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 Multiple Choice QuestionsMultiple Choice Questions

31.

Let α and β be the distinct roots of ax2 + bx + c = 0, then limit as straight x rightwards arrow straight alpha of space fraction numerator 1 minus cos space left parenthesis ax squared plus bx plus space straight c right parenthesis over denominator left parenthesis straight x minus straight alpha right parenthesis squared end fraction space is equal to

  • straight a squared over 2 left parenthesis straight alpha minus straight beta right parenthesis squared
  • 0

  • negative straight a squared over 2 left parenthesis straight alpha minus straight beta right parenthesis squared
  • negative straight a squared over 2 left parenthesis straight alpha minus straight beta right parenthesis squared
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32.

If x is so small that x3 and higher powers of x may be neglected, then fraction numerator left parenthesis 1 plus straight x right parenthesis to the power of 3 divided by 2 end exponent space minus open parentheses 1 plus begin display style 1 half end style straight x close parentheses cubed over denominator left parenthesis 1 minus straight x right parenthesis to the power of 1 divided by 2 end exponent end fraction spacemay be approximated as

  • 1 minus 3 over 8 straight x squared
  • 3 x 6 plus 3 over 8 straight x squared
  • negative 3 over 8 straight x squared
  • negative 3 over 8 straight x squared
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33.

If both the roots of the quadratic equation x2 – 2kx + k2 + k – 5 = 0 are less than 5, then k lies in the interval

  • (5, 6]

  • (6, ∞)

  • (-∞, 4)

  • (-∞, 4)

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34.

If the equation anxn +an-1xn-1 +....... +a1x =0, a1 ≠ 0, n≥2, has a positive root x =  α, then the equation nanxn-1 + (n-1)an-1xn-2 +......+a1 = 0 has a positive root, which is

  • greater than α

  • smaller than α

  • greater than or equal to α

  • greater than or equal to α

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35.

Let z, w be complex numbers such that z iw + = 0 and arg zw = π. Then arg z equals

  • π/4

  • 5π/4

  • 3π/4

  • 3π/4

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36.

If z = x – i y and z1/3 = p+ iq , then fraction numerator begin display style open parentheses straight x over straight p plus straight y over straight q close parentheses end style over denominator left parenthesis straight p squared plus straight q squared right parenthesis end fraction is equal to 

  • 1

  • -2

  • 2

  • 2

421 Views

37.

If (1 – p) is a root of quadratic equation x2 +px + (1-p)=0 , then its roots are

  • 0, 1

  • -1, 2

  • 0, -1

  • 0, -1

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38.

If one root of the equation x2+px+12 =0 is 4, while the equation x2 +px +q = 0 has equal roots, then the value of 'q' is

  • 49/3

  • 4

  • 3

  • 3

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39.

The coefficient of xn in expansion of (1+x)(1-x)n is

  • (n-1)

  • (-1)n(1-n)

  • (-1)n-1(n-1)2

  • (-1)n-1(n-1)2

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40.

If 2a + 3b + 6c =0, then at least one root of the equation ax+ bx+ c = 0  lies in the interval

  • (0,1)

  • (1,2)

  • (2,3)

  • (2,3)


A.

(0,1)

Let space straight f apostrophe left parenthesis straight x right parenthesis space equals space ax squared space plus bx space plus straight c
rightwards double arrow space straight f left parenthesis straight x right parenthesis space equals space ax cubed over 3 plus bx squared over 2 space plus cx space plus straight d
straight f left parenthesis straight x right parenthesis space equals space fraction numerator 2 ax cubed space plus 3 bx squared space plus 6 cx space plus 6 straight d over denominator 6 end fraction
straight f left parenthesis 1 right parenthesis space equals space fraction numerator 2 straight a space plus 3 straight b space plus 6 straight c space plus 6 straight d over denominator 6 end fraction space equals fraction numerator 6 straight d over denominator 6 end fraction space equals space straight d
straight f left parenthesis 1 right parenthesis space space equals fraction numerator 6 straight d over denominator 6 end fraction space equals straight d
straight f left parenthesis 0 right parenthesis space equals space straight f left parenthesis 1 right parenthesis
straight f apostrophe left parenthesis straight x right parenthesis space equals space 0

∴ One of the roots of ax2 + bx + c = 0 lies between 0 and 1.
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