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CBSE

Gujarati JEE Mathematics : દ્વિઘાત સમીકરણ

Multiple Choice Questions

71. સમીકરણ x² - 6x - 2 = 0 નાં બીજ α અને β છે. જો a_m = αⁿ - βⁿ ; n ≥ 1 હોય તો $fraction numerator bold a subscript bold 10 bold minus bold 2 bold a subscript bold 8 over denominator bold 2 bold a subscript bold 9 end fraction$

-6
6
3
-3

72. જો a² + b² + c² = 1 હોય તો ab + bc + ca ∈ ......... જોઈ શકે.

$open square brackets bold 1 bold comma bold 1 over bold 2 close square brackets$
$open square brackets bold minus bold 1 over bold 2 bold comma bold 2 close square brackets$
$open square brackets negative bold 1 over bold 2 comma 1 close square brackets$
[-1, 2]

73. $bold 2 to the power of bold cos bold 2 bold x end exponent bold space bold equals bold space bold 3 bold times bold 2 to the power of bold cos to the power of bold 2 bold space bold x end exponent bold space bold 4$ હોય તો x = ....... જ્યાં x ∈ [0, $bold pi$ ]

0
$bold pi$
$bold pi over bold 4$
$bold pi over bold 2$

74. જો $bold 7 to the power of bold log subscript bold 7 bold left parenthesis bold x to the power of bold 2 bold minus bold 4 bold x bold plus bold 5 bold right parenthesis end exponent bold space bold equals bold space bold x bold space bold minus bold space bold 1$ હોય તો x ની શક્ય કિંમતો ..... હોય.

-3, -2
7
3
2

75. જો $bold x bold space bold equals bold space bold 2 bold space bold plus bold space bold 2 to the power of begin inline style bold 1 over bold 3 end style end exponent bold space bold plus bold space bold 2 to the power of begin inline style bold 2 over bold 3 end style end exponent$ હોય તો x³ - 6x² + 6x ની કિંમત ....... હોય.

76.

જો સમીકરણ px² + qx + r = 0 નાં બીજ α અને β હોય (જ્યાં p ≠ 0) તથા p, q, r સમાંતર શ્રેણીમાં હોય તેમજ $bold 1 over bold alpha bold plus bold 1 over bold beta bold space bold equals bold space bold 4$ હોય તો |α -β| = ........ .

$fraction numerator bold 2 square root of bold 13 over denominator bold 9 end fraction$
$fraction numerator square root of bold 34 over denominator bold 9 end fraction$
$fraction numerator bold 2 square root of bold 17 over denominator bold 9 end fraction$
$fraction numerator square root of bold 61 over denominator bold 9 end fraction$

fraction numerator bold 2 square root of bold 13 over denominator bold 9 end fraction

Tips: -

અહીં આપેલ દ્વિઘાત સમીકરણ px² + qx + r = 0 નાં બીજ α અને β છે.

$bold therefore bold space bold alpha bold space bold plus bold space bold beta bold space fraction numerator bold minus bold q over denominator bold p end fraction$ તથા $bold αβ bold space bold equals bold space bold r over bold p$

વળી, p, q, r સમાંતર શ્રેણીમાં છે. આથી 2q = p + r ... (1)

હવે, $bold 1 over bold alpha bold plus bold 1 over bold beta bold equals bold 4 bold.$ આથી $bold alpha bold space bold plus bold space bold beta bold space bold equals bold space bold 4 bold αβ$

$bold therefore bold space fraction numerator bold minus bold q over denominator bold p end fraction bold space bold equals bold space fraction numerator bold 4 bold r over denominator bold p end fraction bold therefore bold space bold q bold space bold equals bold space bold minus bold 4 bold r bold. bold space bold space bold space bold space bold space bold space bold space bold space bold space bold space bold space bold space bold space bold space bold space bold space bold space bold space bold space bold space bold space bold space bold space bold space bold space bold space bold space bold space bold space bold space bold space bold space bold space bold space bold space bold space bold space bold left parenthesis bold 2 bold right parenthesis bold space bold therefore bold space bold vertical line bold alpha bold space bold minus bold space bold beta bold vertical line bold space bold equals bold space square root of bold left parenthesis bold alpha bold minus bold beta bold right parenthesis to the power of bold 2 end root bold space bold equals bold space square root of bold left parenthesis bold alpha bold plus bold beta bold right parenthesis to the power of bold 2 bold minus bold 4 bold αβ end root bold space bold equals bold space square root of bold q to the power of bold 2 over bold p to the power of bold 2 bold minus fraction numerator bold 4 bold r over denominator bold p end fraction end root bold space bold space bold space bold space bold space bold space bold space bold space bold space bold space bold space bold space bold space bold space bold space bold. bold. bold. bold left parenthesis bold 3 bold right parenthesis bold space$

હવે,(1) તથા (2) પરથી, $fraction numerator bold p bold plus bold r over denominator bold 2 end fraction bold equals bold minus bold 4 bold r$ આથી p + r = -8r

∴ 9r = -p

$bold therefore bold space bold r over bold p bold space bold equals bold space fraction numerator bold minus bold 1 over denominator bold 9 end fraction$ ... (4)

(3) પરથી, $bold vertical line bold alpha bold space bold minus bold space bold beta bold vertical line bold space bold equals bold space square root of fraction numerator bold 16 bold r to the power of bold 2 over denominator bold p to the power of bold 2 end fraction bold minus fraction numerator bold 4 bold r over denominator bold p end fraction end root bold space bold equals bold space square root of fraction numerator bold 16 bold plus bold 36 over denominator bold 81 end fraction end root bold space bold equals bold space fraction numerator bold 2 square root of bold 13 over denominator bold 9 end fraction$

77. જો સમીકરણ x² + px + q = 0 નાં બીજ p અને q હોય, તો p નીચે કિંમત ....... હોઈ શકે.

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

78. જો 0 ≤ x ≤ $bold pi$ માટે $bold 16 to the power of bold sin to the power of bold 2 bold x end exponent bold space bold plus bold space bold 16 to the power of bold cos to the power of bold 2 bold x end exponent bold space bold equals bold space bold 10$ હોય તો x = .......... .