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Gujarati JEE Mathematics : નિશ્વાયક

Multiple Choice Questions

11. 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

12. 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

કોઈક અયુગ્મ પૂર્ણાંક

Tips: -

_{$bold D bold space bold plus bold space bold D subscript bold 1 bold space bold equals bold space 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 bold left parenthesis bold minus bold 1 bold right parenthesis to the power of bold n 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 bold a cell bold minus bold b end cell bold c end table close vertical bar bold space bold equals bold space bold 0 bold space bold therefore bold space bold D bold space bold plus bold space bold left parenthesis bold minus bold 1 bold right parenthesis to the power of bold n bold space open vertical bar table row cell bold a bold plus bold 1 end cell cell bold a bold minus bold 1 end cell bold a row cell bold b bold plus bold 1 end cell cell bold b bold plus bold 1 end cell cell bold minus bold b end cell row cell bold c bold minus bold 1 end cell cell bold c bold minus bold 1 end cell bold c end table close vertical bar bold space bold equals bold space bold 0 bold space bold R subscript bold i bold space bold left right arrow bold C subscript bold i bold space bold જ ્ ય ાં bold space bold D bold space bold equals bold space 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$

$bold therefore bold space bold D bold space bold plus bold space bold left parenthesis bold minus bold 1 bold right parenthesis to the power of bold n bold space open vertical bar table row bold a cell bold a bold plus bold 1 end cell cell bold a bold minus bold 1 end cell row cell bold minus bold b end cell cell bold b bold plus bold 1 end cell cell bold b bold minus bold 1 end cell row bold c cell bold c bold minus bold 1 end cell cell bold c bold plus bold 1 end cell end table close vertical bar bold space bold equals bold space bold 0$} (C₁₃ પછી C₂₃ લેતાં) આથી D + (-1)ⁿ D = 0

અહીં $bold D bold space bold equals bold space 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 equals bold space open vertical bar table row bold a cell bold a bold plus bold 1 end cell cell bold minus bold 2 end cell row cell bold minus bold b end cell cell bold b bold plus bold 1 end cell cell bold minus bold 2 end cell row bold c cell bold c bold minus bold 1 end cell bold 2 end table close vertical bar bold space bold space bold space bold space bold space bold space bold space bold C subscript bold 23 bold space bold left parenthesis bold 1 bold right parenthesis bold space bold space bold space bold equals bold space open vertical bar table row bold a cell bold a bold plus bold 1 end cell cell 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 minus bold 1 end cell row bold c cell bold c bold minus bold 1 end cell bold 1 end table close vertical bar bold space bold equals bold space bold 2 bold space bold space open vertical bar table row bold a cell bold a bold plus bold 1 end cell cell 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 minus bold 1 end cell row bold c cell bold c bold minus bold 1 end cell bold 1 end table close vertical bar bold space bold space bold space bold space bold space bold space bold space bold space bold space bold space bold space bold C subscript bold 31 bold left parenthesis bold 1 bold right parenthesis bold space bold space bold space bold equals bold space bold 2 bold space bold space open vertical bar table row bold 0 bold a cell bold minus bold 1 end cell row cell bold minus bold 2 bold b end cell bold b cell bold minus bold 1 end cell row bold 0 bold c bold 1 end table close vertical bar bold space bold space bold space bold space bold C subscript bold 21 bold left parenthesis bold minus bold 1 bold right parenthesis bold space bold space bold space bold equals bold space bold 4 bold b bold left parenthesis bold a bold space bold plus bold space bold c bold right parenthesis bold space bold not equal to bold space bold 0$

(1 + (-1)ⁿD =0 હોવાથી n અયુગ્મ પૂર્ણાંક છે.

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. 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

15. 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

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

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

18. 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

19. 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

20. 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}

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Gujarati JEE Mathematics : નિશ્વાયક

Multiple Choice Questions

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