to a set with 4 elements?

17 / 20

Describe what is meant by the convergence of a sequence:

18 / 20

Which of the following is an example of a constant function?

19 / 20

Alex owns a vineyard that produces 500 barrels of wine annually. Due to a new irrigation system, the production increases by 5% each year. However, due to soil erosion, 30 barrels of wine are lost each year. Model the vineyard's production as a sequence and determine the sustainability of this plan.

20 / 20

Which characteristic is true for a surjective function?

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Math Lesson 13

1 / 20

Which of the following functions is neither injective nor surjective?

2 / 20

If 𝐶𝑛=1+(−1)𝑛Cn​=1+(−1)n for all integers 𝑛≥0n≥0, what is the value of 𝐶3C3​?

3 / 20

What is the range of the function 𝑓(𝑥)=ln⁡(𝑥) where 𝑥>0?

4 / 20

Find the 8th term of the geometric sequence 4, 12, 36, 108, ….

5 / 20

Which of the following is a sequence?

6 / 20

Which sequence is defined by the explicit formula 𝑎𝑛=(−1)𝑛+1𝑛an​=(−1)n+1n for all integers 𝑛≥0n≥0?

7 / 20

If 𝑓:𝑅→𝑅 is defined by 𝑓(𝑥)=𝑥^2+1, is it surjective?

8 / 20

Identify the geometric sequence from the following options:

9 / 20

What is the common difference of the arithmetic sequence 5, 9, 13, 17, …?

10 / 20

Identify the arithmetic sequence from the following options:

11 / 20

Which of the following is true for a bijective function?

12 / 20

Find the 36th term of the arithmetic sequence whose 3rd term is 7 and 8th term is 17.

13 / 20

Find the 20th term of the arithmetic sequence 3, 9, 15, 21, ….

14 / 20

Which of the following is true for a constant function?

15 / 20

The function 𝑓:𝑅→𝑅 defined by 𝑓(𝑥)=𝑒^𝑥 is:

16 / 20

Find the first four terms of the sequence defined by the formula 𝑏𝑗=1+2𝑗bj​=1+2j for all integers 𝑗≥0j≥0.

17 / 20

Which term of the arithmetic sequence 4, 1, -2, …, is -77?

18 / 20

What does the notation 𝑎𝑛an​ represent in a sequence?

19 / 20

What is the common ratio of the geometric sequence 3, -3/2, 3/4, -3/8, …?

20 / 20

If 𝑓:𝑅→𝑅 is defined by 𝑓(𝑥)=∣𝑥∣, is it injective?

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Math Lesson 14

1 / 20

Determine the sum of the infinite geometric series 13+19+127+⋯.

2 / 20

Which of the following represents a series?

3 / 20

Which of the following is a geometric series?

4 / 20

What symbol is commonly used to denote summation?

5 / 20

Which term of the geometric sequence is 1/8 if the first term is 4 and the common ratio is 1/2?

6 / 20

If 𝑎0=2, 𝑎1=3, 𝑎2=−2, 𝑎3=1, 𝑎4=0, find ∑𝑗=13𝑎𝑗.

7 / 20

Find the sum of the series 5+10+15+...+100.

8 / 20

Find the sum of the first 10 terms of the geometric series 3,9,27,81,...

9 / 20

Convert the summation notation ∑𝑘=15(2𝑘+1) to its expanded form.

10 / 20

Write the series 1+2+3+...+𝑛 in summation notation.

11 / 20

What is the sum of the arithmetic series 7+14+21+28+...+70?

12 / 20

Compute the summation ∑𝑗=04𝑎𝑗 for 𝑎0=2, 𝑎1=3, 𝑎2=−2, 𝑎3=1, and 𝑎4=0.

13 / 20

Write the geometric sequence with positive terms whose second term is 9 and fourth term is 1.

14 / 20

Determine the sum of the arithmetic series 2+5+8+11+...+50.

15 / 20

Find the sum of the arithmetic series 1+3+5+7+...+19.

16 / 20

Find the sum of the geometric series 4+12+36+108+... up to the 6th term.

17 / 20

Find the sum of the first 20 natural numbers.

18 / 20

Convert the expanded form 1+2+3+⋯+10 into summation notation.

19 / 20

Which of the following is the correct expanded form of the summation ∑𝑘=1𝑛𝑘?

20 / 20

What is the 10th term of the geometric sequence 5, 10, 20, 40, …?

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Math Lesson 15

1 / 23

Which of the following is NOT a type of operator in Excel?

2 / 23

Which of the following steps is required to enter a value into a cell in Excel?

3 / 23

What is the result of the Excel formula =5^2?

4 / 23

In Excel, which operator is used for multiplication?

5 / 23

What is the result of 4242?

6 / 23

What is the result of 12×512×5?

7 / 23

What is the square root of 4 in exponent notation?

8 / 23

What is the result of 12−512−5?

9 / 23

What is the result of 12+512+5?

10 / 23

In Excel, what is the reference for a cell in column B and row 15?

11 / 23

Convert 20% to a fraction in Excel.

12 / 23

What is the result of the Excel formula =5*4?

13 / 23

In Excel, how would you start a formula?

14 / 23

Which of the following is a correct Excel formula for subtraction?

15 / 23

Which of the following is NOT one of the five basic arithmetic operations?

16 / 23

Which of the following is a correct Excel formula for addition?

17 / 23

In Excel, what symbol is used for exponentiation?

18 / 23

What is the reference for a range that starts at A1 and ends at D15 in Excel?

19 / 23

Which Excel formula correctly divides 240 by 15?

20 / 23

Which of the following steps is NOT included in starting Microsoft Excel 2000 XP?

21 / 23

What is the result of the Excel formula =20%?

22 / 23

In Excel, which symbol is used for percent?

23 / 23

What does the cell reference B15 indicate in Excel?

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Math Lesson 16

1 / 20

What is the percentage increase when the weight of cotton increases from 3 kg to 15 kg?

2 / 20

If the initial value is 1000 and the final value is 2500, what is the percentage increase?

3 / 20

If an investment has rates of return of 4%, 8%, -10%, and 9% over four years, what will be the value of the investment at the end of the fourth year if the initial investment is 100,000 Rs.?

4 / 20

How do you calculate the percentage change if the original weight of cotton is 3 kg and it increases to 15 kg?

5 / 20

What is the formula to calculate percentage change?

6 / 20

What was the percentage change if Monday’s sales were Rs.1000 and grew to Rs.2500 on Tuesday?

7 / 20

In Excel, how would you calculate the new weight if it increases from 3 kg to 15 kg?

8 / 20

What is the value of the investment at the end of the third year if the value at the end of the second year is 104000 Rs. and the rate of return is 4%?

9 / 20

In Excel, how would you write the formula to calculate the value of the investment at the end of the fourth year if the value at the end of the third year is in cell C52 and the rate of return is in cell C53?

10 / 20

If the value of the investment at the end of the second year is in cell C48, how would you write the formula to calculate the value at the end of the third year if the rate of return is in cell C49?

11 / 20

What is the result of the Excel formula =D20 - D19 if D19 is 15 and D20 is 3?

12 / 20

What is the value of the investment at the end of the fourth year if the value at the end of the third year is 101088 Rs. and the rate of return is 9%?

13 / 20

How would you enter the formula to calculate the percentage change in Excel if the initial value is in cell C4 and the final value is in cell C5?

14 / 20

In Excel, how would you calculate the percentage change for fruit shrinking from 15 kg to 3 kg?

15 / 20

If an employee earns 5000 rupees per month and receives wage increases of 3%, 2%, and 1% in successive years, what will be the salary at the end of the third year?

16 / 20

In Excel, how would you write the formula to calculate the salary in the second year if the initial salary is in cell C35 and the percentage increase is in cell C36?

17 / 20

In Excel, how would you write the formula to calculate the value of the investment at the end of the second year if the initial investment is in cell C46 and the rate of return is in cell C47?

18 / 20

If 15 kg of fresh fruit shrinks to 3 kg of dried fruit, what is the percentage change?

19 / 20

What is the value of the investment at the end of the second year if the initial investment is 100,000 Rs. and the rate of return is 4%?

20 / 20

What is the salary at the end of the second year if the initial salary is 5000 Rs. and the percentage increase is 3%?

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MTH 001 Chapter 17

1 / 16

Calculate the interest on drawings for a partner who withdraws Rs. 3,000 at the middle of each quarter at an interest rate of 8% per annum.

2 / 16

If four bills of Rs. 500 each are due in 10, 20, 30, and 40 days respectively, what is the average due date?

3 / 16

If Rs. 5,000 is borrowed at an annual simple interest rate of 9%, what is the total interest to be paid after 3 years?

4 / 16

Calculate the amount after 3 years on a principal of Rs. 2,000 at a rate of 5% compounded annually.

5 / 16

Three payments of Rs. 600, Rs. 800, and Rs. 1,200 are due in 20, 40, and 60 days, respectively. What is the average due date?

6 / 16

A store offers a seasonal discount of 25% on all items. If a customer buys an item with a list price of Rs. 3,200, how much do they pay after the discount?

7 / 16

Calculate the amount after 4 years on a principal of Rs. 1,000 at a rate of 6% compounded annually.

8 / 16

A store offers a 15% discount on a product with a list price of Rs. 3,000. What is the discount amount?

9 / 16

What is the compound interest on Rs. 2,500 invested at 8% per annum for 2 years, compounded annually?

10 / 16

A product originally priced at Rs. 5,000 is sold for Rs. 4,250 after a discount. What is the discount percentage?

11 / 16

What is the compound interest on Rs. 1,000 invested at 10% per annum for 2 years, compounded annually?

12 / 16

If a partner withdraws Rs. 2,000 at the beginning of every month and the interest rate is 6% per annum, what is the total interest on drawings for the year?

13 / 16

A product has a list price of Rs. 2,500. If a 10% discount is applied, what is the net cost price?

14 / 16

What is the simple interest on a principal amount of Rs. 1,200 at an annual interest rate of 5% for 3 years?

15 / 16

A sum of Rs. 2,000 is invested at a simple interest rate of 6% per annum. How much interest will it earn in 4 years?

16 / 16

Calculate the total amount to be paid after 2 years if the principal is Rs. 500 at an annual interest rate of 8%.

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MTH 001 Chapter 18

1 / 15

What formula is used to calculate the present value of an annuity?

2 / 15

What does the accumulation factor for n periods represent in the context of annuities?

3 / 15

If you invest $1,000 every year at an interest rate of 5% for 5 years, what will be the accumulated value?

4 / 15

What is the relationship between the interest rate and the discount factor?

5 / 15

How does compounding frequency affect the future value of an annuity?

6 / 15

What does algebraic operation on annuities typically involve?

7 / 15

How is the discounted value of an annuity calculated?

8 / 15

In the context of annuities, what is the discount factor?

9 / 15

Which of the following is the correct accumulation factor formula?

10 / 15

What is an annuity?

11 / 15

What happens to the present value of an annuity if the interest rate increases?

12 / 15

If you want to receive $2,000 at the end of each half-year for 10 years with an interest rate of 11% compounded semi-annually, how much should you deposit now?

13 / 15

What is the formula for calculating the accumulated value of an annuity?

14 / 15

What is the primary purpose of an annuity?

15 / 15

What is the present value of an annuity?

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MTH 001 Chapter 19

1 / 15

What is a column matrix?

2 / 15

What does the dimension or order of a matrix refer to?

3 / 15

What is a row matrix?

4 / 15

Which property does the identity matrix share with the number 1 in multiplication?

5 / 15

What is a square matrix?

6 / 15

How is the product of a matrix and the identity matrix similar to multiplying a real number by 1?

7 / 15

What is the result of multiplying a matrix by a zero matrix?

8 / 15

Why are matrices important in business and industry?

9 / 15

What are the elements of the identity matrix I3x3?

10 / 15

How are matrices usually represented?

11 / 15

What is an identity matrix?

12 / 15

Given a 2x2 identity matrix I and a 2x2 matrix A, what is the result of A×I?

13 / 15

If matrix A is a 2x3 matrix and matrix B is a 3x2 matrix, what are the dimensions of the resulting matrix when A is multiplied by B?

14 / 15

What does 'r' stand for in the notation r1c1 used in matrix multiplication?

15 / 15

What is a matrix?

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MTH 001 Chapter 20

1 / 15

What is the sum or difference of two matrices calculated by?

2 / 15

Which condition must be met for two matrices to be added or subtracted?

3 / 15

How is the product of a row and a column obtained in matrix multiplication?

4 / 15

What happens if you try to multiply two matrices where the number of columns of the first matrix is not equal to the number of rows of the second matrix?

5 / 15

If A is a 2x3 matrix and B is a 3x2 matrix, what are the dimensions of AB?

6 / 15

What is the result of multiplying a matrix by a scalar?

7 / 15

What is the result of the product of matrix A = 3x3 and matrix B = 3x2?

8 / 15

What is the inverse of a matrix?

9 / 15

What is the total revenue for a company if the sales matrix S = [20000 5500 10600] [18250 7000 11000] and the price matrix P = [1.60] [2.30] [3.10]?

10 / 15

Given a matrix A and a scalar c = 2, what is the result of cA if A = [1 2] [3 4]?

11 / 15

What is the necessary condition for multiplying two matrices A and B?

12 / 15

What is the condition for two 2x2 matrices to be inverses of each other?

13 / 15

What does it mean if two matrices are overproduced in a company setting?

14 / 15

Given two matrices A = [1 2] [3 4] and B = [2 0] [1 2], what is AB?

15 / 15

How can matrices help in business applications?

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LESSON 21

1 / 20

If the ratio of sugar to water in a solution is 1:4, how much sugar is there in 5 liters of solution?

2 / 20

Find the unknown value in the proportion: 5/x = 10/15.

3 / 20

The ratio of the ages of two brothers is 5:3. If the elder brother is 15 years old, how old is the younger brother?

4 / 20

In a mixture of 8 liters, the ratio of acid to water is 3:5. How much acid is in the mixture?

5 / 20

In a classroom, the ratio of boys to girls is 3:2. If there are 15 boys, how many girls are there?

6 / 20

Find the unknown value in the proportion: 4/x = 8/24.

7 / 20

In a certain class, the ratio of passing grades to failing grades is 7:5. How many students out of 48 failed the course?

8 / 20

If the ratio of cats to dogs in a pet shop is 3:4 and there are 24 dogs, how many cats are there?

9 / 20

Find the unknown value in the proportion: 4x + 5/3 = x + 7/2.

10 / 20

If the ratio of red balls to blue balls in a bag is 7:3 and there are 21 red balls, how many blue balls are there?

11 / 20

A mixture contains milk and water in the ratio 5:3. If there is 10 liters of water, what is the total volume of the mixture?

12 / 20

In a punch recipe, the ratio of mango juice, apple juice, and orange juice is 3:2:1. If you have 3 liters of mango juice, how much apple juice do you need?

13 / 20

If the ratio of the circumference of two circles is 2:3, what is the ratio of their radii?

14 / 20

If the ratio of mango juice, apple juice, and orange juice in a punch recipe is 3:2:1.5 and you have 250 milliliters of orange juice, how much mango juice is needed?

15 / 20

A punch recipe has a ratio of mango juice, apple juice, and orange juice as 4:3:2. If you have 1 liter of orange juice, how much punch can you make?

16 / 20

In a punch recipe, the ratio of grape juice, lemon juice, and orange juice is 2:3:5. If you have 2 liters of grape juice, how much punch can you make?

17 / 20

Find the unknown value in the proportion: 6/x = 9/27.

18 / 20

In a punch recipe, the ratio of mango juice, apple juice, and orange juice is 3:2:1. If you have 2 liters of orange juice, how much punch can you make?

19 / 20

The ratio of the lengths of two rectangles is 4:5. If the length of the first rectangle is 20 cm, what is the length of the second rectangle?

20 / 20

A punch recipe has a ratio of mango juice, apple juice, and orange juice as 2:1:3. If you have 6 liters of apple juice, how much punch can you make?

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LESSON 22

1 / 20

Statistics is crucial for handling:

2 / 20

Data obtained from scientific inquiry include:

3 / 20

A variable that varies with an individual or object is called:

4 / 20

Which scale of measurement includes ordering or ranking of measurements?

5 / 20

Which of the following is also known as Quantitative Analysis?

6 / 20

Statistics deals with variability that:

7 / 20

Statistics has applications in which of the following fields?

8 / 20

What does the term "statistic" refer to in the third sense?

9 / 20

Which of the following is a function of statistics?

10 / 20

Which of the following describes the first meaning of the word "statistics"?

11 / 20

A continuous variable is one that:

12 / 20

In its second meaning, statistics is defined as:

13 / 20

Which of the following is a quantitative variable?

14 / 20

Which characteristic of statistics deals with aggregates rather than individual observations?

15 / 20

What is a discrete variable?

16 / 20

In statistics, what does an observation often mean?

17 / 20

Which statement is true about the use of statistics in decision making?

18 / 20

Which scale of measurement classifies observations into mutually exclusive qualitative categories?

19 / 20

Which scale of measurement has a true zero point and allows for meaningful ratios?

20 / 20

What does the science of statistics enable us to do?

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LESSON 23

1 / 20

Which chart would best show the proportion of students from different schooling mediums in a college?

2 / 20

In which situation would you use a bivariate frequency table?

3 / 20

In the context of data representation, what does "univariate" mean?

4 / 20

In a simple bar chart, what does the width of each bar represent?

5 / 20

When should a component bar chart be used instead of a multiple bar chart?

6 / 20

What does the term "frequency" refer to in a frequency table?

7 / 20

If you want to represent the total number of male and female students in different schools, which type of chart would you use?

8 / 20

How do you determine the angle at which to cut a pie chart for a category?

9 / 20

In a component bar chart, what does the upper part of each bar represent?

10 / 20

What type of data is best represented by a pie chart?

11 / 20

What type of chart is useful for comparing two different kinds of information?

12 / 20

When constructing a component bar chart, what is depicted by the different sections of each bar?

13 / 20

If you have data for a company's turnover for five years, which chart is most suitable for visual representation?

14 / 20

For which type of data would a histogram be an appropriate representation?

15 / 20

What information does the height of a bar in a simple bar chart convey?

16 / 20

How is a component bar chart different from a simple bar chart?

17 / 20

What does a multiple bar chart consist of?

18 / 20

What is the first step in creating a frequency table for qualitative data?

19 / 20

Which chart would be most appropriate to compare the import and export values of a country over multiple years?

20 / 20

How is the percentage calculated in a frequency table?

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LESSON 24

1 / 20

Given a frequency distribution, how do you identify the median class?

2 / 20

If a car travels distances of 30 miles at 60 mph and 30 miles at 40 mph, what is the harmonic mean of the speeds?

3 / 20

What is the median of the following data set: 5, 8, 12, 15, 18?

4 / 20

When should the mode be used in a business context?

5 / 20

Why is the median sometimes preferred over the mean?

6 / 20

Calculate the arithmetic mean for the following data set: 5, 10, 15, 20, 25. Provide steps.

7 / 20

In a frequency distribution, how is the median calculated?

8 / 20

The median is defined as:

9 / 20

Given the following data set: 2, 3, 3, 3, 5, 7, 7, 8, what is the mode?

10 / 20

In a frequency distribution, the arithmetic mean is represented as:

11 / 20

The arithmetic mean of a data set is calculated as:

12 / 20

Using the formula for the mode, calculate the mode for the given class boundaries: l = 35.95, h = 3, fm = 14, f1 = 10, f2 = 6.

13 / 20

What is the mode in a data set?

14 / 20

Given the following frequency distribution, calculate the mode:

Class Interval   Frequency
0-10   5
10-20   7
20-30   10
30-40   8
40-50   6

15 / 20

Which of the following is a desirable property of the mode?

16 / 20

What is the relationship between mean, median, and mode in a perfectly symmetrical distribution?

17 / 20

In the example of EPA mileage ratings, which value represents the mode if the highest frequency is in the 36.0-38.9 class?

18 / 20

Which of the following is true about the arithmetic mean?

19 / 20

Calculate the arithmetic mean of the following data: 10, 20, 30, 40, 50.

20 / 20

Which measure of central tendency is not affected by skewed distributions?

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LESSON 25

1 / 20

What is the third quartile (Q3) if the total number of observations is 40?

2 / 20

How is the first decile (D1) calculated?

3 / 20

Which class interval contains the median in the EPA mileage rating example?

4 / 20

The first quartile (Q1) divides the dataset into which proportion?

5 / 20

In a positively skewed distribution, which of the following is true?

6 / 20

What is the lower class boundary of the median class in the EPA mileage rating example?

7 / 20

Which empirical relation between the mean, median, and mode is correct?

8 / 20

Why are percentile rankings useful?

9 / 20

In a perfectly symmetrical distribution, how do the mean, median, and mode relate to each other?

10 / 20

Which of the following is not a quantile?

11 / 20

The graphical representation used to locate quantiles is called:

12 / 20

What is the value of n/2 in the EPA mileage rating example?

13 / 20

If the mean of a dataset is 60 and the median is 50, which statement is likely true?

14 / 20

What is the cumulative frequency just before the median class in the EPA mileage rating example?

15 / 20

What is the formula for calculating the median in a frequency distribution?

16 / 20

What is the second quartile also known as?

17 / 20

In an open-ended frequency distribution, which class is often problematic for computing the median?

18 / 20

What does the empirical relation between the mean, median, and mode indicate in a unimodal distribution?

19 / 20

Which of the following is a true statement about percentiles?

20 / 20

The median is preferred over the mean in cases where:

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LESSON 26

1 / 20

What is the geometric mean of the values 2, 4, 8, and 16?

2 / 20

If the arithmetic mean of two numbers is 8 and the harmonic mean is 6, what is the geometric mean?

3 / 20

If a car travels 120 miles at speeds of 30 mph and 60 mph for equal distances, what is the harmonic mean of the speeds?

4 / 20

What is the mid-range of a data set with the smallest value x0 and the largest value xm?

5 / 20

Which formula correctly represents the geometric mean (G)?

6 / 20

What is the harmonic mean of the values 10 and 40?

7 / 20

If the harmonic mean of two numbers is 12 and one of the numbers is 18, what is the other number?

8 / 20

Which measure of central tendency is most affected by extreme values?

9 / 20

When is the harmonic mean most appropriate to use?

10 / 20

Which formula correctly represents the harmonic mean (HM)?

11 / 20

When dealing with average speeds, why is the harmonic mean preferred over the arithmetic mean?

12 / 20

What is the relationship between arithmetic mean (AM), geometric mean (GM), and harmonic mean (HM)?

13 / 20

How is the mid-quartile range calculated?

14 / 20

The harmonic mean (HM) of a set of n positive values X1, X2, …, Xn is defined as:

15 / 20

Which of the following measures of central tendency is also known as the mid-hinge?

16 / 20

The geometric mean (G) of a set of n positive values X1, X2, …, Xn is defined as:

17 / 20

For which type of data is the geometric mean most suitable?

18 / 20

Given the values 3, 9, and 27, what is the geometric mean?

19 / 20

What is the logarithm of the geometric mean (G) of values X1, X2, …, Xn?

20 / 20

In which scenario is the geometric mean preferred over the arithmetic mean?

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LESSON 27

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What does a high coefficient of dispersion indicate about a dataset?

2 / 20

Which measure of dispersion is defined as half the difference between the first and third quartiles?

3 / 20

What is the relative measure of range known as?

4 / 20

Which measure of dispersion is calculated as the average of the absolute deviations from the mean?

5 / 20

Which of the following is a relative measure of dispersion?

6 / 20

Which measure of dispersion takes into account every data point in a dataset?

7 / 20

What is the coefficient of quartile deviation if Q1 = 25 and Q3 = 75?

8 / 20

If the coefficient of dispersion for one dataset is 0.3 and for another dataset is 0.5, what can be inferred?

9 / 20

Which measure of dispersion is most appropriate for skewed data?

10 / 20

Which of the following statements about the quartile deviation is true?

11 / 20

If the range of a dataset is 50 and the mid-range is 30, what are the smallest and largest values?

12 / 20

How is the range of a dataset calculated?

13 / 20

Which of the following is NOT true about the range as a measure of dispersion?

14 / 20

What does the term "dispersion" refer to in statistics?

15 / 20

Which of the following is an absolute measure of dispersion?

16 / 20

What is the formula for calculating the coefficient of dispersion?

17 / 20

What is the main disadvantage of using the range as a measure of dispersion?

18 / 20

What is the coefficient of quartile deviation if Q1 = 40 and Q3 = 80?

19 / 20

In which scenario is the quartile deviation particularly useful?

20 / 20

If Q1 = 20 and Q3 = 60, what is the quartile deviation?

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LESSON 28

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Which of the following is a measure of relative dispersion?

2 / 20

If the mean of a dataset is 40 and the standard deviation is 8, what is the coefficient of variation?

3 / 20

What is the mean deviation of the number of fatalities in motorway accidents for the given data set?

4 / 20

What is the variance of the number of fatalities in the given data set?

5 / 20

Which of the following statements about standard deviation is true?

6 / 20

How is the variance of a dataset calculated?

7 / 20

What does a high coefficient of variation indicate?

8 / 20

What is the shortcut formula for calculating the standard deviation in case of grouped data?

9 / 20

Which formula is used for calculating the standard deviation in case of grouped data?

10 / 20

What is the variance of a dataset if the standard deviation is 5?

11 / 20

What is the coefficient of variation?

12 / 20

Which of the following measures of dispersion is also known as mean absolute deviation?

13 / 20

Which measure of dispersion involves each and every data value in its computation?

14 / 20

What does a high standard deviation indicate about a dataset?

15 / 20

What is the formula for the mean deviation of a frequency distribution?

16 / 20

For which purpose is the coefficient of variation particularly useful?

17 / 20

What is the primary advantage of using the standard deviation over the range?

18 / 20

What is the standard deviation of the number of fatalities in the given data set?

19 / 20

Which measure of dispersion is represented in the same unit as the original data?

20 / 20

What is the coefficient of variation for a dataset with a mean of 50 and a standard deviation of 10?

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LESSON 29

1 / 20

Which of the following best describes a simple event?

2 / 20

What is the sample space for rolling a six-sided die?

3 / 20

What is the probability of drawing an Ace from a standard deck of 52 cards?

4 / 20

What is the probability of drawing an Ace or a King from a standard deck of 52 cards?

5 / 20

What is the number of ways to select 3 officers (president, secretary, and treasurer) from a club of 4 members?

6 / 20

What is the formula for the number of permutations of r objects selected from n distinct objects?

7 / 20

Which of the following events are mutually exclusive?

8 / 20

Which of the following sets of events is exhaustive in a single toss of a fair coin?

9 / 20

How many ways can a committee of 4 be formed from a group of 10 people?

10 / 20

What is the complement of the event 'drawing a heart' from a deck of 52 cards?

11 / 20

What is the number of permutations of the letters in the word 'SCHOOL'?

12 / 20

In a standard deck of 52 cards, what is the probability of drawing either a king or a diamond?

13 / 20

What is the probability of drawing either a heart or a spade from a standard deck of 52 cards?

14 / 20

If the outcome of a random experiment is unpredictable, the experiment is called:

15 / 20

If two events cannot occur simultaneously, they are said to be:

16 / 20

What is the number of ways to draw a hand of 5 cards from a well-shuffled ordinary deck of 52 cards?

17 / 20

If events A and B are such that P(A) = 0.5 and P(B) = 0.3, and A and B are mutually exclusive, what is P(A or B)?

18 / 20

What is the sample space for tossing a coin twice?

19 / 20

How many permutations can be made from the word 'committee'?

20 / 20

What is the number of combinations of 3 objects selected from 10 distinct objects?

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LESSON 30

1 / 20

What is the probability that a randomly drawn card from a standard deck of 52 cards is a 10?

2 / 20

If a fair coin is tossed three times, what is the probability of getting at least one head?

3 / 20

In a relative frequency approach, if an event occurs 30 times out of 100 trials, what is the probability of the event?

4 / 20

Which concept states that the probability of an event is the limit of its relative frequency as the number of trials approaches infinity?

5 / 20

What does subjective probability rely on?

6 / 20

Which of the following best defines mutually exclusive events?

7 / 20

If a card is drawn from a deck of 52 cards, what is the probability that it is a red card?

8 / 20

Which type of probability involves using prior knowledge and subjective judgment?

9 / 20

Which of the following sets of events is collectively exhaustive?

10 / 20

Which of the following is a characteristic of exhaustive events?

11 / 20

What is the probability of drawing a queen from a standard deck of 52 cards?

12 / 20

What is the classical definition of probability?

13 / 20

What does it mean for events to be equally likely?

14 / 20

Which of the following probabilities is an example of subjective probability?

15 / 20

Which of the following is a limitation of the classical definition of probability?

16 / 20

Which definition of probability is based on the proportion of times an event occurs in a large number of trials?

17 / 20

If an event has a probability of 0.6, what is the probability of its complement?

18 / 20

In the context of probability, what does the term 'sample space' refer to?

19 / 20

What is an example of mutually exclusive events?

20 / 20

When a fair coin is tossed, what is the probability of getting either a head or a tail?

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LESSON 31

1 / 20

If events A and B are mutually exclusive, what is P(A∪B) according to the axiomatic definition of probability?

2 / 20

In the context of probability, what does the symbol P(A∩B) represent?

3 / 20

What does the axiomatic definition of probability require for any event Ei?

4 / 20

In the context of probability, what does the symbol P(A∪B) represent?

5 / 20

Which axiom of the axiomatic definition of probability states that the probability of the sample space S is 1?

6 / 20

What is the probability of drawing either a king or a heart from a standard deck of 52 cards?

7 / 20

What is the relative frequency of male births if there are 359,881 male live births out of a total of 700,335 births?

8 / 20

Which axiom of the axiomatic definition of probability ensures that the probability of a certain event is 1?

9 / 20

If P(A∩B)=0, what does this indicate about events A and B?

10 / 20

Which of the following best describes empirical probability?

11 / 20

According to the axiomatic definition of probability, if P(A∪B)=1, what can be inferred about events A and B?

12 / 20

Which of the following best describes the relative frequency definition of probability?

13 / 20

If events A and B are not mutually exclusive, how do you calculate P(A∪B)?

14 / 20

If a coin is tossed 4 times, what is the probability of getting at least one head?

15 / 20

What does the relative frequency definition of probability rely on?

16 / 20

If A and B are any two events in a sample space S, what is P(A∪B) according to the addition law?

17 / 20

Which of the following is true according to the law of complementation?

18 / 20

What is the probability of getting exactly one head in two flips of a fair coin?

19 / 20

What is the probability of getting heads in a fair coin toss experiment performed 10,000 times with 5067 heads?

20 / 20

What does the law of complementation state?

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LESSON 32

1 / 20

In the context of probability, which of the following best describes the multiplication theorem for dependent events?

2 / 20

What type of probability is represented by the total proportion of male births?

3 / 20

What is the relationship between joint probability, conditional probability, and marginal probability?

4 / 20

After transferring two balls to the second bag, what is the probability of drawing a white ball from the second bag if two white balls were transferred?

5 / 20

What is the probability that a randomly selected birth from the provided data is a female live birth?

6 / 20

If the probability of event A occurring is 0.4 and the probability of event B occurring is 0.5, and A and B are independent events, what is the probability of either event A or event B occurring?

7 / 20

Using the provided data, what is the marginal probability of a live birth?

8 / 20

What is the multiplication theorem of probability for independent events?

9 / 20

What is the probability of event B occurring given that event A has already occurred if events A and B are independent?

10 / 20

Which term refers to the probability of a single event occurring, without consideration of any other events?

11 / 20

What is the conditional probability of stillbirth given that the baby is male, based on the provided data?

12 / 20

What does it mean if the probability of the intersection of two events A and B is zero, P(A∩B)=0?

13 / 20

What are two events A and B called if the occurrence of one event does not affect the probability of the other?

14 / 20

In the example with the two bags of balls, what is the probability of transferring one white ball and one black ball from the first bag to the second bag?

15 / 20

According to the axiomatic definition of probability, if P(A∪B)=1, what can be inferred about events A and B?

16 / 20

What is the probability of getting exactly one head in two flips of a fair coin?

17 / 20

If the probability of event A occurring is 0.5 and the probability of event B occurring is 0.6, and A and B are independent events, what is P(A∩B)?

18 / 20

In the example with two bags of balls, what is the probability that two white balls are transferred from the first bag to the second bag?

19 / 20

If two events A and B are dependent, which of the following is true?

20 / 20

What does the joint probability of events A and B represent?

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/75

PAST PAPERS FOR FINALS

1 / 75

Which of the following is a proposition?

2 / 75

After transferring two balls to the second bag, what is the probability of drawing a white ball from the second bag if two white balls were transferred?

3 / 75

The component bar chart should be used when we have information regarding totals and their components:

4 / 75

The probability of a sure event is:

5 / 75

If E and F are mutually exclusive events such that P(E) = 0.4 and P(F) = 0.5 then the probability of either event occurring is:

6 / 75

The probability of a sure event is:

7 / 75

What is the probability of getting exactly one head in two flips of a fair coin?

8 / 75

According to the axiomatic definition of probability, if P(A∪B)=1, what can be inferred about events A and B?

9 / 75

Probability of a sure event is:

10 / 75

The order of elements in a set does not matter:

11 / 75

What is the conditional probability of stillbirth given that the baby is male, based on the provided data?

12 / 75

If E and F are mutually exclusive events such that P(E) = 0.4 and P(F) = 0.5 then the probability of either event occurring is:

13 / 75

According to De Morgan’s Law, ¬(p∧q)=

14 / 75

What are two events A and B called if the occurrence of one event does not affect the probability of the other?

15 / 75

What is the relationship between joint probability, conditional probability, and marginal probability?

16 / 75

The harmonic mean is the reciprocal of the arithmetic mean of the reciprocals of the values.

17 / 75

In the example with the two bags of balls, what is the probability of transferring one white ball and one black ball from the first bag to the second bag?

18 / 75

Which one of the following is the negation of the statement "If Tanveer lives in Lahore then he does not live in Pakistan"?

19 / 75

The probability of getting a tail when a coin is tossed is:

20 / 75

In the context of probability, which of the following best describes the multiplication theorem for dependent events?

21 / 75

Mode is the measure which always exists in any numerical data:

22 / 75

If events A and B are not mutually exclusive, how do you calculate P(A∪B)?

23 / 75

What does the joint probability of events A and B represent?

24 / 75

The combination of 5 objects taken 2 at a time from a set of 5 distinct objects is:

25 / 75

A conditional statement is logically equivalent to its:

26 / 75

Which measure of central tendency is defined as the value that occurs most frequently in a data set?

27 / 75

List price of a shirt is Rs 450. If the discount rate is 20%, calculate the discount amount of the shirt.

28 / 75

If the probability of event A occurring is 0.4 and the probability of event B occurring is 0.5, and A and B are independent events, what is the probability of either event A or event B occurring?

29 / 75

If P(A∩B)=0, what does this indicate about events A and B?

30 / 75

If two events A and B are dependent, which of the following is true?

31 / 75

The component bar chart should be used when we have information regarding totals and their components:

32 / 75

What is the probability that a randomly selected birth from the provided data is a female live birth?

33 / 75

Standard deviation is always measured from:

34 / 75

The chart below is a ___ graph.

35 / 75

According to the axiomatic definition of probability, if P(A∪B)=1, what can be inferred about events A and B?

36 / 75

The difference between the upper and the lower class boundaries of a class is known as:

37 / 75

If the probability of event A occurring is 0.5 and the probability of event B occurring is 0.6, and A and B are independent events, what is P(A∩B)?

38 / 75

Probability of a sure event is:

39 / 75

What type of probability is represented by the total proportion of male births?

40 / 75

What is the probability of event B occurring given that event A has already occurred if events A and B are independent?

41 / 75

Which one of the following sets is finite?

42 / 75

The number of permutations of 4 objects taken 3 at a time from a set of 5 distinct objects is:

43 / 75

Which term refers to the probability of a single event occurring, without consideration of any other events?

44 / 75

If A is a sure event, then P(A) =

45 / 75

The conjunction of two statements p and q is denoted by:

46 / 75

What is the probability of getting exactly one head in two flips of a fair coin?

47 / 75

In a symmetric distribution:

48 / 75

Which of the following is a proposition?

49 / 75

What is the probability that a randomly selected birth from the provided data is a female live birth?

50 / 75

When relative changes in some variable quantity are to be averaged, which mean is the appropriate average to use?

51 / 75

A major disadvantage of the mean is that it is affected by:

52 / 75

According to the axiomatic definition of probability, if P(A∪B)=1, what can be inferred about events A and B?

53 / 75

If two events A and B are independent, then the probability of both events occurring together is given by:

54 / 75

Which axiom of the axiomatic definition of probability ensures that the probability of a certain event is 1?

55 / 75

The conjunction of two statements p and q is denoted by:

56 / 75

When 3 coins are tossed simultaneously, the probability of getting 3 heads is:

57 / 75

A 20 m long rope is cut to the length of 15m. What is the percentage decrease?

58 / 75

What is the probability of getting an odd number on a fair die?

59 / 75

When two coins are tossed simultaneously, the probability of at least one head is:

60 / 75

What is the simple interest earned on Rs.3000 invested at 8% per annum for 6 months?

61 / 75

What does it mean if the probability of the intersection of two events A and B is zero, P(A∩B)=0?

62 / 75

If A is a sure event, then P(A) =

63 / 75

What is the probability of getting exactly one head in two flips of a fair coin?

64 / 75

If R={(a,a),(b,b),(c,c)} is a relation on the set A={a,b,c}:

65 / 75

The probability of vowel letters from the word STATISTICS is:

66 / 75

In the context of probability, what does the symbol P(A∩B) represent?

67 / 75

A conditional statement is logically equivalent to its:

68 / 75

What does it mean if the probability of the intersection of two events A and B is zero, P(A∩B)=0?

69 / 75

In the example with two bags of balls, what is the probability that two white balls are transferred from the first bag to the second bag?

70 / 75

The sample space of rolling two dice consists of how many possible outcomes?

71 / 75

If A = {1, 2, 3, 4} and B = {5, 6, 7}, then A∩B is:

72 / 75

Third Quartile = Q3 =

73 / 75

Smoking habits of the residents of an area are:

74 / 75

Using the provided data, what is the marginal probability of a live birth?

75 / 75

What is the multiplication theorem of probability for independent events?

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MATH-001 EXAM 1

1 / 50

What is the definition of an algebraic expression?

2 / 50

What is the face value of a share of stock?

3 / 50

What is the formula to calculate the future value of an annuity?

4 / 50

How do you calculate the compound interest earned on Rs. 750 at 12% per annum for 8 years?

5 / 50

What is the meaning of 'face value' of a stock?

6 / 50

What does the accumulation factor represent in the context of an annuity?

7 / 50

What is the function of a middleman in merchandising?

8 / 50

What is the formula for the price-earnings ratio?

9 / 50

How do you calculate return on investment (ROI)?

10 / 50

What is a dividend?

11 / 50

What is the first step in solving a linear equation?

12 / 50

What is the result of multiplying a matrix by a scalar?

13 / 50

What does the coefficient of dispersion measure?

14 / 50

What is an identity matrix?

15 / 50

What is a middleman in the context of merchandising?

16 / 50

What is the discounted value or present worth of an annuity?

17 / 50

How is the percent increase in price calculated?

18 / 50

How do you add two matrices?

19 / 50

How do you calculate the simple interest for a principal of Rs. 500 at a rate of 11% for 4 years?

20 / 50

What is the requirement for multiplying two matrices?

21 / 50

How is multiplication of polynomials typically represented?

22 / 50

How do you calculate the future value of an annuity?

23 / 50

In a punch recipe, the ratio of mango juice, apple juice, and orange juice is 3:2:1. If you have 1.5 liters of orange juice, how much punch can you make?

24 / 50

What is a polynomial?

25 / 50

How is earnings per share (EPS) calculated?

26 / 50

How many liters of mango juice are required if the punch ratio of mango juice, apple juice, and orange juice is 3:2:1.5 and you have 500 liters of orange juice?

27 / 50

What is the purpose of using matrices in business?

28 / 50

What is the formula to calculate the present value of an annuity?

29 / 50

What are current assets?

30 / 50

What does the discounted value of an annuity represent?

31 / 50

What is the multiplicative inverse of a real number x?

32 / 50

What does a 50% dividend mean if the face value of a share is Rs 10?

33 / 50

How do you simplify (48a – 32ab)/8a?

34 / 50

How is the net price calculated after a trade discount?

35 / 50

How do you calculate the net price after a trade discount?

36 / 50

What is the definition of a stock?

37 / 50

How many students failed if the ratio of passing to failing grades is 7:5 in a class of 36 students?

38 / 50

How is simple interest calculated?

39 / 50

What is the formula for calculating the accumulation factor?

40 / 50

How is the stock yield typically represented?

41 / 50

What is the primary purpose of calculating return on investment (ROI)?

42 / 50

What is the compound interest formula?

43 / 50

What is a discount factor?

44 / 50

What is the purpose of a trade discount?

45 / 50

If a bank increases its interest rate from 7% to 9%, what is the percent increase in the interest rate?

46 / 50

What is a proportion in mathematics?

47 / 50

What is an annuity?

48 / 50

What is the term for a series of fixed payments made at regular intervals?

49 / 50

If a bank reduces its interest rate from 9% to 7%, what is the percent decrease in the interest rate?

50 / 50

What is the term for stock currently held by investors?

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/50

MATH-001 EXAM 1

1 / 50

How many liters of mango juice are required if the punch ratio of mango juice, apple juice, and orange juice is 3:2:1.5 and you have 500 liters of orange juice?

2 / 50

What is the term for a series of fixed payments made at regular intervals?

3 / 50

What is the formula to calculate the future value of an annuity?

4 / 50

What is the multiplicative inverse of a real number x?

5 / 50

What is the formula for the price-earnings ratio?

6 / 50

How is earnings per share (EPS) calculated?

7 / 50

What is the discounted value or present worth of an annuity?

8 / 50

How do you add two matrices?

9 / 50

What is the function of a middleman in merchandising?

10 / 50

How do you calculate return on investment (ROI)?

11 / 50

How is the net price calculated after a trade discount?

12 / 50

What is the definition of an algebraic expression?

13 / 50

What is the requirement for multiplying two matrices?

14 / 50

What does the discounted value of an annuity represent?

15 / 50

What is the term for stock currently held by investors?

16 / 50

What is a polynomial?

17 / 50

In a punch recipe, the ratio of mango juice, apple juice, and orange juice is 3:2:1. If you have 1.5 liters of orange juice, how much punch can you make?

18 / 50

How is the percent increase in price calculated?

19 / 50

What is an identity matrix?

20 / 50

What is the result of multiplying a matrix by a scalar?

21 / 50

What is a middleman in the context of merchandising?

22 / 50

How is multiplication of polynomials typically represented?

23 / 50

What is the compound interest formula?

24 / 50

What is a discount factor?

25 / 50

How do you calculate the simple interest for a principal of Rs. 500 at a rate of 11% for 4 years?

26 / 50

What is the face value of a share of stock?

27 / 50

If a bank reduces its interest rate from 9% to 7%, what is the percent decrease in the interest rate?

28 / 50

How many students failed if the ratio of passing to failing grades is 7:5 in a class of 36 students?

29 / 50

How do you simplify (48a – 32ab)/8a?

30 / 50

What is a proportion in mathematics?

31 / 50

What is the formula to calculate the present value of an annuity?

32 / 50

How do you calculate the compound interest earned on Rs. 750 at 12% per annum for 8 years?

33 / 50

What is the primary purpose of calculating return on investment (ROI)?

34 / 50

What is the meaning of 'face value' of a stock?

35 / 50

If a bank increases its interest rate from 7% to 9%, what is the percent increase in the interest rate?

36 / 50

What does the coefficient of dispersion measure?

37 / 50

What is the definition of a stock?

38 / 50

What is a dividend?

39 / 50

How do you calculate the future value of an annuity?

40 / 50

What is the first step in solving a linear equation?

41 / 50

What is the formula for calculating the accumulation factor?

42 / 50

What is the purpose of using matrices in business?

43 / 50

What does a 50% dividend mean if the face value of a share is Rs 10?

44 / 50

What are current assets?

45 / 50

What does the accumulation factor represent in the context of an annuity?

46 / 50

What is an annuity?

47 / 50

How do you calculate the net price after a trade discount?

48 / 50

How is simple interest calculated?

49 / 50

How is the stock yield typically represented?

50 / 50

What is the purpose of a trade discount?

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Mid Term Past Papers

MTH-PAST PAPERS

1 / 77

For sets A and B, if A ⊂ B , then

2 / 77

A relation R on the set of Natural numbers N is defined as:
For all , aRb iff is odd. Is R reflexive?

3 / 77

Let ( a_1, a_2, a_3, ldots, a_n ) be an arithmetic sequence. Is the sum of the sequence ( S_n = frac{n}{2}(a_1 + a_n) ) true?

4 / 77

The nth term of a Geometric Progression (G.P.) is:

5 / 77

?  {x}

6 / 77

The nth term of an A. P (Arithmetic Progression) is:

7 / 77

Let ( A = {1, 2, 3} ) and ( B = {{1,2}, 3} ). Is ( A cup B = {1, 2, 3} ) true?

8 / 77

Is the singleton set ( {x} ) a subset of itself?

9 / 77

The nth term of an Arithmetic Progression (A.P.) is:

10 / 77

In Sets ordered pairs order of elements doesn’t matters.

11 / 77

From the truth table, for ( p leftrightarrow q ) to be true, both ( p ) and ( q ) must have the same truth values.

12 / 77

Which relation defines a function from X = {2, 4, 5} to Y = {1, 2, 4, 6}?
R1 = {(2, 4), (4, 1)}
R2 = {(2, 4), (4, 1), (4, 2), (5, 6)}
R3 = {(2, 4), (4, 1), (5, 6)}

13 / 77

Let A= {0,1,2,3,} and R is a relation defined on A such that R = { (0,0)) (0, 1), (1, 1) ( 1,0) (2,2) (0,3) (3,3) (3,0) } Is R reflexive and symmetric but not Transitive.

14 / 77

An argument is invalid if the conclusion is false when all the premises are:
A) 2
B) False
C) True
D) 3

15 / 77

The above relation shows _______________.

16 / 77

Let p be the statement ‘you study’ and q be the statement ‘you pass the exams’. Express the following proposition as an English sentence: ( p → q )
A) If you study, then you pass the exams.
B) If you do not study, then you pass the exams.
C) If you study, then you do not pass the exams.
D) If you do not study, then you do not pass the exams.

17 / 77

Given an initial value of 1200 and a final value of 1500, which represents an increase of 300, what is the percentage change?

18 / 77

The nth term of an A. P (Arithmetic Progression) is:

19 / 77

Let A = {1, 2, 3} and B = {{1,2}, 3} then A  B = {1, 2, 3}

20 / 77

The above relation shows _______________.

21 / 77

?% of 360=129.6

22 / 77

A ∩ B is a ---------of A.

23 / 77

Let T be a relation from set A to B, if T is symmetric then its inverse will be:
A) Symmetric
B) Antisymmetric
C) Transitive

24 / 77

Let a1,a2,a3,…,an be an arithmetic sequence, then sum of the sequence Sn =

25 / 77

If A is a subset of the universal set U, then:
A) f
B) A
C) U

26 / 77

If A = {1, 2, 3, 4} then A is proper subset of A.

27 / 77

An argument is invalid if the conclusion is false when all the premises are:
A) False
B) True

28 / 77

If ( p ) is a proposition, its negation is denoted by:

29 / 77

Which of the relations defines a function from X = {2, 4, 5} to Y = {1, 2, 4, 6}?
A) R1 = {(2, 4), (4, 1)}
B) R2 = {(2, 4), (4, 1), (4, 2), (5, 6)}
C) R3 = {(2, 4), (4, 1), (5, 6)}
D) R2 and R3

30 / 77

Let A = {1, 2, 3, 4} and B = {5, 6, 7}, then ( A ∩ B ) =
A) {1, 2, 3, 4}
B) {5}
C) {1, 2, 3, 4, 5, 6, 7}

31 / 77

The sets {1, b} and {b, 1} are equal.
A) True
B) False

32 / 77

In ordered pairs, does the order of elements matter?

33 / 77

The percentage change is calculated as:

34 / 77

The intersection of sets ( A ) and ( B ), denoted by ( A cap B ), is a _______ of ( A ).

35 / 77

An argument is valid if the conclusion is true when all the premises are:
A) True
B) Not given
C) False

36 / 77

Given two sets ( A ) and ( B ), if ( A subseteq B ), then which of the following statements is true?

37 / 77

In ordered pairs order of elements matters.

38 / 77

If the initial value is 120 and the final value is 200, representing an increase of 80, what is the percentage change?

39 / 77

{x}⊂ {x}

40 / 77

If ( A = {1, 2, 3, 4} ), then is ( A ) a proper subset of itself?

41 / 77

The text concatenation operator is used to

42 / 77

What percent of 36 is 5?

43 / 77

The final statement is called…….

44 / 77

Consider the following diagram
Then the shaded region is

45 / 77

Union of any two sets satisfies commutative law.

46 / 77

Let A = {1, 2, 3} and B = {{1,2}, 3} then A  B

47 / 77

If A = {1, 2, 3, 4} then A is proper subset of A.

48 / 77

Let f : R R  be defined by f (x) = 4x +2 Prove that f is an onto function.

49 / 77

Find the sum of first five terms of following geometric series:
1 + 4 + 16 +

50 / 77

Which of them is a statement?

51 / 77

The proposition ( p ↔ q ) is called:
A) Negation
B) Conjunction
C) Disjunction
D) Biconditional

52 / 77

If a relation is given by R = {(0, 1), (1, 2), (3, 4)}, then the range of R is:
A) {0, 1, 3}
B) {1, 2, 3}
C) {2, 3, 4}
D) {1, 2, 4}

53 / 77

Initial value = 120
Final value = 200
Increase = 80
% Change =

54 / 77

Television sale $300.The sale price 20% less than regular price. What is regular price?

55 / 77

According to De Morgan’s law:
~ (p ∧ q) ≡
A) ~ p ∧ q
B) ~ p ∧ ~ q
C) ~ p ∨ ~ q
D) p ∧ ~ q

56 / 77

From the truth table, for p q to be true, if both p and q must have the same truth values.

57 / 77

The nth term of an G. P (Geometric Progression) is:

58 / 77

The above relation shows _______________.

59 / 77

How many functions are there from set A to set B?
A) M.n
B) n.m

60 / 77

Conjunction of two statements p and q is denoted by

61 / 77

If p is a proposition then its negation is denoted by

62 / 77

Name the quadrant in which these points is located.
1. (5, 2)
2. (-3, -1)
3. (-2, 3)
4. (6, 0)
5. (0, -2)
6. (4, -3)

63 / 77

Initial value = 1200
Final value = 1500
increase= 300
% Change =

64 / 77

The conjunction of two statements ( p ) and ( q ) is denoted by:

65 / 77

What is the negation of the statement ‘Today is Friday’?
A) Today is Saturday.
B) Today is not Friday.
C) Today is Thursday.

66 / 77

For the two sets A and B, A ∩ B is a _______ of A:
A) Super set
B) Subset
C) Complement set

67 / 77

Percentage change =

68 / 77

If A and B are two sets such that A ∩ B = A ∪ B, then:
A) A and B are disjoint sets
B) A and B are super sets
C) A and B are equal sets
D) Order of elements in a set does not matter

69 / 77

Find the sum of first five terms of following geometric series:
1 + 4 + 16 + ……………

70 / 77

Let A= {a, b, c} and R is a relation defined on A such that R = {(a, b), (b, a), (a, a)}
Is R reflexive and symmetric? Justify your answer.

71 / 77

If no element is common in two sets A and B, then the sets are called:
A) Exhaustive sets
B) Dissimilar sets
C) Disjoint sets

72 / 77

If P and Q are proposition, P is true and Q is false, then P → Q is

73 / 77

If R and S are reflexive, then R ∩ S is:
A) Reflexive
B) Transitive
C) Symmetric

74 / 77

Consider the Venn diagram below, where the shaded area represents a particular set operation. Which of the following options correctly identifies the shaded region?

75 / 77

Let a1,a2,a3,…,an be an arithmetic sequence, then sum of the sequence Sn = n(a +a ) / 2

76 / 77

If ( P ) and ( Q ) are propositions with ( P ) being true and ( Q ) being false, what is the truth value of ( P rightarrow Q )?

77 / 77

Let f : → be defined by f (x) = 2x - 3
Show that f is an onto function.

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Quiz 1 - SPRING 2024

MTH 001 Quiz

1 / 18

(Amna is good in English grammar) denoted by S and (Amna is good in philosophy.) denoted by N. The statement that (Amna is not good in English neither in philosophy) can be translated in logical expression.

 

2 / 18

If A is a subset of B then A intersection B = ___.

 

3 / 18

Consider that q and p are statements such that q→p is false. Then the truth value of q→∼p is __________.

 

4 / 18

If there are two variables p and q in a logical statement then the truth table will have _________ rows.

 

5 / 18

The set of English alphabets is _____________.

 

6 / 18

Sets Z and N are infinite

 

7 / 18

Let p and q be two statements, then the conjunction of p and q is denoted as ____________.

 

8 / 18

Consider that q and p are statements such that q→p is false. Then the truth value of q∨p is ….

 

9 / 18

A tautology is a statement form that is ____________ regardless of the truth values of the statement variables.

 

10 / 18

How is a set defined in mathematics?

 

11 / 18

The negation of (x < 3) is:

12 / 18

An argument is invalid if all the premises are true but the …. Is false

 

13 / 18

How we can translate P or Q and not p and q in logical expression

 

14 / 18

The Set of Natural Numbers starts from:

 

15 / 18

The set of rational numbers between 5 and 9 is __________.

 

16 / 18

The truth value of ~p∧q when p is true and q is false is:

 

17 / 18

Translate the statement “If you have fever, then you will miss the final match” in logical expression.

 

18 / 18

An Argument is valid if the conclusion is true when all the premises are false.

 

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Quiz 02

MTH 001 Quiz 2

1 / 20

Every relation is a function.

2 / 20

If f is a function from A={a,b,c} to B={1,2,3} such that f={(a,3),(b,1),(c,2)}, then f is called _____.

3 / 20

Given that AxA = { (1,1), (1,2), (1,3), (2,1), (2,2), (2,3), (3,1), (3,2), (3,3)}, which of the given relations is a function?

4 / 20

How many functions are there from a set with two elements to a set with three elements?

5 / 20

Let N = {-5, -4, 0, 6, 8} and O = {-4, 0, 8, 9}. Then N intersection O = _____.

6 / 20

The graph of y=x2 is a straight line.

7 / 20

A function f:X→Y defined by f(x)=a, where a is a constant and x belongs to X, will be one-to-one if and only if ____.

8 / 20

The graph of y=x2 is a straight line.

9 / 20

Let A = {1,2,3}. Which of the following is a symmetric relation on A?

10 / 20

Let R be a relation from A to B. The set of all first elements of the ordered pairs which belong to R is called ____ of R.

11 / 20

Which of the following relations on the set A={1,2,3,4} is transitive?

12 / 20

A relation R on a set A is transitive if and only if:

13 / 20

In the directed graph of a symmetric relation, if there is an arrow going from one point to a second, there must be:

14 / 20

Which of the following relations on the set A={1,2,3,4} is symmetric?

15 / 20

A relation R on a set A is symmetric if and only if:

16 / 20

Consider the relation R on A={1,2,3} represented by the matrix:
[1 1 0
0 1 1
1 0 0]

17 / 20

Which of the following is a directed graph of a reflexive relation?

18 / 20

A relation R on a set A is not reflexive if and only if:

19 / 20

Which of the following relations on the set A={1,2,3,4} is reflexive?

20 / 20

A relation R on a set A is reflexive if and only if:

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MATH QUIZ

1 / 10

The graph of y=x^2 is a straight line.

2 / 10

Every relation is a function.

3 / 10

Let N = {-5, -4, 0, 6, 8} and O = {-4, 0, 8, 9}. Then N intersection O = _______________.

4 / 10

How many functions are there from a set with two elements to a set with three elements?

5 / 10

The graph of y=x^2 is a straight line.

6 / 10

Given that AxA = { (1,1), (1,2), (1,3), (2,1), (2,2), (2,3), (3,1), (3,2), (3,3)}, which of the given relations is a function?

7 / 10

If f is a function from A={a,b,c} to B={1,2,3} such that f={(a,3),(b,1),(c,2)}, then f is called _______________.

8 / 10

Let A = {1,2,3}. Which of the following is a symmetric relation on A?

9 / 10

Let R be a relation from A to B. The set of all first elements of the ordered pairs which belong to R is called ____________ of R.

10 / 10

A function f:X→Y defined by f(x)=a, where a is a constant and x belongs to X, will be one-to-one if and only if ____________.

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