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MATHS [5 marks sum]

Representing 3-D in 2-D · ICSE STD 8 (ICSE - English Medium). 2 questions.

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[5 marks sum]

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Question 15 Marks
Using Euler's formula, find the values of $x,y,z.$
  
  Faces Vertices Edges
$(i)$ $x$ $15$ $20$
$(ii)$ $6$ $y$ $8$
$(iii)$ $14$ $26$ $z$
Answer
$(i)F + V - E = 2$
$\Rightarrow x + 15 - 20 = 2$
$\Rightarrow x - 5 = 2 $
$\Rightarrow x = 2 + 5 = 7$
$(ii)F + V - E = 2$
$\Rightarrow 15 + y - 26 = 2$
$\Rightarrow y - 11 = 2$
$\Rightarrow y = 2 + 11 $
$\Rightarrow y = 13$
$(iii)F + V - E = 2$
$\Rightarrow 14 + 26 - Z = 2$
$\Rightarrow -Z = 2 - 40 $
$\Rightarrow Z = 38$
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Question 25 Marks
Verify Euler’s formula for the following three$-$dimensional figures:
Answer
$(i)$Number of vertices $= 6$
Number of faces $= 8$
Number of edges $= 12$
Using Euler formula,
$F + V – E = 2$
$8 + 6 – 12 = 2$
$2 = 2$ Hence proved.
$(ii)$ Number of vertices $= 9$
Number of faces $= 8$
Number of edges $= 15$
Using, Euler’s formula,
$F + V – E = 2$
$9 + 8 – 15 = 2$
$2 = 2$ Hence proved.
$(iii)$ Number of vertices $= 9$
Number of faces $= 5$
Number of edges $= 12$
Using, Euler’s formula,
$F + V – E = 2$
$9 + 5 – 12 = 2$
$2 = 2$ Hence proved.
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