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Gujarati JEE Mathematics : ગણ, સંબંધ અને વિધેય

Multiple Choice Questions

61. જો X = {1, 2, 3, 4, 5}, Y = {1, 3, 5, 7, 9} હોય, તો નીચેનામાંથી શું સત્ય છે ?

S₁ = {(x,a)|a=x+2, x ∈ X, a ∈ Y} સંબંધ દર્શાવે પરંતુ X થી Y પરનું વિધેય નથી.
S₂ = {(1,1), (2,1),(3,3),(4,3),(5,5)} એ X થી Y પરનો સંબંધ દર્શાવે તથા વિધેય છે.
S₃ = {(1,1), (1,3), (3,5), (3,7), (5,7)} એ X થી Y પરનો સંબધ દર્શાવે તથા પરંતુ વિધેય નથી.
S₄ = {(1,3), (2,5), (4,7), (5,9), (3, 1) } એ X થી Y પરનો સંબંધ દર્શાવે તથા વિધેય છે.

62. f:(-1, 1) → B, f(x) = tan^-1 $fraction numerator bold 2 bold x over denominator bold 1 bold minus bold x to the power of bold 2 end fraction$ એ એક-એક તથા વ્યાપ્ત વિધેય હોય તો B =

$open square brackets 0 comma space straight pi over 2 close square brackets$
$open parentheses fraction numerator negative straight pi over denominator 2 end fraction comma straight pi over 2 close parentheses$
$open parentheses 0 comma space pi over 2 close parentheses$
$open square brackets fraction numerator italic minus pi over denominator 2 end fraction comma pi over 2 close square brackets$

63.

જો A = {1, 2, 3, 4},B = {3, 4, 5} હોય, તો A થી B પરના એક-એક વિધેયોની સંખ્યા તથા વ્યાપ્ત વિધેયોની સંખ્યા અનુક્રમે ....... અને ...... મળે. જો A = {3, 4, 5}, B = {1, 2, 3, 4} હોય, તો વ્યાપ્ત વિધેયોની સંખ્યા ...... મળે.

36, 0, 0
0, 0, 36
0, 36, 0
36, 6, 0

64. જો 3^x = 4^x-1 હોય, તો x = .......

$fraction numerator 2 space log subscript 3 space 2 over denominator 2 space log subscript 3 space 2 space minus space 1 end fraction$
$fraction numerator 2 over denominator 2 minus log subscript 2 space 3 end fraction$
$fraction numerator 1 over denominator 1 space minus space log subscript 4 space 3 end fraction$
$fraction numerator 2 space log subscript 2 space 3 over denominator 2 space log subscript 2 space 3 space minus 1 end fraction$

fraction numerator 2 space log subscript 3 space 2 over denominator 2 space log subscript 3 space 2 space minus space 1 end fraction

fraction numerator 2 over denominator 2 minus log subscript 2 space 3 end fraction

fraction numerator 1 over denominator 1 space minus space log subscript 4 space 3 end fraction

Tips: -

અહીં 3^x = 4x^x-1 આપેલ છે.

$bold therefore bold space bold log subscript bold 2 bold space bold 3 to the power of bold x bold space bold equals bold space bold log subscript bold 2 bold space bold 4 to the power of bold x bold minus bold 1 end exponent bold therefore bold space bold x bold space bold log subscript bold 2 bold space bold 3 bold space bold equals bold space bold left parenthesis bold x bold space bold minus bold space bold 1 bold right parenthesis bold space bold log subscript bold 2 bold space bold 4 bold space bold space bold space bold space bold space bold space bold space bold space bold space bold space bold space bold space bold space bold space bold space bold space bold equals bold space bold left parenthesis bold x bold space bold minus bold 1 bold right parenthesis bold space bold log subscript bold 2 bold space bold 2 to the power of bold 2 bold space bold equals bold space bold 2 bold left parenthesis bold x bold space bold minus bold space bold 1 bold right parenthesis bold space bold log subscript bold 2 bold space bold 2 bold space bold space bold space bold space bold space bold space bold space bold space bold space bold space bold space bold space bold space bold space bold space bold equals bold left parenthesis bold x bold space bold minus bold space bold 1 bold right parenthesis bold space bold space bold space bold space bold space bold space bold space bold space bold space bold space bold space bold space bold space bold space bold space bold space bold space bold equals bold space bold 2 bold x bold space bold minus bold space bold 2$

$bold therefore bold space bold x bold space bold equals bold space bold 2 bold x bold space bold minus bold space bold x bold space bold log subscript bold 3 bold space bold 3 bold space bold equals bold space bold x bold left parenthesis bold 2 bold minus bold log subscript bold 2 bold space bold 3 bold right parenthesis$

$bold therefore bold space bold x bold space bold equals bold space fraction numerator bold 2 over denominator bold 2 bold minus bold log subscript bold 2 bold space bold 3 end fraction$ જવાબ B

$bold equals bold space fraction numerator bold 2 over denominator bold 2 bold minus begin display style fraction numerator bold 1 over denominator bold log subscript bold 3 bold space bold 2 end fraction end style end fraction$

$bold therefore bold space bold x bold space bold equals bold space fraction numerator bold 2 bold space bold log subscript bold 3 bold space bold 2 over denominator bold 2 bold space bold log subscript bold 3 bold space bold 2 bold minus bold 1 end fraction$ જવાબ A

$bold therefore bold space bold x bold space bold equals fraction numerator bold log subscript bold 3 bold space bold 2 to the power of bold 2 over denominator bold log subscript bold 3 bold space bold 2 to the power of bold 2 bold space bold minus bold space bold 1 end fraction$

$bold equals bold space fraction numerator bold log subscript bold 3 bold space bold 4 over denominator bold log subscript bold 3 bold space bold 4 bold space bold minus bold space bold 1 end fraction bold equals bold space fraction numerator bold 1 bold divided by bold log subscript bold 4 bold space bold 3 over denominator begin display style fraction numerator bold 1 over denominator bold log subscript bold 4 bold space bold 3 end fraction end style bold minus bold 1 end fraction$

$bold equals bold space fraction numerator bold 1 over denominator bold 1 bold minus bold log subscript bold 4 bold space bold 3 end fraction$ જવાબ C

65. જો f: R →S, f(x) = sin x - $square root of bold 3$ cos + 1 વ્યાપ્ત વિધેય હોય, તો S = .......... .

[0, 3]
[-1, 3]
[-1, 1]
[0, 1]

66. વિધેય એ ........ અંતરાલમાં વ્યાખ્યાયિત થાય.

$left parenthesis 0 comma straight pi right parenthesis$
$open parentheses fraction numerator bold minus bold pi over denominator bold 2 end fraction bold comma bold pi over bold 2 close parentheses$
$left square bracket 0 comma space straight pi over 2 right parenthesis$
$left parenthesis 0 comma space straight pi right square bracket$

67. કોઈ રિક્ત ગણ માટે, n[P{P{P(P( $up diagonal strike bold 0$ ))}}] = .......

68.

બે અરિક્ત ગણ X તથા Y માટે, f : X → Y એ એક-એક વિધેય છે જો A ⊂ X તથા B ⊂ Y માટે, f(A) = {f(x) | x ∈ A} અને f^-1(B) = {x ∈ | f(x) ∈ B} હોય, તો .......

f^-1(f(A))=A
f^-1(f(A)) ⊄ A
f(f^-1(N))=B
અપેલ પૈકી એક પણ નહી

69. જો f(x) = cos [ $bold pi to the power of bold 2$ ] x ; જ્યાં [x] એ પૂર્ણાંક ભાગ વિધેય દર્શાવે છે, તો નીચેનામાંથી કયું સત્ય છે ?

$bold f open parentheses bold pi over bold 4 close parentheses bold space bold equals bold space bold 1$
$bold f bold left parenthesis bold minus bold pi bold right parenthesis bold space bold equals bold space bold 0$
$bold f bold left parenthesis bold pi bold right parenthesis bold space bold equals bold space bold 1$
$bold f open parentheses bold pi over bold 2 close parentheses bold space bold equals bold space bold minus bold 1$

70.

જો f(x) = sin $open curly brackets bold pi over bold 6 bold sin bold space open parentheses bold pi over bold 2 bold space bold sin bold space bold x close parentheses close curly brackets bold space bold semicolon$ x ∈ R તથા g(x) = $bold pi over bold 2 bold space bold sinx bold space bold semicolon$ x ∈ R, (fog) (x) તથા (gof) (x) ને f(g(x)) તથા g(f(x)) થી દર્શાવીએ તો નીચેનામાંથી શું સત્ય બને ?

f નો વિસ્તાર $open square brackets fraction numerator bold minus bold 1 over denominator bold 2 end fraction bold comma bold 1 over bold 2 close square brackets$ છે
fog નો વિસ્તાર $open square brackets fraction numerator bold minus bold 1 over denominator bold 2 end fraction bold comma bold 1 over bold 2 close square brackets$ છે.
$table row cell table row bold lim row cell bold x bold rightwards arrow bold m end cell end table bold space fraction numerator bold f bold left parenthesis bold x bold right parenthesis over denominator bold g bold left parenthesis bold x bold right parenthesis end fraction end cell end table bold space bold equals bold space bold space bold pi over bold 6$
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Gujarati JEE Mathematics : ગણ, સંબંધ અને વિધેય

Multiple Choice Questions

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