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

Let p and q be any two logical statements and $r : p \rightarrow (\sim p \vee q)$. If r has a truth value F, then the truth values of p and q are respectively

  1. F, F

  2. T, T

  3. F, T

  4. T, F
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation
p q $\sim$p $\sim$ p $\vee$ q r
T T F T T
F F T T T
T F F F F
F T T T T

$\therefore$ Clearly from above able, If r has a truth value F, then the truth values of p and core T and F respectively.

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

State, whether the is given the statement, is True or False.
$\sim [(p \vee \sim q) \rightarrow (p \wedge \sim q)] \equiv (p \vee \sim q) \wedge (\sim \vee q)$

  1. True

  2. False
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation
 $p$  $q$  $\sim q$  $(p\vee \sim q)$  $(p\wedge \sim q)$ $(p\vee \sim q)\rightarrow (p\wedge \sim q)$  $\sim[(p\vee \sim q)\rightarrow (p\wedge \sim q)]$
 $T$  $T$  $F$  $T$  $F$  $F$  $T$
 $T$  $F$  $T$  $T$  $T$  $T$   $F$
 $F$  $T$  $F$  $F$  $F$  $T$   $F$
 $F$  $F$  $T$  $T$  $F$  $F$ $T$
 $p$  $q$  $\sim q$  $(p\vee \sim q)$ $\sim p$  $(\sim p \vee q)$ $(p\vee \sim q)\wedge (\sim p \vee q)$
 $T$  $T$  $F$  $T$  $F$  $T$  $T$
 $T$  $F$  $T$  $T$  $F$  $F$  $F$
 $F$  $T$  $F$   $F$  $T$  $T$  $F$
 $F$  $F$  $T$  $T$  $T$  $T$  $T$
Multiple choice maths proofs in mathematics implications principle of mathematical induction different forms of theoretical statements

State whether the following statements is True or False?
$p \leftrightarrow q \equiv (p \wedge q) \vee (\sim p \wedge \sim q)$

  1. True

  2. False
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation
given statement 
$p\leftrightarrow q=(p\wedge q)\vee (\sim p\wedge \sim q)$
taking RHS
$(p\wedge q)\vee (\sim p\wedge \sim q)$
$(p\wedge q)\vee (\sim(p\wedge  q))$
$(p\wedge q)\wedge(p\wedge  q)$
$p\leftrightarrow q$
It is true
Multiple choice maths proofs in mathematics implications principle of mathematical induction different forms of theoretical statements

Let p,q be statements. Negation of statement $p \leftrightarrow  ~ q$, is

  1. $~ q \rightarrow p$
  2. $ ~ p v q$
  3. $p \leftrightarrow q$
  4. $p \rightarrow q$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation
p q ~q $p \leftrightarrow ~q$ $~(p \leftrightarrow ~q)$ $~q \rightarrow p$ $~p v q$ $p \leftrightarrow q$ $p \rightarrow q$
T T F F T T T T T
T F T T F T F F F
F T T T F T T F T
F F F F T F T T T
Multiple choice maths parts and whole finding the whole when a fraction is given multiplication of a fraction multiplication of a fractions

Which point on the number line most likely represent $-2\cfrac { 5 }{ 8 } $?

  1. On the left of $-3$
  2. On the right of $-2$
  3. Between $-2$ and $-3$
  4. In the middle of $-2$ and $-3$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation
$-2\cfrac { 5 }{ 8 } $

$-2\cfrac { 5 }{ 8 } =(-1)2\cfrac { 5 }{ 8 } $

$=-1\times \cfrac { 21 }{ 8 } $

$=-2.625$

It lies in between $-3$ and $-2$
Multiple choice maths number systems existence of irrational numbers irrational numbers properties of irrational numbers

$A,B,C$ and $D$ are all different digits between $0$ and $9$. If $AB+DC=7B\ (AB,DC$ and $7B$ are two digit numbers), then the value of $C$ is

  1. $0$
  2. $1$
  3. $2$
  4. $3$
  5. $5$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

AB + DC = 7B. (10A + B) + (10D + C) = 70 + B. 10A + 10D + C = 70. A + D + C/10 = 7. Since A, D, C are digits, C must be 0 for the equation to hold with integer digits A and D. If C=0, A+D=7.

Multiple choice maths rational and irrational numbers existence of irrational numbers irrational numbers properties of irrational numbers

If $\sqrt{a}$ is an irrational number, what is a? 

  1. Rational

  2. Irrational
  3. $0$

  4. Real
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Consider the given irrational number$\sqrt{a}$ ,

Definition  of rational number- which number can be write in the form of $\dfrac{p}{q}$ but $q\ne 0$ is called rational number.

Hence, $a=\dfrac{a}{1}$

That why  $a$ is rational number

 

Hence, this is the answer.

Multiple choice maths rational and irrational numbers existence of irrational numbers irrational numbers properties of irrational numbers

Which of the following is irrational

  1. $\sqrt {\dfrac{4}{9}} $
  2. $\dfrac{4}{5}$
  3. $\sqrt 7 $
  4. $\sqrt {81} $
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation
A $=\sqrt{\dfrac{4}{9}}=\dfrac{2}{3}$         Rational

B $=\dfrac{4}{5}$                       Rational

C $=\sqrt7$                     Irrational

D $=\sqrt{81}=9$          Rational