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Gujarati JEE Mathematics : દ્વિઘાત સમીકરણ

Multiple Choice Questions

41.

સમીકરણ (x - a) (x- b) + (x - b) (x - c) + (x - c) (x - a) = 0 નાં બીજ હંમેશાં ........ હોય. (a≠b)

વાસ્ત્વવિક અસમાન
સમાન
અવાસ્તવિક સંકર
શુદ્વ કાલ્પનિક

42.

જો સમીકરણો x² + ax + b = 0 અને x² + bx + a = 0 નું એક બીજ સમાન હોય તો a + b ની કિંમત ......... હોય. (a ≠ b)

2
1
-1
0

43.

જો ax + by = 1 હોય અને સમીકરણ px² + qy² = 1 ને માત્ર એક જ બીજ હોય તો નીચેનામાંથી કયું સત્ય બને ?

a²b² = pq
$bold a to the power of bold 2 over bold p bold space bold plus bold space bold b to the power of bold 2 over bold q bold space bold equals bold space bold 1$
$bold x bold space bold equals bold space fraction numerator bold minus bold a over denominator bold p end fraction$
આપેલ પૈકી એક પણ નહી

44. ચલ x માં દ્વિઘાત સમીકરણ (cos p - 1)x² + cos px + sin p = 0 નાં બીજ વાસ્તવિક હોય તો, p ∈ .....

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

45.

$bold log subscript bold 10 bold space bold a bold space bold plus bold space bold log subscript bold 10 bold space square root of bold a bold space bold plus bold space bold log subscript bold 10 bold space scriptbase square root of bold a end scriptbase presuperscript bold 4 bold space bold plus bold space bold. bold. bold. bold space bold equals bold space bold b bold space bold greater than bold space bold 0$ હોય તથા $fraction numerator begin display style bold sum from bold n bold minus bold 1 to bold b of end style bold left parenthesis bold 2 bold n bold minus bold 1 bold right parenthesis over denominator begin display style bold sum from bold n bold equals bold 1 to bold b of end style bold left parenthesis bold 3 bold n bold plus bold 1 bold right parenthesis end fraction bold space bold equals bold space begin inline style fraction numerator bold 20 over denominator bold 7 bold space bold log subscript bold 10 bold space bold a end fraction end style$ હોય તો a = ...... .

1000
100
100000
10

46.

x ∈ R માટે જો દ્વિઘાત બહુપદી f(x) = ax² + bx + c > 0 હોય તો g(x) = f(x) + f'(x) + f"(x) ....... થાય. x ∈ R.

g(x) = 0
g(x) < 0
g(x) ≥0
g(x) > 0

47.

a, b, c ∈ R; a ≠ 0 માટે જો સમીકરણ a²x² + bx + c = 0 નું એક બીજ α હોય તથા સમીકરણ a² x² - bx - c = 0 નું એક બીજ β હોય જ્યાં 0 < α < β હોય તો સમીકરણ a²x² + 2bx + 2c = 0 નું બીજ γ હંમેશા નીચેનામાંથી ........ નું સમાધાન કરે.

$bold alpha bold space bold less than bold space bold gamma bold space bold less than bold space bold beta$
$bold gamma bold space bold equals bold space bold alpha bold space bold plus bold space bold beta over bold 2$
$bold gamma bold space bold equals bold space bold space fraction numerator bold alpha bold space bold plus bold beta over denominator bold 2 end fraction$
$bold gamma bold space bold equals bold space bold space fraction numerator bold alpha bold space over denominator bold 2 end fraction bold plus bold beta$

48.

જો α, β એ સમીકરણ x² + px + q = 0 નાં બીજ હોય અને α⁴, β⁴ એ સમીકરણ x² - rx + s = 0 નાં બીજ હોય તો સમીકરણ x² - 4qx + 2q² - r = 0 નાં બીજ હંમેશાં ........ હોય.

બે સમાન અને વાસ્તવિક
બે ભિન્ન અને વાસ્તવિક
અવાસ્તવિક સંકર
એક વાસ્તવિક અને એક શુદ્વ કાલ્પનિક

49. x ∈ R માટે $bold 3 to the power of bold 72 bold space open parentheses bold 1 over bold 3 close parentheses to the power of bold x bold space open parentheses bold 1 over bold 3 close parentheses to the power of square root of bold x end exponent bold space bold greater than bold space bold 1$ હોય તો x ∈ ........ .

[0, 64]
(6, 64)
(0, 64]
[0, 64)

[0, 64)

Tips: -

bold 3 to the power of bold 72 bold space open parentheses bold 1 over bold 3 close parentheses to the power of bold x bold space open parentheses bold 1 over bold 3 close parentheses square root of bold x bold space bold greater than bold space bold 1 bold space bold therefore bold space bold 3 to the power of bold 72 bold left parenthesis bold 3 to the power of bold minus bold 1 end exponent bold right parenthesis to the power of bold x bold space bold left parenthesis bold 3 to the power of bold minus bold 1 end exponent bold right parenthesis to the power of square root of bold x end exponent bold space bold greater than bold 1 bold therefore bold space bold 3 to the power of bold 72 bold space bold minus bold space bold x bold space bold minus square root of bold x end exponent bold space bold greater than bold space bold 3 to the power of bold 0 bold therefore bold space bold 72 bold space bold minus bold space bold x bold space bold minus bold space square root of bold x bold space bold greater than bold space bold 0 bold therefore bold space bold x bold space bold plus bold space square root of bold x bold space bold minus bold space bold 72 bold space bold less than bold space bold 0 bold space

bold therefore bold space bold left parenthesis square root of bold x bold space bold minus bold space bold 8 bold right parenthesis bold space bold left parenthesis square root of bold x bold space bold plus bold space bold 9 bold right parenthesis bold space bold less than bold space bold 0 bold. 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 minus bold 9 bold space bold less than bold space square root of bold x bold space bold space bold less than bold space bold 8 bold space bold therefore bold space bold 0 bold space bold less or equal than bold space square root of bold x bold space bold less than bold space bold 8 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 આથ ી bold space bold 0 bold space bold less or equal than bold space bold x bold space bold less than bold space bold 64 bold space bold therefore bold space bold x bold space bold element of bold space bold left square bracket bold 0 bold comma bold space bold 64 bold right parenthesis

50.

જો દ્વિઘાત સમીકરણ (x - a) (x - b) - k = 0 નાં બીજ c તથા d હોય તો a તથા b બીજવાળું દ્વિઘાત સમીકરણ ....... મળે.

(x - c) (x - d) + k = 0
(x + c) (x + d) - k = 0
(x - c) (x - d) - k = 0
(x + c) (x + d) + k = 0

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Gujarati JEE Mathematics : દ્વિઘાત સમીકરણ

Multiple Choice Questions

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