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Gujarati JEE Mathematics : નિશ્વાયક

Multiple Choice Questions

11. જો l, m અને n એ કોઈ સમગુણોત્તર શ્રેણીના p, q અને r મા પદ હોય તથા l > 0, m >0, n >0 તો $open vertical bar table row cell bold log bold space bold l end cell bold p bold 1 row cell bold log bold space bold m end cell bold q bold 1 row cell bold log bold space bold n end cell bold r bold 1 end table close vertical bar bold space bold equals bold space bold. bold. bold. bold. bold. bold. bold. bold. bold space bold.$

Tips: -

l = AR^p-1, m = AR^q-1, n = AR^r-1લો.

log l = log A + (p - 1) log R, log m = log A + (1 - 1) log તથા log n = log A + (r - 1) log R

$bold therefore bold space bold D bold space bold equals bold space open vertical bar table row cell bold log bold space bold A bold plus bold left parenthesis bold p bold plus bold 1 bold right parenthesis bold log bold space bold R end cell bold p bold 1 row cell bold log bold space bold A bold plus bold left parenthesis bold q bold minus bold 1 bold right parenthesis bold log bold space bold R end cell bold q bold 1 row cell bold log bold space bold A bold plus bold left parenthesis bold r bold minus bold 1 bold right parenthesis bold log bold space bold R end cell bold r bold 1 end table close vertical bar bold equals bold space open vertical bar table row cell bold log bold space bold A end cell bold p bold 1 row cell bold log bold space bold A end cell bold q bold 1 row cell bold log bold space bold A end cell bold r bold 1 end table close vertical bar bold space bold plus bold space bold log bold space bold R bold space open vertical bar table row cell bold p bold minus bold 1 end cell bold p bold 1 row cell bold q bold minus bold 1 end cell bold q bold 1 row cell bold r bold minus bold 1 end cell bold r bold 1 end table close vertical bar bold space bold equals bold space bold log bold space bold A bold space open vertical bar table row bold 1 bold p bold 1 row bold 1 bold q bold 1 row bold 1 bold r bold 1 end table close vertical bar bold space bold plus bold space bold log bold space bold R bold space open vertical bar table row bold p bold p bold 1 row bold q bold q bold 1 row bold r bold r bold 1 end table close vertical bar bold equals bold space bold 0 bold space bold plus bold space bold 0 bold space bold equals bold space bold 0 bold space$

12. જો a₁, a₂, ..., a_n સમગુણોત્તર શ્રેણીમાં હોય અને a_i > 0, i ≥ 1 તો $open vertical bar table row cell bold log bold space bold a subscript bold n end cell cell bold log bold space bold a subscript bold n bold plus bold 1 end subscript end cell cell bold log bold space bold a subscript bold n bold plus bold 2 end subscript end cell row cell bold log bold space bold a subscript bold n bold plus bold 3 end subscript end cell cell bold log bold space bold a subscript bold n bold plus bold 4 end subscript end cell cell bold log bold space bold a subscript bold n bold plus bold 5 end subscript end cell row cell bold semicolon bold pg bold space bold a subscript bold n bold plus bold 6 end subscript end cell cell bold log bold space bold a subscript bold n bold plus bold 7 end subscript end cell cell bold log bold space bold a subscript bold n bold plus bold 8 end subscript end cell end table close vertical bar space equals space......... space. space$

13.

ધારો a, b, c કે કોઇ પણ વાસ્તવિક સંખ્યાઓ છે. બધી જ શૂન્ય ન હોય તેવી વાસ્તવિક સંખ્યાઓ x, y, z માટે x = cy + bz; y = az + cx અને z = ay + bx છે. તો a² + c² + 2abc = .......

-1
0
1
2

14. ધારો કે a, b, c ∈ R - {0} તથા a + c ≠ 0 જો $open vertical bar table row bold a cell bold a bold plus bold 1 end cell cell bold a bold minus bold 1 end cell row cell bold minus bold b end cell cell bold b bold plus bold 1 end cell cell bold b bold minus bold 1 end cell row bold c cell bold c bold minus bold 1 end cell cell bold c bold plus bold 1 end cell end table close vertical bar bold space bold plus bold space open vertical bar table row cell bold a bold plus bold 1 end cell cell bold b bold plus bold 1 end cell cell bold c bold minus bold 1 end cell row cell bold a bold minus bold 1 end cell cell bold b bold minus bold 1 end cell cell bold c bold plus bold 1 end cell row cell bold left parenthesis bold minus bold 1 bold right parenthesis to the power of bold n bold plus bold 2 end exponent bold a end cell cell bold left parenthesis bold minus bold 1 bold right parenthesis to the power of bold n bold minus bold 1 end exponent bold b end cell cell bold left parenthesis bold minus bold 1 bold right parenthesis to the power of bold n bold c end cell end table close vertical bar bold space bold equals bold space bold 0$ તો n નું મૂલ્ય .....

શૂન્ય
કોઇક શૂન્યેતર યુગ્મ પૂર્ણાંક
અસંમેય સંખ્યા
કોઈક અયુગ્મ પૂર્ણાંક

15. સુરેખ સમીકરણની સંહતિ નીચે મુજબ છે:
x₁+ 2x₂ + x₃ = 3
2x₁ + 3x₂ + x₃ = 3
3x₁ + 5x₂ + 2x₃ = 1 સંહતિના ઉકેલોની સંખ્યા ....... છે.

3
એક
થી વધુ
શૂન્ય

16. નીચેની સમીકરણ સંહિતનો ઉકેલ અનન્ય હોય, તો k ની કિંમતનો ગણ ........ છે.
x - ky + z = 0
kx + 3y - kz = 0
3x + y - z = 0

R - {2, 3}
R - {-3}
R - {2}
{2, 3}

17. k ની કેટલી કિંમતો માટે સમીકરણ સંહિત
(k + 1)x + 8y = 4k
kx + (k+3)y = 3k-1 ને એક પણ ઉકેલ નથી.

2
1
3
અનંત

18. જો a + b + c = 0 હોય, તો $open vertical bar table row cell bold a bold minus bold x end cell bold c bold b row bold c cell bold b bold minus bold c end cell bold a row bold b bold a cell bold c bold minus bold x end cell end table close vertical bar bold space bold equals bold space bold 0$ નો એક ઉકેલ ....... છે.

1
2
0
a² + b² + c²

19. જો a ≠ p, b ≠ q, c ≠ r અને $open vertical bar table row bold p bold b bold c row bold a bold q bold c row bold a bold b bold r end table close vertical bar bold space bold equals bold space bold 0$ હોય, તો $fraction numerator bold p over denominator bold p bold minus bold a end fraction bold space bold plus bold space fraction numerator bold q over denominator bold q bold minus bold b end fraction bold space bold plus bold space fraction numerator bold r over denominator bold r bold minus bold c end fraction$ નું મૂલ્ય ....... છે.

2
-1
1
0

20. જો a, b, c ભિન્ન અને ધન સંખ્યાઓ હોય, તો $open vertical bar table row bold a bold b bold c row bold b bold c bold a row bold c bold a bold b end table close vertical bar$ નું મૂલ્ય ........ છે.

ધન
ઋણ
શૂન્ય
સંકર

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Gujarati JEE Mathematics : નિશ્વાયક

Multiple Choice Questions

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