Tag: maths

Questions Related to maths

A square is inscribed in the circle $x^2+y^2-10x- 6y +30=0$. One side of the square is parallel to $y=x+3$. Then which of the following can be a vertex of the square

maths fun with numbers some special sequences triangular numbers properties and patterns of perfect squares

  1. $(3, 3)$

  2. $(7, 3)$

  3. $(5, 5)$

  4. $(1, 1)$

Show answer
Correct Option: A,B

For real number $a,b,c$ and $d$ , if $a^2+b^2=4$ and $c^2+d^2=1$, then possible value of $ac+bd$ is / are 

maths fun with numbers some special sequences triangular numbers properties and patterns of perfect squares

  1. $2$

  2. $3$

  3. $1$

  4. $\dfrac{1}{4}$

Show answer
Correct Option: A,C,D
Explanation:

According to Cauchy-Schwarz inequality:


$(ac+bd)^2\le(a^2+b^2)(c^2+d^2)$      $(\forall a, b, c, d \in \mathbb{R})$

Substituting the values here gives:

$(ac+bd)^2\le4$

$(ac+bd)^\le2$

Therefore, it can take all values given in the options except 3.

What is the least number that must be added to $594$ to make sum a perfect square?

maths fun with numbers some special sequences triangular numbers properties and patterns of perfect squares

  1. $13$

  2. $29$

  3. $31$

  4. $33$

Show answer
Correct Option: C
Explanation:

First calculate the square-root of $594$

$\sqrt{594}\approx 24.37$

The whole number larger than $24.37$ is $25$

and $(25)^{2}=625$

Now, $625$ is a perfect square.

So, the least number that must be added to $594$ to make sum a perfect square is $=625-594=31$

A rectangle with integer side length has perimeter $10$. What is the greatest numbers of these rectangles that can be cut from a piece of paper with width $24$ and length $60$?

maths fun with numbers some special sequences triangular numbers properties and patterns of perfect squares

  1. $144$

  2. $180$

  3. $240$

  4. $360$

  5. $480$

Show answer
Correct Option: D
Explanation:

If a rectangle has perimeter 10 then the sum of its length and width is 5, giving two choice with integer sides:

$(i)2\times3$ rectangle of area $6$
$(ii)1\times4$ rectangle of area $4$
The piece of paper has area $24\times60=1400$
This can be divided into $12\times20=240$ rectangles with sides $2\times3$
It can be divided into $24\times15=360$ recatngles with sides $1\times4$
So, the greatest number of rectangles is $360$

Fourth roots of $193-4\sqrt{2178}$ is

maths fun with numbers some special sequences triangular numbers properties and patterns of perfect squares

  1. $(7-\sqrt{2})$

  2. $(5-\sqrt{2})$

  3. $(3-\sqrt{2})$

  4. $(10-\sqrt{7})$

Show answer
Correct Option: C
Explanation:
According to Question

$(193-4\sqrt{2178} )^{1/4}$

$=(193-4\sqrt{11\times11\times3\times3\times2} )^{1/4}$

$=(193-4\times11\times3\sqrt2 )^{1/4}$

$=(121+72-132\sqrt{2} )^{1/4}$

$=(11^2+(6\sqrt2)^2-2\times11\times6\sqrt2)^{1/4}$                          $Using\ a^2+b^2-2ab=(a-b)^2$

$=(11-6\sqrt2)^{2\times0.25}$

$=(9+2-6\sqrt{2})^{0.5}$

$=(3^2+\sqrt{2}\ ^2-2\times3\times\sqrt{2})^{0.5}$                          $Using\ a^2+b^2-2ab=(a-b)^2$

$=(3-\sqrt{2})^{0.5\times2}$

$=3-\sqrt{2}$

$C$ is the right answer

The value of $1^{2}+3^{2}+5^{2}+.....25^{2}$ is:

maths fun with numbers some special sequences triangular numbers properties and patterns of perfect squares

  1. $1728$

  2. $1456$

  3. $2925$

  4. $1469$

Show answer
Correct Option: C
Explanation:

$1^2+3^2+5^2+....+25^2$

$\Rightarrow$  $(1^2+2^2+3^2+4^2+....+25^2)-(2^2+4^2+6^2+8^2....24^2)$

$\Rightarrow$  $(1^2+2^2+3^2+4^2+...25^2)-[(2\times 1)^2+(2\times 2)^2+(2\times 3)^2+....(2\times 12)^2]$

$\Rightarrow$  $(1^2+2^2+3^2+....+25^2)-2^2(1^2+2^2+3^2+.....12^2)$

$1^2+2^2+3^2+...+n^2=\dfrac{n(n+1)(2n+1)}{6}$

$\Rightarrow$  $\dfrac{25(25+1)(2\times 25+1)}{6}-2^2\dfrac{12(12+1)(2\times 12+1)}{6}$

$\Rightarrow$  $\dfrac{25\times 26\times 51}{6}-4\times\dfrac{12\times 13\times 25}{6}$

$\Rightarrow$  $25\times 13\times 17-4\times 2\times 13\times 25$

$\Rightarrow$  $25\times 13(17-8)$

$\Rightarrow$  $25\times 13\times 9$

$\Rightarrow$  $2925$

Is it possible for the square of a number to end with 5 zeroes?
State true or false.

maths fun with numbers some special sequences triangular numbers properties and patterns of perfect squares

  1. True

  2. False

Show answer
Correct Option: B
Explanation:

The square of a number with 'n' number of zeroes, will have $ 2 \times n $ zeroes. Hence, there cannot be $ 5 $ zeroes in a square of a number as it would be an even number.

If the square of a number ends with $10$ zeroes, how many zeroes will the number have at the end?

maths fun with numbers some special sequences triangular numbers properties and patterns of perfect squares

  1. $5$

  2. $15$

  3. $30$

  4. $40$

Show answer
Correct Option: A
Explanation:

If the square of a number has $ 10 $ zeroes, the number will have $ 10 \div 2 = 5 $ zeroes.

If a number ends with 3 zeroes, how many zeroes will its square have at the end ?

maths fun with numbers some special sequences triangular numbers properties and patterns of perfect squares

  1. 3

  2. 4

  3. 6

  4. 1

Show answer
Correct Option: C
Explanation:

If a number ends with $ 3 $ zeroes, then its square will have $ 3 \times 2 = 6 $ zeroes.

Express $49$ as the sum of $7$ odd numbers.
Express $121$ as the sum of $11$ odd numbers.

maths fun with numbers some special sequences triangular numbers properties and patterns of perfect squares

  1. $1+3+5+7+9+11$
    $1+3+5+7+9+11+13+15+19$

  2. $1+3+5+7+9+11+13$
    $1+3+5+7+9+11+13+15+19+21$

  3. $1+3+5+7+9+11$
    $1+3+5+7+9+11+13+15+19+21$

  4. $1+3+5+7+9+11+13$
    $1+3+5+7+9+11+13+15+19$

Show answer
Correct Option: B
Explanation:

$49=7^2=$ sum of first $7$ odd numbers.
So, $49=1+3+5+7+9+11+13$.
Similarly, $121=11^2=$ sum of first $11$ odd numbers. 
So, $121=1+3+5+7+9+11+13+15+19+21$