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