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CBSE

Gujarati JEE Mathematics : લક્ષ-સાતત્ય અને વિકલન

Multiple Choice Questions

101.

ધારો કે, $bold f bold left parenthesis bold x bold right parenthesis bold space bold equals bold space open curly brackets table attributes columnalign left columnspacing 1.4ex end attributes row cell bold xnsin bold x over bold 1 end cell cell bold x bold space bold # bold space bold 0 end cell row bold 0 cell bold x bold space bold equals bold 0 end cell end table close$ જો f(x) એ x = 0 આગળ સતત હોય, પરંતુ વિકલનીય ન હોય તો

n = 0
n∈(0,1]
n∈[1, ∞)
n∈(-∞, 0)

102.

જો xy = e - e^y તો $open parentheses fraction numerator bold d to the power of bold 2 bold y over denominator bold dx to the power of bold 2 end fraction close parentheses subscript bold x bold equals bold 0 end subscript bold space bold equals bold space bold. bold. bold. bold. bold. bold. bold. bold.$

e²
$bold 1 over bold e$
$bold 1 over bold e to the power of bold 3$
$bold 1 over bold e to the power of bold 2$

103.

bold f bold left parenthesis bold x bold right parenthesis bold space bold equals bold space fraction numerator bold vertical line bold x bold vertical line over denominator bold vertical line bold x bold vertical line bold space bold plus bold space bold 2016 end fraction

એ કેટલા બિંદુએ વિકલનીય નથી ?

0
1
2
અનંત

104.

bold જ ો bold space square root of bold r bold space bold equals bold space bold ae bold space to the power of bold theta bold space bold cot bold space bold alpha end exponent bold space bold ત ો bold space fraction numerator bold d to the power of bold 2 bold r over denominator bold dθ to the power of bold 2 end fraction bold space bold minus bold space bold 4 bold space bold r bold space bold cot to the power of bold 2 bold space bold alpha bold space bold equals bold space bold. bold. bold. bold. bold. bold. bold. bold space bold left parenthesis bold a bold comma bold space bold alpha bold space bold element of bold space bold R bold right parenthesis

0
1
$bold 1 over bold r$
r

105.

bold જ ો bold space bold f bold left parenthesis bold x bold right parenthesis bold space bold equals bold space open parentheses bold sum from bold k bold equals bold 0 to bold n of bold a to the power of bold k bold space bold vertical line bold x bold minus bold 1 bold vertical line to the power of bold k close parentheses bold space bold જ ્ ય ાં bold space bold a subscript bold k bold greater than bold 0 bold comma bold space bold ત ો bold space bold f bold left parenthesis bold x bold right parenthesis bold space bold એ bold space

R પર વિકલનીય છે.
R પર સતત છે. પરંતુ x = 1 આગળ વિકલનીય નથી.
x = 1 આગળ સતત નથી.
R પર સતત છે, પરંતુ x = 0 આગળ વિકલનીય નથી.

106.

bold જ ો bold space bold y bold space bold equals bold space bold sin bold space bold x bold times bold space bold sin bold space bold 2 bold x bold space bold times bold space bold sin bold space bold 3 bold x bold space bold. bold. bold. bold. bold space bold comma bold space bold sin bold space bold nx bold comma bold space bold ત ો bold space bold dy over bold dx bold space bold equals bold space bold. bold. bold. bold. bold. bold. bold. bold space

$bold sum from bold k bold equals bold 1 to bold n of bold space bold k bold space bold cot bold space bold kx$
$bold sum from bold k bold equals bold 1 to bold n of bold space bold k bold space bold tan bold space bold kx$
$bold y bold space bold sum from bold k bold equals bold 1 to bold n of bold space bold k bold space bold cot bold space bold kx bold space bold right square bracket$
$bold y bold space bold sum from bold k bold equals bold 1 to bold n of bold space bold k bold space bold tan bold space bold kx bold space$

107.

bold જ ો bold space bold f bold left parenthesis bold x bold right parenthesis bold space bold equals bold space fraction numerator bold sin to the power of bold 2 bold space bold x over denominator bold 1 bold plus bold cot bold space bold x bold space end fraction bold space bold plus bold space fraction numerator bold cos to the power of bold 2 bold space bold x over denominator bold 1 bold plus bold tan bold space bold x bold space end fraction bold space bold ત ો bold space bold f bold apostrophe bold left parenthesis bold x bold right parenthesis bold space bold equals bold space bold. bold. bold. bold. bold. bold. bold. bold. bold space

sin 2x
-sin 2x
cos 2x
-cos 2x

108.

જો f(x + y + z) = f(x) f(y) f(z), ∀x, y ∈ R, f(4) = 4, f(0) = 2 તો f'(4), f(4), f(0) એ ....

આપેલ તમામ સત્ય છે.
સમાંતર શ્રેણીમાં છે.
સમગુણોત્તર શ્રેણીમાં છે.
સ્વરિત શ્રેણીમાં છે.

સમગુણોત્તર શ્રેણીમાં છે.

Tips: -

f(x + y+ z) = f(x) f(y) f(z)

સમીકરણ (1)માં x = 4, y = 0, x = 0 મુૂલતાં

f(4) = f(4)f(0) f(0)

∴ f(0) = ± 1

ફરીથી સમીકરણ (1) માં x = 4, y = -2, z = -2 મૂકતાં

f(0) = f(4) [f(-2)]² ≥ 0 = 4 [f(-2)]² ≥0.

∴ f(0) # -1 આથી f(0) = 1

સમીકરણ (1)માં y = 4, z = 0 મૂકતાં,

f(x + 4) = f(x) f(4) f(0) 4 f(x)

∴ f'(x + 4) = 4 f'(x), આથી f'(4) = 4 f'(0) = 8

∴ f'(4) f(4), f(0) સમગુણોત્તર શ્રેણીમાં છે.