Tag: proofs in mathematics

Quizzes (2) Questions (121)

Questions Related to proofs in mathematics

Multiple choice maths proofs in mathematics implications principle of mathematical induction different forms of theoretical statements

Write the converse and contrapositive of the statement
"If a dog is barking,then it will not bite"
$(i)$Converse of the statement:If a dog will bite then the dog is barking.
$(ii)$Contrapositive of the statement:If a dog will bite then the dog is not barking.

  1. $(i)$True $(ii)$False
  2. $(i)$True $(ii)$True
  3. $(i)$False $(ii)$False
  4. $(i)$False $(ii)$True
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

"If a dog is barking,then it will not bite"
$(i)$Converse of the statement:If a dog will not bite then the dog is barking.
$(ii)$Contrapositive of the statement:If a dog will bite then the dog is not barking.

Multiple choice maths proofs in mathematics implications principle of mathematical induction different forms of theoretical statements

Which of the following is true about the converse and contrapositive of the statement
"If two triangles are congruent, then their areas are equal."
(i) Converse of the statement :
If the areas of the two triangles are equal, then the triangles are congruent.
(ii) Contrapositive of the statement:
If the areas of the two triangles are not equal, then the triangles are not congruent.

  1. (i) True (ii) False

  2. (i) False (ii) True

  3. (i) True (ii) True

  4. (i) False (ii) False
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

"If two triangles are congruent, then their areas are equal."
(i) Converse of the statement :
If the areas of the two triangles are equal, then the triangles are congruent.
(ii) Contrapositive of the statement:
If the areas of the two triangles are not equal, then the triangles are not congruent.

Multiple choice maths proofs in mathematics implications principle of mathematical induction different forms of theoretical statements

$p\wedge q$ is logically equivalent to

  1. $\sim(p\rightarrow\sim q)$
  2. $(p\rightarrow\sim q)$
  3. $(\sim p\rightarrow\sim q)$
  4. $(\sim p\rightarrow q)$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation
$p$ $q$ $p\wedge q$ $\sim(p\rightarrow\sim q)$
$T$ $T$ $T$ $T$
$T$ $F$ $T$ $T$
$F$ $F$ $T$ $T$
$F$ $T$ $T$ $T$

$(p\wedge q)\longrightarrow $$\sim(p\rightarrow\sim q)$ is a tautology.

Multiple choice maths proofs in mathematics implications principle of mathematical induction different forms of theoretical statements

The contrapositive of $p \to \left( { \sim q \to  \sim r} \right)$ is 

  1. $\left( { \sim q \wedge r} \right) \to \sim p$
  2. $\left( {q \wedge \sim r} \right) \to \sim p$
  3. $p \to \left( { \sim r \vee q} \right)$
  4. $p \wedge \left( {q \vee r} \right)$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Contraceptive of $a \rightarrow b$ is $\sim b \rightarrow \sim a$


Then

Contraceptive of $p \rightarrow (\sim q \rightarrow \sim r )$

$\equiv \sim (\sim q \rightarrow \sim r) \rightarrow \sim p$

$\equiv \sim (q \wedge \sim r) \rightarrow \sim p [a \rightarrow b \equiv \sim a \wedge b]$

$\equiv (\sim q \vee r) \rightarrow \sim p $  [Demorgas law]

$A$ is correct.

Multiple choice maths proofs in mathematics implications principle of mathematical induction different forms of theoretical statements

Statement I : if p is false statement and q is true statement, then $ \sim \,p\, \wedge \,q$ is true
Statement II : $ \sim \,p\, \wedge \,q$ is equivalent to $ \sim \left( {pV \sim \,q\,} \right)$

  1. Statement I is true and Statement II is the correct explanation for statement.

  2. Statement I is true and Statement II is true. Statement II is not the correct explanation for the Statement I.

  3. Statement I is true but Statement II is false.

  4. Statement I is false but Statement II is true
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation
p q $ \sim p$ $ \sim q$ (i)$ \sim \,p\, \wedge \,q$ $p\,\, \vee \, \sim \,q$ (ii)$ \sim \left( {p\,\, \vee \, \sim \,q} \right)$
T T F F F T F
T F F T F T F
F T T F T F T
F F T T F T F

p is false
q is true
$ \sim p\, \wedge \,q$ is true
from (i) & (ii)
$ \sim p\, \wedge \,q$ = $ \sim \left( {p\,\, \vee  \sim \,\,q} \right)$
Statement (i) & (ii) is correct 
$ \sim \left( {p\,\, \vee  \sim \,\,q} \right)\, = \, \sim \,p\, \wedge \, \sim \,\left( { \sim \,q} \right)$
$ = \, \sim \,p\, \wedge \,\,q$

Multiple choice maths proofs in mathematics implications principle of mathematical induction different forms of theoretical statements

$ \sim (p \vee q) \vee ( \sim p \wedge q)$ is logically euivalent to 

  1. $ \sim p$

  2. p

  3. q
  4. $ \sim q$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

${ \sim  }({ p }{ \vee  }{ q }){ \vee  }({ \sim  }{ p }{ \wedge  }{ q })\ ={ \sim  }({ U }){ \vee  }({ q }{ \wedge  }{ q })\ ={ \phi  }{ \vee  }({ q })\ ={ q }$


Where U is universal set , $\phi $ is nulll set and p and q are two disjoint sets 
Correct option is C

Multiple choice maths proofs in mathematics implications principle of mathematical induction different forms of theoretical statements

Cost of a diamond varies directly as the square of its weight.A diamond broke into four pieces with their weight in the ratio $1:2:3:4$ If the loss in the total value of the diamond was $Rs.\ 70000$. Find the prices of the original diamond.

  1. $10000\ Rs$
  2. $100000\ Rs$
  3. $1000000\ Rs$
  4. $2000000\ Rs$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Let, the weights of the four pieces of the diamond $x,2x,3x,4x$ respectively.

$\therefore$ Total weight of the original diamond$=x+2x+3x+4x=10x$
Let, price of the original diamond$=k{ (100{ x }^{ 2 }) },where\quad k\quad is\quad constant$
$\therefore$ Cost of four pieces$=k({ x }^{ 2 }+4{ x }^{ 2 }+9{ x }^{ 2 }+16{ x }^{ 2 })\ =k(30{ x }^{ 2 })$
Loss in the total value of the diamond,
$k(100{ x }^{ 2 })-k(30{ x }^{ 2 })=70000\ \Rightarrow k(70{ x }^{ 2 })=70000\ \Rightarrow k{ x }^{ 2 }=Rs.1000$
Hence, price of the original diamond$=k(100{ x }^{ 2 })\ =1000\times 100=Rs.100000.$