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 Multiple Choice QuestionsMultiple Choice Questions

1. integral fraction numerator dx over denominator cos space straight x space minus space sin space straight x end fraction is equal to
  • fraction numerator 1 over denominator square root of 2 end fraction space log space open vertical bar tan open parentheses straight x over 2 minus fraction numerator begin display style straight pi end style over denominator 8 end fraction close parentheses close vertical bar space plus straight C
  • fraction numerator 1 over denominator square root of 2 end fraction space log space open vertical bar cot open parentheses straight x over 2 close parentheses close vertical bar space plus straight C
  • fraction numerator 1 over denominator square root of 2 end fraction space log space open vertical bar tan open parentheses straight x over 2 minus fraction numerator 3 straight pi over denominator 8 end fraction close parentheses close vertical bar space plus straight C
  • fraction numerator 1 over denominator square root of 2 end fraction space log space open vertical bar tan open parentheses straight x over 2 plus fraction numerator 3 straight pi over denominator 8 end fraction close parentheses close vertical bar space plus straight C

D.

fraction numerator 1 over denominator square root of 2 end fraction space log space open vertical bar tan open parentheses straight x over 2 plus fraction numerator 3 straight pi over denominator 8 end fraction close parentheses close vertical bar space plus straight C
integral fraction numerator dx over denominator cos space straight x space minus space sin space straight x end fraction space equals space fraction numerator 1 over denominator square root of 2 end fraction integral fraction numerator 1 over denominator cos begin display style space end style begin display style open parentheses straight x space plus straight pi over 4 close parentheses end style end fraction space dx

equals space fraction numerator 1 over denominator square root of 2 end fraction integral space sec space open parentheses space straight x plus straight pi over 4 space close parentheses dx
space equals space fraction numerator 1 over denominator square root of 2 end fraction space log space open vertical bar space tan space open parentheses straight x over 2 space plus fraction numerator 3 straight pi over denominator 8 end fraction close parentheses close vertical bar space plus straight C
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2.

The value of integral subscript 0 superscript pi divided by 2 end superscript fraction numerator left parenthesis sin space x space plus space cos space x right parenthesis squared over denominator square root of 1 plus space sin space 2 x end root end fraction space d x space i s

  • 0

  • 1

  • 2

  • 3


C.

2

integral subscript 0 superscript straight pi divided by 2 end superscript fraction numerator left parenthesis sin space straight x space plus cos space straight x right parenthesis squared over denominator square root of left parenthesis sin space straight x space plus space cos space straight x right parenthesis squared end root end fraction space dx
space equals space integral subscript 0 superscript straight pi divided by 2 end superscript space left parenthesis sin space straight x space plus space cos space straight x right parenthesis space dx
space equals space vertical line minus space cos space straight x space plus space sin space straight x vertical line subscript 0 superscript straight pi divided by 2 end superscript space equals space 2
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3.

Ifintegral subscript 0 superscript straight pi space xf space left parenthesis sin right parenthesis space dx space equals space straight A space integral subscript 0 superscript straight pi divided by 2 end superscript space straight f space left parenthesis sin space straight x right parenthesis space dx comma space then space straight A space is

  • 0

  • π

  • π/2

  • π/4


B.

π

Letlet space straight I space equals space integral subscript 0 superscript straight pi space xf space left parenthesis sin space straight x right parenthesis space dx
space equals space integral subscript 0 superscript straight pi space left parenthesis straight pi minus straight x right parenthesis space straight f space left parenthesis sin space straight x right parenthesis space dx
space equals space straight pi integral subscript 0 superscript straight pi space straight f space left parenthesis sin space straight x right parenthesis dx minus 1
left parenthesis since space straight f space left parenthesis 2 straight a minus straight x right parenthesis space equals space straight f left parenthesis straight x right parenthesis right parenthesis
rightwards double arrow space straight I space equals space straight pi integral subscript 0 superscript straight pi divided by 2 end superscript space straight f space left parenthesis sin right parenthesis space dx
rightwards double arrow space straight I subscript 2 divided by straight I subscript 1 space equals space 2
 =π

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4.

If integral space fraction numerator sin space straight x over denominator sin space left parenthesis straight x minus straight alpha right parenthesis end fraction space dx space equals space Ax space plus space straight B space log space sin space left parenthesis straight x minus straight alpha right parenthesis space plus straight C comma space space then space value space left parenthesis straight A comma space straight B right parenthesis is

  • (sinα, cosα)

  • (cosα, sinα)

  • (- sinα, cosα)

  • (- cosα, sinα)


B.

(cosα, sinα)

Put x – α = t

rightwards double arrow space integral fraction numerator sin space left parenthesis straight alpha space plus straight t right parenthesis over denominator sin space straight t end fraction space dt space equals space sin space straight alpha space integral space cot space dt space plus space cos space straight alpha space integral dt
space equals space cos space straight alpha space left parenthesis straight x minus straight alpha right parenthesis space plus space sin space straight alpha space ln space vertical line sin space straight t vertical line space plus straight c
straight A space equals space cos space straight alpha space comma space straight B space equals space sin space straight alpha

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5.

The value of integral subscript negative 2 end subscript superscript 3 space vertical line 1 minus straight x squared vertical line dx space is

  • 28/7

  • 28/3

  • 7/3

  • 1/3


B.

28/3

integral subscript negative 2 end subscript superscript negative 1 end superscript space left parenthesis straight x squared minus 1 right parenthesis space dx space plus integral subscript negative 1 end subscript superscript 1 space left parenthesis 1 minus straight x squared right parenthesis space dx space plus integral subscript 1 superscript 3 left parenthesis straight x squared minus 1 right parenthesis space dx space equals space straight x cubed over 3 minus right enclose straight x subscript negative 2 end subscript superscript 1 space plus straight x minus right enclose straight x cubed over 3 end enclose subscript negative 1 end subscript superscript 1 plus straight x cubed over 3 minus right enclose right enclose straight x end enclose subscript 1 superscript 3
space equals space 28 over 3 space
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6.

If straight f left parenthesis straight x right parenthesis space equals space fraction numerator straight e to the power of straight x over denominator 1 plus straight e to the power of straight x end fraction. space straight I subscript 1 space equals space integral subscript straight f left parenthesis negative straight a right parenthesis end subscript superscript straight f left parenthesis straight a right parenthesis end superscript space xg open curly brackets straight x open parentheses 1 minus straight x close parentheses close curly brackets dx space and space straight I subscript 2 space equals space integral subscript straight f left parenthesis negative straight a right parenthesis end subscript superscript straight f left parenthesis straight a right parenthesis end superscript space straight g open curly brackets straight x open parentheses 1 minus straight x close parentheses close curly brackets dx then the value of I2/I1 is

  • 2

  • -2

  • 1

  • -1


A.

2

f(-a) + f(a) = 1

straight I subscript 1 space equals space integral subscript straight f left parenthesis negative straight a right parenthesis end subscript superscript straight f left parenthesis straight a right parenthesis end superscript space xg left curly bracket straight x left parenthesis 1 minus straight x right parenthesis right curly bracket space dx
space equals space integral subscript straight f left parenthesis negative straight a right parenthesis end subscript superscript straight f left parenthesis straight a right parenthesis end superscript space left parenthesis 1 minus straight x right parenthesis space straight g space left curly bracket space straight x space left parenthesis 1 minus straight x right parenthesis right curly bracket dx
open parentheses because space integral subscript straight a superscript straight b space straight f space left parenthesis straight x right parenthesis space dx space equals space integral subscript straight a superscript straight b space straight f space left parenthesis straight a plus straight b minus straight x right parenthesis space dx close parentheses
2 straight I subscript 1 space equals space integral subscript straight f left parenthesis negative straight a right parenthesis end subscript superscript straight f left parenthesis straight a right parenthesis end superscript space straight g left curly bracket straight x space left parenthesis 1 minus straight x right parenthesis right curly bracket space dx space equals space straight I subscript 2
rightwards double arrow space straight I subscript 2 divided by straight I subscript 1 space space equals space 2

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7.

Let f (x) be a non−negative continuous function such that the area bounded by the curve y = f (x), x−axis and the ordinates x = π/4  and x = β >  π/4 open parentheses straight beta space sin space straight beta space plus space straight pi over 4 space cos space straight beta space plus space square root of 2 straight beta close parentheses. Then f (π/2) is

  • open parentheses straight pi over 4 plus square root of 2 minus 1 close parentheses
  • open parentheses straight pi over 4 minus square root of 2 plus 1 close parentheses
  • open parentheses 1 minus straight pi over 4 minus square root of 2 plus 1 close parentheses
  • open parentheses 1 minus space straight pi over 4 plus space square root of 2 close parentheses

D.

open parentheses 1 minus space straight pi over 4 plus space square root of 2 close parentheses

Given that 
integral subscript straight pi divided by 4 end subscript superscript straight beta space straight f left parenthesis straight x right parenthesis space dx space equals space straight beta space sin space straight beta space plus space straight pi over 4 space cos space straight beta space plus space square root of 2 straight beta
Differentiating space. straight w. straight r. space straight t space straight beta
straight f left parenthesis straight beta right parenthesis space equals space straight beta space cos space straight beta space plus space sin space straight beta space minus space straight pi over 4 space sin space straight beta space plus space square root of 2
space straight f open parentheses straight pi over 2 close parentheses space equals space open parentheses 1 minus straight pi over 4 close parentheses space sin space straight pi over 2 space plus space square root of 2 space equals space 1 minus straight pi over 4 space plus space square root of 2

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8.

The value of integral subscript negative straight pi end subscript superscript straight pi space fraction numerator cos squared space straight x over denominator 1 plus space straight a to the power of straight x end fraction space dx comma space straight a greater than 0 space is

  • a π

  • π/2

  • π/a


B.

π/2

integral subscript negative straight pi end subscript superscript straight pi space fraction numerator cos squared over denominator 1 plus straight a to the power of straight x end fraction space dx space equals space integral subscript 0 superscript straight pi space cos squared space straight x space dx space equals space straight pi over 2
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9. integral space open curly brackets fraction numerator left parenthesis log space straight x minus 1 right parenthesis over denominator 1 plus left parenthesis log space straight x right parenthesis squared end fraction close curly brackets squared space dx space is space equal space to
  • fraction numerator log space straight x over denominator left parenthesis log space straight x right parenthesis squared plus 1 end fraction plus space straight C
  • fraction numerator straight x over denominator straight x plus 1 end fraction space plus straight C
  • fraction numerator xe to the power of straight x over denominator 1 plus straight x squared end fraction plus straight C
  • fraction numerator straight x over denominator left parenthesis log space straight x right parenthesis squared space plus space 1 end fraction plus space straight C

D.

fraction numerator straight x over denominator left parenthesis log space straight x right parenthesis squared space plus space 1 end fraction plus space straight C
integral fraction numerator left parenthesis log space straight x minus 1 right parenthesis squared over denominator left parenthesis 1 plus left parenthesis log space straight x right parenthesis squared right parenthesis squared end fraction dx
space equals space integral space open square brackets fraction numerator 1 over denominator left parenthesis 1 plus left parenthesis log space straight x right parenthesis squared right parenthesis end fraction minus fraction numerator 2 space log space straight x over denominator left parenthesis 1 plus left parenthesis log space straight x right parenthesis squared right parenthesis squared end fraction close square brackets space dx
space equals integral open square brackets fraction numerator straight e to the power of straight t over denominator 1 plus straight t squared end fraction minus fraction numerator 2 straight t space straight e to the power of straight t over denominator left parenthesis 1 plus straight t squared right parenthesis squared end fraction close square brackets space dt space put space log space straight x space equals space straight t
rightwards double arrow space dx space equals straight e to the power of straight t space dt
integral straight e to the power of straight t space open square brackets fraction numerator 1 over denominator 1 plus straight t squared end fraction minus fraction numerator 2 straight t over denominator left parenthesis 1 plus straight t squared right parenthesis squared end fraction close square brackets dt
space equals space fraction numerator straight e to the power of straight t over denominator 1 plus straight t squared end fraction space plus straight c equals space fraction numerator straight x over denominator 1 plus space left parenthesis log space straight x right parenthesis squared end fraction space plus straight c
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10.

Let f : R → R be a differentiable function having f (2) = 6, f′ (2) =(1/48) . Then limit as straight x space rightwards arrow 2 of space integral subscript 6 superscript straight f left parenthesis straight x right parenthesis end superscript space fraction numerator 4 straight t cubed over denominator straight x minus 2 end fraction dt space equals

  • 24

  • 36

  • 12

  • 18


D.

18

limit as straight x space rightwards arrow 2 of space integral subscript 0 superscript straight f left parenthesis straight x right parenthesis end superscript fraction numerator 4 straight t cubed over denominator straight x minus 2 end fraction dt
Applying space straight L space hospital space rule
limit as straight x rightwards arrow 2 of space open square brackets 4 straight f space left parenthesis straight x right parenthesis squared space straight f apostrophe space left parenthesis straight x right parenthesis close square brackets space equals space 4 straight f space left parenthesis 2 right parenthesis cubed space straight f apostrophe space left parenthesis 2 right parenthesis
space equals space 4 space straight x space 6 cubed space straight x 1 over 48 space equals space 18
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