Tag: some special sequences

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Questions Related to some special sequences

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

The expression $(x + 1)(x + 2)(x + 3)(x + 4) + 1$ is a 

  1. perfect square

  2. cube

  3. quartic polynomial

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

Solving

$(x+1)(x+2)(x+3)(x+4)+1$
$Multiplying\ first\ bracket\ with\ last\ and\ second\ to\ the\ third\ one$
$(x^2+5x+4)(x^2+5x+6)+1$
$Replacing\ x^2+5x+4\ by\ 'B'$
$(B)(B+2)+1$
$B^2+2B+1$
$(B+1)^2=(x^2+5x+5)^2$
Hence $L.H.S.$ is the $Perfect\ Square$ of $(x^2+5x+5)$



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

$\cfrac { { \left( 963+476 \right)  }^{ 2 }+{ \left( 963-476 \right)  }^{ 2 } }{ \left( 973\times 963+476\times 476 \right)  } =$?

  1. $1449$
  2. $497$
  3. $2$
  4. $4$

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

Given Exp.$=\cfrac { { \left( a+b \right)  }^{ 2 }+{ \left( a-b \right)  }^{ 2 } }{ \left( { a }^{ 2 }+{ b }^{ 2 } \right)  } =\cfrac { 2\left( { a }^{ 2 }+{ b }^{ 2 } \right)  }{ \left( { a }^{ 2 }+{ b }^{ 2 } \right)  } =2$

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

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

  1. $(3, 3)$
  2. $(7, 3)$
  3. $(5, 5)$
  4. $(1, 1)$
Reveal answer Fill a bubble to check yourself
A,B Correct answer
Explanation

The circle equation x^2+y^2-10x-6y+30=0 simplifies to (x-5)^2+(y-3)^2=4, so the center is (5,3) and radius is 2. A square inscribed in this circle has a diagonal of 4. Since the side is parallel to y=x+3 (slope 1), the vertices are at distance sqrt(2) from the center along lines with slopes 1 and -1. Calculating these points yields (3,3) and (7,3) as valid vertices.

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

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 

  1. $2$
  2. $3$
  3. $1$
  4. $\dfrac{1}{4}$
Reveal answer Fill a bubble to check yourself
A,C,D Correct answer
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.

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

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

  1. $13$
  2. $29$
  3. $31$
  4. $33$
Reveal answer Fill a bubble to check yourself
C Correct answer
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 $=625-594=31$

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

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 $60$?

  1. $144$
  2. $180$
  3. $240$
  4. $360$
  5. $480$
Reveal answer Fill a bubble to check yourself
D Correct answer
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 $360$

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

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

  1. $(7-\sqrt{2})$
  2. $(5-\sqrt{2})$
  3. $(3-\sqrt{2})$
  4. $(10-\sqrt{7})$
Reveal answer Fill a bubble to check yourself
C Correct answer
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

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

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

  1. $1728$
  2. $1456$
  3. $2925$
  4. $1469$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

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

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

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

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

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

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

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

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

$\Rightarrow$  $25\times 13(17-8)$

$\Rightarrow$  $25\times 13\times 9$

$\Rightarrow$  $2925$