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CBSE

Gujarati JEE Mathematics : લક્ષ-સાતત્ય અને વિકલન

Multiple Choice Questions

151. વિધેય 2tan3x - 3tan2x + 12tan + 3 એ ........

વધતું વિધેય છે.
ઘટતું વિધેય છે.
$open parentheses bold 0 bold comma bold space bold pi over bold 4 close parentheses$ માં વધતું તથા $open parentheses bold pi over bold 4 bold comma bold pi over bold 2 close parentheses$ માં ઘટતું વિધેય છે.
$open parentheses bold 0 bold comma bold space bold pi over bold 4 close parentheses$ માં ઘટતું તથા $open parentheses bold pi over bold 4 bold comma bold pi over bold 2 close parentheses$ માં વધતું વિધેય છે.

152.

જો વક્ર 2y³ = ax² + x³ નો બિંદુ (a, c) આગળનો સ્પર્શક અક્ષો પર α અને β અંતઃખંડો કાપતા હોય તથા જો α² + β² = 61 હોય તો |a| = .....

153.

a = કયા ગણનો સભ્ય હોય તો વિધેય

$bold f bold left parenthesis bold x bold right parenthesis bold space bold equals bold space open parentheses fraction numerator square root of bold a bold space bold plus bold space bold 4 end root over denominator bold 1 bold minus bold a end fraction bold minus bold 1 close parentheses$ x⁵ - 3x + log⁵ એ R પર ઘટતું વિધેય થશે ?

(-∞, ∞)
(1, ∞)
$Error converting from MathML to accessible text.$
$Error converting from MathML to accessible text.$

154. વિધેય f(x) = $bold sum from bold k bold space bold equals bold space bold 1 bold A to bold 2015 of$ 2(x - k) ને x ની કઈ કિંમત માટે ન્યુનત્તમ મળે ?

0
1000
1008
2015

155. વક્ર y = x³ - 6x² + 9x + 4 0 ≤ x ≤ 5 ના સ્પર્શકના મહત્તમ ઢાળનું મૂલ્ય ..... છે.

156.

વક્ર y² = x(2 - x)² ના બિંદુ (1, 1) આગળનો સ્પર્શક જો વક્રને ફરીથી બિંદુ P માં મળે તો P એ ......

$open parentheses bold 9 over bold 4 bold comma bold 3 over bold 8 close parentheses$
(4,4)
(-1,2)
(-1, -1‌)

157.

bold g bold left parenthesis bold x bold right parenthesis bold space bold equals bold space fraction numerator bold 1 over denominator bold log bold left parenthesis bold 1 bold space bold plus bold space bold x bold right parenthesis end fraction bold space bold minus bold space bold 1 over bold x bold comma bold space bold x bold space bold greater than bold space bold 0 bold space bold ત ો bold comma bold space

-∞ < g(x) < 0
0 < g(x) < 1
1 < g(x) < 2
-1 < g(x) < 0

0 < g(x) < 1

Tips: -

x > 0 માટે f(x) = log (1 + x) લો.

[0, x] અંતરાલ પર મધ્યકમાન પ્રમેય ઉપયોગ કરતાં, c ∈(0, x) એવો મળે કે જેથી

$fraction numerator bold log bold left parenthesis bold 1 bold space bold plus bold space bold x bold right parenthesis bold space bold minus bold space bold log bold space bold left parenthesis bold x bold space bold minus bold space bold 0 bold right parenthesis bold space over denominator bold x bold space bold minus bold space bold c end fraction bold space bold equals bold space bold f bold apostrophe bold left parenthesis bold c bold right parenthesis bold space bold equals bold space fraction numerator bold 1 over denominator bold 1 bold space bold plus bold space bold c end fraction fraction numerator bold log bold left parenthesis bold 1 bold space bold plus bold space bold x bold right parenthesis over denominator bold x end fraction bold space bold equals bold space fraction numerator bold 1 over denominator bold 1 bold plus bold c end fraction bold less than bold 1 bold log bold left parenthesis bold 1 bold space bold plus bold space bold x bold right parenthesis bold space bold less than bold space bold x bold. bold space bold આથ ી bold space fraction numerator bold 1 over denominator bold log bold left parenthesis bold 1 bold space bold plus bold space bold x bold right parenthesis end fraction bold space bold minus bold 1 over bold x bold greater than bold 0 bold space bold space bold space bold space bold space bold space bold space bold space bold space bold space bold આથ ી bold space bold space bold log bold left parenthesis bold x bold right parenthesis bold space bold greater than bold space bold 0 bold space bold વળ ી bold comma bold space bold c bold space bold element of bold space bold left parenthesis bold 0 bold comma bold space bold x bold right parenthesis bold. 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 આથ ી bold space bold c bold space bold less than bold space bold x bold space fraction numerator bold 1 over denominator bold 1 bold plus bold c end fraction bold space bold greater than bold space fraction numerator bold 1 over denominator bold 1 bold space bold plus bold space bold x end fraction bold space bold space bold space bold આથ ી bold space bold space fraction numerator bold log bold left parenthesis bold 1 bold space bold plus bold space bold x bold right parenthesis over denominator bold x end fraction bold space bold greater than bold space fraction numerator bold 1 over denominator bold 1 bold plus bold x end fraction bold log bold left parenthesis bold 1 bold plus bold x bold right parenthesis bold space bold greater than bold space fraction numerator bold x over denominator bold 1 bold space bold plus bold space bold x end fraction fraction numerator bold 1 over denominator bold log bold left parenthesis bold 1 bold plus bold x bold right parenthesis bold space end fraction bold space bold less than bold space fraction numerator bold 1 bold space bold plus bold space bold x over denominator bold 1 bold space end fraction bold. 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 આથ ી bold space bold g bold left parenthesis bold x bold right parenthesis bold space bold less than bold space bold 1 bold space$

158.

જ્યાં a > 0. જે અંતરાલમાં f'(x) ચુસ્ત વધતુ6 વિધેય હોય તેની લંબાએ L(a) છે. $fraction numerator bold 1 over denominator bold L bold left parenthesis bold a bold right parenthesis end fraction bold equals bold. bold. bold.$