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Questions Related to maths

Multiple choice irrational numbers irrational numbers and their operations rational and irrational numbers real numbers (rational and irrational numbers) maths

State whether the given statement is true/false.
An irrational number between two numbers $\dfrac{1}{7}$ and $\dfrac{2}{7}$ is $0.1501500 15000...$ .

  1. True

  2. False
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation
Let us first find the decimal forms of the given numbers as follows: 
 
$\dfrac { 1 }{ 7 } =0.\overline { 142857 } ,\dfrac { 2 }{ 7 } =0.\overline { 285714 }$

We find a number which is non-terminating non-recurring lying between them.
So, we can find infinite many such numbers. For example, $0.150150015000...$ and $0.20200200020000....$

Hence, an irrational number between two numbers $\dfrac {1}{7}$ and $\dfrac {2}{7}$ is $0.150150015000...$
Multiple choice maths set language different sets de morgan's law de morgan's law for set theory

Let $A$ and $B$ are two finite sets such that $n(A)=3$ and $n(B)=4$ then  the number of elements in $A\Delta B$.

  1. $2$
  2. $7$
  3. $5$

  4. can not be determined
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

Now, we have,

$A\Delta B=(A-B)\cup(B-A)$.
But it is impossible to find the number of elements in the set $A\Delta B$ as the sets $A$ and $B$ are not given explicitly. 

Multiple choice maths set language different sets de morgan's law de morgan's law for set theory

$A\cup B=A\cap B$ if and only if

  1. A is an empty set

  2. B is an empty set

  3. Both A and B are empty sets

  4. Both A and B are non-empty sets
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Solution:- Lets assume A is an empty set and B=$\left{ a,b \right}$

Now $A\cup B=\left{ a,b \right}$  and $A\cap B=\oslash $, so in all cases other than C , the condition is not satisfied. So C is the correct answer.

Multiple choice maths set language different sets de morgan's law de morgan's law for set theory

If A and B be two sets such that n(A) = 15, n(B) =25, then number of possible values of $n(A\Delta B)$(symmetric difference of  A and B) is

  1. 30

  2. 16

  3. 26

  4. 40
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation
$n(A \triangle B)= n (A \cup B)- n (A \cap B)$
for $n$ (A \triangle B)$ to be max. $n (A \cap B)=0$
We know, that 
$n(A \cup B)= n (A)+ n (B)- n (A \cap B) = 15+25-0=40$
$\Rightarrow n (A \triangle B)_{max} = 40-0 =40$
For minimum value of $n (A \triangle B)$
$n (A \cup B)$ should be min, $n (A \cap B)$ should be max.
$n (A \triangle B)$ min $=25-15= 10$
So. value of 
$n (A \triangle  B)= n (A \cup B)- n(A \cap B)$ lies om the set
${10,11,12,......, 3,9,40}$
Now, when $n (A \triangle B)$ is max. i.e. when 
$n( A \cup B )=40$ & $n (A \cap B)=0$
If we decrease $n (A \cup B)$ by $1$ then $n (A \cap B)$
Will increase by $1$
$n (A \triangle B)=39-1= 38$
Similarly on for the decrease of $1$ you will get in $(A \triangle B)$ as $36$ and $30$ so on.
Hence 
Range of $n (A \triangle B)= {10,12,14,16,18,20,......,38,40}$ 
$=16$ values 
Multiple choice maths set language different sets de morgan's law de morgan's law for set theory

If $A=\left {x\epsilon C: x^2=1\right }$ and $B=\left {x\epsilon C: x^4=1\right }$, then $A\Delta B$ is equal to

  1. $\left \{-1, 1\right \}$
  2. $\left \{-1, 1, i, -i\right \}$
  3. $\left \{-i, i\right \}$

  4. None of these
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

$x^2=1\Rightarrow x=-1, 1.\therefore A=\left {-1, 1\right }$
$x^4=1\Rightarrow x^2=-1, 1$
$\Rightarrow x=-i, i, -1, 1.\therefore B=\left {-i, i, -1, 1\right }$
$\therefore A\Delta B=(A-B)\cup (B-A)=\phi \cup \left {-i, i\right }=\left {-i, i\right }$.

Multiple choice maths part number dividing fractions division of a fractions division of a fraction

Evaluate: $\dfrac{(3\dfrac{2}{3})^{2} -(2\dfrac{1}{2})^{2}}{(4\dfrac{3}{2})^{2} -(3\dfrac{1}{3})^{2}}$ $\div$ $\dfrac{3\dfrac{2}{3} -2\dfrac{1}{2}}{4\dfrac{3}{2} -3\dfrac{1}{3}}$

  1. $\dfrac{35}{53}$
  2. $\dfrac{37}{53}$
  3. $\dfrac{42}{59}$
  4. $\dfrac{47}{60}$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

$\dfrac{(3\frac{2}{3})^2-(2\frac{1}{2})^2}{(4\frac{3}{2})^2-(3\frac{1}{3})^2} \div \dfrac{3\frac{2}{3}-2\frac{1}{2}}{4\frac{3}{2}-3\frac{1}{3}}$


$=\dfrac{(3\frac{2}{3}+2\frac{1}{2})(3\frac{2}{3}-2\frac{1}{2})}{(4\frac{3}{2}+3\frac{1}{3})(4\frac{3}{2}-3\frac{1}{3})} \div \dfrac{3\frac{2}{3}-2\frac{1}{2}}{4\frac{3}{2}-3\frac{1}{3}}$

$=\dfrac{3\frac{2}{3}+2\frac{1}{2}}{4\frac{3}{2}+3\frac{1}{3}}$

$=\dfrac{\dfrac{11}{3}+\dfrac{5}{2}}{\dfrac{11}{2}+\dfrac{10}{3}}$

$=\dfrac{\dfrac{22+15}{6}}{\dfrac{33+20}{6}}$

$=\dfrac{37}{53}$

Multiple choice maths part number dividing fractions division of a fractions division of a fraction

Solve $\left[\dfrac{170}{3} +\dfrac{6}{7}\right] \div \left[\dfrac{2}{7} \times \dfrac{11}{2}\right]$

  1. $\dfrac{1208}{3\times 11}$
  2. $\dfrac{1208}{11}$
  3. $\dfrac{1208}{3}$
  4. $\dfrac{1208}{9\times 11}$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

$\left[\dfrac{170}{3} +\dfrac{6}{7}\right] \div \left[\dfrac{2}{7} \times \dfrac{11}{2}\right]$


$=\left[ \dfrac{1190+18}{21}\right] \div \left[ \dfrac{11}{7}\right]$


$=\left[ \dfrac{1190+18}{21}\right] \times \left[ \dfrac{7}{11}\right]$

$=\left[ \dfrac{1208}{11\times 3}\right]$

Multiple choice maths part number dividing fractions division of a fractions division of a fraction

Divide the difference of $\dfrac{1}{5}$ and $\dfrac{2}{7}$ by $\dfrac{2}{7}$.

  1. $\dfrac{1}{10}$
  2. $\dfrac{21}{10}$
  3. $\dfrac{3}{10}$
  4. $\dfrac{4}{10}$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Difference of  $\dfrac{1}{5}$  and $\dfrac{2}{7}$ is  


$\dfrac{2}{7} - \dfrac{1}{5} = \dfrac{2\times5 - 1\times7}{35} =\dfrac{10 - 7}{35} =\dfrac{3}{35}$

Now divide $\dfrac{3}{35}$  by $\dfrac{2}{7}$  

we have, $\dfrac{3}{35}÷\dfrac{2}{7} =\dfrac{3}{35}\times \dfrac{7}{2} =\dfrac{3}{10}$

Ans: $\dfrac{3}{10}$

Multiple choice maths part number dividing fractions division of a fractions division of a fraction

Simplify $35\times 6\dfrac{1}{14}$(approximately)$=$

  1. $220.5$
  2. $220$
  3. $212$
  4. $231$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation
$6 \cfrac{1}{14} = \cfrac{14 \times 6 + 1}{14} = \cfrac{85}{14}$
$\therefore \; 35 \times 6\cfrac{1}{14} = 35 \times \cfrac{85}{14} = \cfrac{5 \times 85}{2} = \cfrac{425}{2} = 212.5 \approx 212$
Hence, 212 is the correct answer.
Multiple choice maths part number dividing fractions division of a fractions division of a fraction

A ribbon of length $5\dfrac{1}{4}$m is cut in to small pieces each of length $\dfrac{3}{4}$ m number of pieces will be

  1. $5$
  2. $6$
  3. $7$
  4. $8$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

$ \Rightarrow  $ let length l $ = 5\dfrac{1}{4}$ m 

$ l = 5+\dfrac{1}{4} = \dfrac{21}{4}m $

& Small n pieces of length $ x = \dfrac{3}{4}$m 

So, $ l = nx $

$ n = \dfrac{l}{x} = \dfrac{21}{4}\times \dfrac{4}{3} = 7 $

So, there are 7 pieces of length $ \dfrac{3}{4} $