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Class 10 Class 12

નિશ્વાયક

જો, w એ 1 નું ઘન મૂળ હોય, અને w ≠ 1 $open vertical bar table row bold 1 cell bold 1 bold plus bold i bold plus bold w to the power of bold 2 end cell cell bold w to the power of bold 2 end cell row cell bold 1 bold minus bold i end cell cell bold minus bold 1 end cell cell bold w to the power of bold 2 bold minus bold 1 end cell row cell bold minus bold i end cell cell bold minus bold 1 bold plus bold w bold minus bold i end cell cell bold minus bold 1 end cell end table close vertical bar bold space bold equals bold space bold. bold. bold. bold. bold. bold. bold. bold. bold space$

જો s = p + q + r, તો $open vertical bar table row cell bold s bold plus bold r end cell bold p bold q row bold r cell bold s bold plus bold p end cell bold q row bold r bold p cell bold s bold plus bold q end cell end table close vertical bar$ ની કિંમત ....... છે.

s³
2s³
2s²
3s³

જો 1, w, w² એ 1 નાં ઘન મુળ હોય, તો $open vertical bar table row bold 1 cell bold w to the power of bold n end cell cell bold w to the power of bold 2 bold n end exponent end cell row cell bold w to the power of bold n end cell cell bold w to the power of bold 2 bold n end exponent end cell bold 1 row cell bold w to the power of bold 2 bold n end exponent end cell bold 1 cell bold w to the power of bold n end cell end table close vertical bar bold space bold equals bold space bold. bold. bold. bold. bold. bold. bold. bold. bold. bold. bold.$

જો $open vertical bar table row bold a bold b cell bold ax bold plus bold by end cell row bold b bold c cell bold ax bold plus bold cy end cell row cell bold ax bold plus bold by end cell cell bold bx bold plus bold cy end cell bold 0 end table close vertical bar bold space bold equals bold space bold 0$ અને ax² + 2bxy + cy² ≠ 0 તો.........

a, b, cસમગુણોત્તર શ્રેણીમાં છે.
a, b, c સમાંતર શ્રેણીમાં છે.
a, b, c સ્વરિત શ્રેણીમાં છે.
a, b, c એ સમગુણોત્તર, સમાંતર કે સ્વરિત શ્રેણીમાં નથી.

a, b, cસમગુણોત્તર શ્રેણીમાં છે.

Tips: -

$open vertical bar table row bold a bold b cell bold ax bold plus bold by end cell row bold b bold c cell bold ax bold plus bold cy end cell row cell bold ax bold plus bold by end cell cell bold bx bold plus bold cy end cell bold 0 end table close vertical bar bold space bold equals bold space bold 0 bold therefore bold space open vertical bar table row bold a bold b bold 0 row bold b bold c bold 0 row cell bold ax bold plus bold by end cell cell bold bx bold plus bold y end cell cell bold minus bold ax to the power of bold 2 bold minus bold 2 bold bxy bold minus bold cy to the power of bold 2 end cell end table close vertical bar bold space bold equals bold space bold 0 bold space bold therefore bold space bold left parenthesis bold minus bold ax to the power of bold 2 bold space bold minus bold space bold 2 bold bxy bold space bold minus bold space bold cy to the power of bold 2 bold right parenthesis bold space bold left parenthesis bold ac bold space bold minus bold space bold b to the power of bold 2 bold right parenthesis bold space bold equals bold space bold 0 bold space bold therefore bold space bold ac bold space bold minus bold space bold b to the power of bold 2 bold space bold equals bold space bold 0 bold space bold therefore bold space bold ac bold space bold equals bold space bold b to the power of bold 2$

∴ a, b, c સમગુણોત્તર શ્રેણીમાં છે.

જો $bold D bold space bold equals bold space open vertical bar table row bold 1 bold 1 bold 1 row bold 1 cell bold 1 bold plus bold x end cell bold 1 row bold 1 bold 1 cell bold 1 bold plus bold y end cell end table close vertical bar$ જ્યાં x ≠ 0, y ≠ 0 તો D એ ....... x, y ∈ N

x અને y વડે વિભાજ્ય છે.
x વડે વિભાજ્ય છે પરંતુ y વડે વિભાજ્ય નથી.
x વડે વિભાજ્ય નથી પરંતુ y વડે વિભાજ્ય છે.
x અને y પૈકી એક પણ વડે વિભાજ્ય નથી.

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