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

$\sqrt{21-4\sqrt{5}+8\sqrt{3}-4\sqrt{15}}=$...........

  1. $\sqrt{5}-2+2\sqrt{3}$
  2. $\sqrt{5}-\sqrt{4}-\sqrt{12}$
  3. $-\sqrt{5}+\sqrt{4}+\sqrt{12}$
  4. $-\sqrt{5}-\sqrt{4}+\sqrt{12}$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

The expression is sqrt(21 - 4*sqrt(5) + 8*sqrt(3) - 4*sqrt(15)). This is of the form sqrt((a+b+c)^2) = |a+b+c|. Expanding (sqrt(5) - 2 - 2*sqrt(3))^2 gives 5 + 4 + 12 - 4*sqrt(5) - 4*sqrt(15) + 8*sqrt(3) = 21 - 4*sqrt(5) + 8*sqrt(3) - 4*sqrt(15). Thus the square root is |sqrt(5) - 2 - 2*sqrt(3)|, which equals -sqrt(5) + 2 + 2*sqrt(3).

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

State whether the following statements are true or false. 
$\sqrt {n}$ is not irrational if n is a perfect square

  1. True

  2. False
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

False ,

$\sqrt{4}=2$ where 2 is a rational number.Here n is perfect square the  $\sqrt{n}$ is rational number 
$\sqrt{5}=2.236..$ is not rational  number But it is irrational number . here n is not a perfect square the  $\sqrt{n}$ is  irrational  number
So $\sqrt{n}$ is not irrational number if n is perfect square

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

If $p$ is prime, then $\sqrt {p}$ is:

  1. Composite number

  2. Rational number

  3. Positive integer

  4. Irrational number
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

SInce, we know that prime numbers are those which are never perfect square and not divisible by any other number except by itself.
which are $2,3,5,7,...$
Clearly, if $p$ is prime then $\sqrt p $ is irrational number.
Option $D$ is correct. 

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

$6+\sqrt{2}$ is a rational number.

  1. True

  2. False
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Let's assume that $6+\sqrt2$ is rational..... 

then 

$6+\sqrt2 = p/q $

$\sqrt2 =( p-6q)/(q) $ 

now take $p-6q$ to be P and $q$ to be Q........where P and Q are integers 

which means, $\sqrt2= P/Q$...... 

But this contradicts the fact that $\sqrt2$ is rational 

So our assumption is wrong and $6+\sqrt2$ is irrational.

Multiple choice maths equation reducing simple equations to simpler form solving linear equations solution of a linear equation in one variable

Sabarmati express take 18 second to pass completely through a stations $162$m long and $15 second $ through another station $120m$ long. The length of the sabarmathi express os 

  1. $132m$
  2. $100m$
  3. $80m$
  4. $90m$
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

Let the length of the train be L and its speed be v. We have (L + 162) / 18 = v and (L + 120) / 15 = v. Equating the two expressions gives 15(L + 162) = 18(L + 120), which simplifies to 3L = 270, giving L = 90 meters.

Multiple choice maths equation reducing simple equations to simpler form solving linear equations solution of a linear equation in one variable

The manufacturer of a certain item can sell all he can produce at the selling price of $Rs. 60$ each. It costs him $Rs. 40$ in materials and labour to produce each item and he has overhead expenses of $Rs. 3000$ per week in order to operate the plant. The number of units he should produce and sell in order to make a profit of at least $Rs\,1000$ per week, is : 

  1. $200$
  2. $250$
  3. $300$
  4. $400$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Let the no of units sold be $x$ per week.

$60x$ = sales should include all the expenses and profit required to balance things out.
$60x = 1000+40x+3000$

$20x = 4000$
$x =200$

Multiple choice maths equation reducing simple equations to simpler form solving linear equations solution of a linear equation in one variable

The value of $x$ for which $\cfrac{x-3}{4}--x< \cfrac{x-1}{2}-\cfrac{x-2}{3}$ and $2-x> 2x-8$

  1. $[-1,10/3]$
  2. $(1,10/3)$
  3. $R$

  4. none of these
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

$\dfrac{x-3}{4}-x<\dfrac{x-1}{2}-\dfrac{x-2}{3}$


$\dfrac{-3{x}-3}{4}<\dfrac{x+1}{6}$


$\implies x>-1$

$2-x>2{x}-8\implies x<\dfrac{10}{3}$

$\implies x\in (-1,10/3)$