Tag: maths

Questions Related to maths

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

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

  1. even

  2. odd

  3. prime

  4. none of these

Show answer
Correct Option: B
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.

$\displaystyle \frac {11}{4}$ is a number between

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

  1. $1 \ and \ 2$

  2. $2 \ and \ 3$

  3. $3\  and \ 4$

  4. $11\  and \ 12$

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Correct Option: B
Explanation:

$\displaystyle \frac {11}{4}\, =\, 2\displaystyle \frac {3}{4}$
So $\displaystyle \frac {11}{4}$ lies between $2$ and $3.$

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

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

  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

Show answer
Correct Option: A
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 $A$

(0 , - 3 ) lies on _______ .

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

  1. Positive x- axis

  2. Negative x-axis

  3. Positive y-axis

  4. Negative y- axis

Show answer
Correct Option: D
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.

The number of surjections from $A = {1, 2,.....n}, n \leq 2$, onto B = {a, b} is

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

  1. $^nP _2$

  2. $2^n - 2$

  3. $2^n - 1$

  4. none of these

Show answer
Correct Option: B
Explanation:

A = {1, 2, 3, ........, n}

B = {a, b}
A has n elements.
B has 2 elements.
$\therefore$ No. of surjection is $2^{n}-2$.

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

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

  1. $2a$

  2. $2b$

  3. $3a$

  4. $4a$

Show answer
Correct Option: A
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, $x^3+3bx=2a$.

Estimate the cube root of the number $23.$

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

  1. $2.6$

  2. $2.1$

  3. $2.8$

  4. $1.4$

Show answer
Correct Option: C
Explanation:
$\sqrt[3]{\dfrac{23 \times 1000}{1000}}$
$\sqrt[3]{23000} \div 10$
Now, find the closest cube root of $23000.$
$28^3 = 21952,$ therefore we can say that $\sqrt[3]{23}$ $\sim$ $\dfrac{28}{10} = 2.8$

Find the cube root of the number $120.$

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

  1. $4.1$

  2. $4.2$

  3. $4.7$

  4. $4.9$

Show answer
Correct Option: D
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 $'D'.$

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

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

      1. $2.08$

      2. $2.19$

      3. $2.34$

      4. $2.51$

      Show answer
      Correct Option: A
      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 $\sqrt[3]{9}$ $\sim$ $\dfrac{208}{100} \sim 2.08$

      State whether true or false:
      If $x^3 = 11$, then $x=\sqrt[3] {11}$
      maths cube and cube root estimation of cube root estimation of cube roots cubes and cube roots

      1. True

      2. False

      Show answer
      Correct Option: A
      Explanation:
      TRUE:
      Odd exponents preserve the sign of the original expression. Therefore, if ${x}^{3}$ is positive, then $x$ must itself be positive. IF ${x}^{3}=11$, then $x$ must be $\sqrt [ 3 ]{ 11 } $