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CBSE

Gujarati JEE Mathematics : લક્ષ-સાતત્ય અને વિકલન

Multiple Choice Questions

121.

જો વક્ર xy + ax + by = 0 ને (1, 1) આગળનો સ્પર્શક X-અક્ષ સાથે tan-1 2 માપનો ખૂણો બનાવે, તો $fraction numerator bold a bold space bold plus bold space bold b bold space over denominator bold ab bold space end fraction bold space bold equals bold space bold. bold. bold. bold. bold. bold. bold. bold space$

0
1
$bold 1 over bold 2$
$fraction numerator bold minus bold 1 over denominator bold 2 end fraction$

122.

અરિક્ત ગણ A, B માટે f : A → B અને g : B → A એવાં વિધેય છે જ્યાં f(g(x)) = x, ∀ x ∈ B. નીચેનામાંથી કયા વિધાન સત્ય (T) અને મિથ્યા (F) છે ?

(1) વિધેય f એક-એક વિધેય છે.
(2) વિધેય f વ્યાત્પ વિધેય છે.
(3) વિધેય g એક-એક વિધેય છે.
(4) વિધેય g વ્યાપ્ત વિધેય છે.

FTFT
TTFF
TFTT
FTTF

123.

વિધેય f : (0, ∞) → (0, ∞) માટે,

(1) f(ab) = f(a) f(b) અને
(2) $bold lim with bold x bold rightwards arrow bold infinity below$ f(x) = c, (જ્યાં ક # 0) પ્રકારનું છે. f(4) = ....

124.

bold lim with bold n bold rightwards arrow bold infinity below bold space open parentheses root index bold 3 of bold n to the power of bold 2 bold space bold minus bold space bold n to the power of bold 3 bold space end root bold plus bold space bold n close parentheses bold space bold equals bold space bold. bold. bold. bold. bold. bold. bold space

$bold minus bold 1 over bold 3$
$bold 2 over bold 3$
$bold minus bold 2 over bold 3$
$fraction numerator begin display style bold 1 end style over denominator begin display style bold 3 end style end fraction$

125.

f(x) = sin x + cos x, 0 ≤ x ≤ 2 $bold pi$ એ ...... અંતરાલમાં ચુસ્ત ઘટતું વિધેય છે.

$open parentheses fraction numerator bold 5 bold pi over denominator bold 4 end fraction bold comma bold 2 bold pi close parentheses$
$bold left parenthesis bold 0 bold comma bold space bold 2 bold pi bold right parenthesis$
$open parentheses bold 0 bold comma bold pi over bold 4 close parentheses$
$open parentheses bold pi over bold 4 bold comma bold pi over bold 4 close parentheses$

126.

જો a < b < c, f(x) એ (a, c) પર ચુસ્ત રીતે વધતું વિધેય હોય અને f(x) એ [a, c] પર સતત હોય તો .......

(b - c) f(a) + (c - b) f(b) > (c - a) f(c)
(b - a) f(c) + (c - b) f(a) > (c - a) f(b)
(b - a) f(c) + (c - b) f(a) < (c - a) f(b)
(b - c) f(a) + (c - b) f(b) > (c - a) f(c)

(b - a) f(c) + (c - b) f(a) > (c - a) f(b)

Tips: -

[a, b] તેમજ [b, c] પર મધ્યકમાન પ્રમેયનો ઉપયોગ કરતાં,

$fraction numerator bold f bold left parenthesis bold b bold right parenthesis bold space bold minus bold space bold f bold left parenthesis bold a bold right parenthesis over denominator bold b bold space bold minus bold space bold a end fraction bold space bold equals bold space bold f bold apostrophe bold left parenthesis bold alpha bold right parenthesis bold space bold તથ ા bold space fraction numerator bold f bold left parenthesis bold c bold right parenthesis bold space bold minus bold space bold f bold left parenthesis bold b bold right parenthesis bold space over denominator bold c bold space bold minus bold space bold b end fraction bold equals bold space bold f bold apostrophe bold left parenthesis bold beta bold right parenthesis bold comma bold space bold space bold space bold space bold space bold space bold space bold alpha bold space bold element of bold space bold left parenthesis bold a bold comma bold space bold b bold right parenthesis bold space bold beta bold space bold element of bold space bold left parenthesis bold b bold comma bold space bold c bold right parenthesis bold space$

વળે, f' એ ચુસ્ત રીતે વધતુ વિધેય હોવાથી, f'(b) > f'(α)

$fraction numerator bold f bold left parenthesis bold c bold right parenthesis bold space bold minus bold space bold f bold left parenthesis bold b bold right parenthesis over denominator bold c bold space bold minus bold space bold b end fraction bold space bold greater than bold space fraction numerator bold f bold left parenthesis bold b bold right parenthesis bold space bold minus bold space bold f bold left parenthesis bold a bold right parenthesis bold space over denominator bold b bold space bold minus bold space bold a bold space end fraction$

(b - c) f(c) _ (c - b) f(a) > (c - a) f(b)

127.

bold જ ો bold space bold v bold space bold e to the power of bold u over bold v to the power of bold 3 end exponent bold space bold equals bold space bold 1 bold space bold ત ો bold space bold space bold. bold. bold. bold. bold. bold. bold. bold. bold. bold. bold. bold. bold. bold. bold. bold. bold. bold.

$bold v bold space fraction numerator bold d to the power of bold 2 bold u over denominator bold dv to the power of bold 2 end fraction bold space bold plus bold space bold 2 bold space bold du over bold dv bold space bold plus bold space bold 3 bold v to the power of bold 2 bold space bold equals bold space bold 0 bold space$
$bold v bold space fraction numerator bold d to the power of bold 2 bold u over denominator bold dv to the power of bold 2 end fraction bold space bold plus bold space bold 2 bold space bold du over bold dv bold space bold equals bold space bold space bold 3 bold v to the power of bold 2 bold space$
$bold v bold space fraction numerator bold d to the power of bold 2 bold u over denominator bold dv to the power of bold 2 end fraction bold space bold minus bold space bold 2 bold space bold du over bold dv bold space bold plus bold space bold 3 bold v to the power of bold 2 bold space bold equals bold space bold 0 bold space$
$fraction numerator bold d to the power of bold 2 bold u over denominator bold dv to the power of bold 2 end fraction bold space bold minus bold space bold 2 bold space bold du over bold dv bold space bold equals bold space bold 3 bold v to the power of bold 2 bold space$