Tag: maths

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Multiple choice maths four operations whole number operations on the number line whole numbers on number line introduction to multiples and factors

$n^2+n+1$ is a or an ______ number for all $n\in N$

  1. even

  2. odd

  3. prime

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

Consider $ {n}^{2} + n = n(n+1) $ 

We know that if $ n $ is a number , then $ n  +1 $ will be its consecutive number

And product of a number and its consecutive number is always even. For example, $ 2 \times 3 = 6 ; 9 \times 10 = 90 $

And as  $ {n}^{2} + n$ is an even number.  Then
$ {n}^{2} + n + 1 $ will be the next consecutive number of the even number , which is an odd number.

Hence, $ {n}^{2} + n + 1 $ will always be an odd number for all natural numbers.

Multiple choice maths four operations whole number operations on the number line whole numbers on number line introduction to multiples and factors

Select the correct order for defining the following terms:
I - natural number
II - imaginary number
III - rational number
IV - integer

  1. I, IV, III, II

  2. I, II, III, IV

  3. I, III, II, IV

  4. IV, I, III, II

  5. I, IV, II, III
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation
  • Here natural numbers are subset of integers , integers are subset of rational numbers and rational numbers are subset of imaginary numbers
  • Therefore the correct order of defining them is shown in option $A$
Multiple choice maths four operations whole number operations on the number line whole numbers on number line introduction to multiples and factors

(0 , - 3 ) lies on _______ .

  1. Positive x- axis

  2. Negative x-axis

  3. Positive y-axis

  4. Negative y- axis
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

Given Coordinate of Point $P$ are $(0 , -3)$


$x-coordinate = 0$
$y-coordinate = -3$

$\Rightarrow$ Point $P$ lies of y-axis

Also, As $y-coordinate < 0$
Point $P (0 , -3)$ lies on Negative y- axis.

Multiple choice maths cube and cube root estimation of cube root estimation of cube roots cubes and cube roots

If $x = \sqrt [3]{a + \sqrt {a^{2} - b^{3}}} + \sqrt [3]{a - \sqrt {a^{2} - b^{3}}}$ then $x^{3} + 3bx = $ ____________.

  1. $2a$
  2. $2b$
  3. $3a$
  4. $4a$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation
Given,
$x = \sqrt [3]{a + \sqrt {a^{2} - b^{3}}} + \sqrt [3]{a - \sqrt {a^{2} - b^{3}}}$.......(1).
Now cubing both sides we get,
$x^3=a+\sqrt{a^2-b^3}+a-\sqrt{a^2-b^3}-3$$\sqrt [3]{a + \sqrt {a^{2} - b^{3}}}  \sqrt [3]{a - \sqrt {a^{2} - b^{3}}}$$( \sqrt [3]{a + \sqrt {a^{2} - b^{3}}} + \sqrt [3]{a - \sqrt {a^{2} - b^{3}}})$
or, $x^3=2a-3bx$ [ Using (1)and $ (\sqrt [3]{a + \sqrt {a^{2} - b^{3}}})(\sqrt [3]{a - \sqrt {a^{2} - b^{3}}})=\sqrt[3]{a^2-(a^2-b^3)}=b$]
or, $x^3+3bx=2a$.
Multiple choice maths cube and cube root estimation of cube root estimation of cube roots cubes and cube roots

Find the cube root of the number $120.$

  1. $4.1$
  2. $4.2$
  3. $4.7$
  4. $4.9$
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation
We use the Babylonian Algorithm for cube roots here
According to the algorithm, the cube root is given by the formula 
$x _{n+1}=\dfrac{\left (2x _n+\left (\dfrac N{x _{n^2}}\right )\right )}{3}$
where,
  • $N$ is the number for which cube root is to be found
  • $x _{n}$ is the initial approximation of the cube root
  • $x _{n+1}$ is the subsequent improvement on the cube root 

In this case,
$N = 120$
    $x _0 =4$ since $4^3<40 <5^3$

      $ \therefore$ $x _1 = \dfrac{\left ((2\times4)+\left (\dfrac {120} {4^2}\right )\right )}{3} = \dfrac{\left (8+\left (\dfrac {120}{16}\right )\right )}{3}=4.9$

      $\Rightarrow x _2 =\dfrac{ \left (2\times4.9+\left (\dfrac {120}{(4.9)^2}\right )\right )}{3} = \dfrac{\left (9.8+\left (\dfrac {120}{24.01} \right )\right )}{3}= \dfrac{(9.8+4.99)}{3} = 4.9$

      We can see the value stabilizes around $4.9$. 

      Hence the answer is $'D'.$
      Multiple choice maths cube and cube root estimation of cube root estimation of cube roots cubes and cube roots

      What is the approximate value of the cube root of the number $9?$

      1. $2.08$
      2. $2.19$
      3. $2.34$
      4. $2.51$
      Reveal answer Fill a bubble to check yourself
      A Correct answer
      Explanation

      First multiply and divide by $1,000,000,$ we get
      $\sqrt[3]{\dfrac{9\times 1000,000}{1000,000}}$
      $\sqrt[3]{9,000,000} \div 100$
      Now find the closest cube root of $9,000,000.$
      $208^3 = 8,998,912,$ therefore we can say that $\sqrt[3]{9}$ $\sim$ $\dfrac{208}{100} \sim 2.08$