Important Questions of Continuity and Differentiability Mathematics | Zigya

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 Multiple Choice QuestionsShort Answer Type

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51. Determine the value of the constant k so that the function f defined as
straight f left parenthesis straight x right parenthesis equals open curly brackets table attributes columnalign left end attributes row cell fraction numerator tan space 2 straight x over denominator straight x end fraction comma space straight x not equal to 0 end cell row cell space space space space space space space straight k space space space space space space comma space straight x equals 0 end cell end table close
is space continous space at space straight x equals 0
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52.

Find k so thatstraight f left parenthesis straight x right parenthesis equals open curly brackets table attributes columnalign left end attributes row cell fraction numerator sin space kx over denominator straight x end fraction space comma space straight x not equal to 0 end cell row cell space 4 plus straight x space space space space space comma space straight x equals 0 end cell end table close

is continuous at x = 0.

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53. Find A so that f(x) is continuous at x = 0.
straight f left parenthesis straight x right parenthesis equals open curly brackets table attributes columnalign left end attributes row cell fraction numerator 1 minus cos space 6 straight x over denominator 4 space straight x squared end fraction comma space if space straight x not equal to 0 end cell row cell space space space space space space space space space straight A space space space space space space space space space space comma space if space straight x equals 0 end cell end table close
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54. Find A so that f(x) is continuous at x = 0.
straight f left parenthesis straight x right parenthesis equals open curly brackets table attributes columnalign left end attributes row cell fraction numerator 1 minus cos space 8 straight x over denominator 4 straight x squared end fraction comma space if space straight x not equal to 0 end cell row cell space space space space space space space space space straight A space space space space space space space space space space comma space if space straight x equals 0 end cell end table close
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55. Determine the value of the constant k so that the function
straight f left parenthesis straight x right parenthesis equals open curly brackets table attributes columnalign left end attributes row cell fraction numerator 1 minus cos 2 straight x over denominator 2 straight x squared end fraction comma space straight x not equal to 0 end cell row cell space space space space space space space space space space space space space space straight k space space space space comma space straight x equals 0 end cell end table close
is continuous at x = 0.
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56. Determine the value of constant A so that the function
straight f left parenthesis straight x right parenthesis equals open curly brackets table attributes columnalign left end attributes row cell fraction numerator 1 minus cos space straight x over denominator straight x squared end fraction comma space straight x not equal to 0 end cell row cell space space space space space space space space space space straight A space space space space space space comma space straight x equals 0 end cell end table close
is continous at x=0
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57. Find A so that f(x) is continuous at x = 0.
straight f left parenthesis straight x right parenthesis equals open curly brackets table attributes columnalign left end attributes row cell fraction numerator 1 minus cos space 2 straight x over denominator 4 straight x squared end fraction comma space if space straight x not equal to 0 end cell row cell space space space space space space space space space space straight A space space space space space space space space comma space if space straight x equals 0 end cell end table close
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58. Given straight f left parenthesis straight x right parenthesis equals open curly brackets table attributes columnalign left end attributes row cell fraction numerator 1 minus cos space 2 straight x over denominator straight x squared end fraction comma space straight x not equal to 0 end cell row cell space space space space space space space space space space space space space straight k space space space space space space comma space straight x equals 0 end cell end table close
If f(x) is continuous at x = 0, find the value of k.
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59. GIven straight f left parenthesis straight x right parenthesis equals open curly brackets table attributes columnalign left end attributes row cell fraction numerator 1 minus cos space 4 straight x over denominator 8 straight x squared end fraction comma space straight x not equal to 0 end cell row cell space space space space space space space space space space space space space straight k space space space space space space space comma straight x equals 0 end cell end table close
If f(x) is continuous at x = 0, find the value of k
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60. If f(x) is continuous at .x = 0, find a, where
straight f left parenthesis straight x right parenthesis equals open curly brackets table attributes columnalign left end attributes row cell fraction numerator 1 minus cos space ax over denominator straight x space sinx end fraction comma space straight x not equal to 0 end cell row cell space space space space space space space space space space space space 1 half space space space space comma space straight x equals 0 end cell end table close
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