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 Multiple Choice QuestionsMultiple Choice Questions

1.

Area included between curves y = x2 - 3x + 2 and y = - x2 + 3x - 2 is

  • 16 sq unit

  • 12 sq unit

  • 1 sq unit

  • 13 sq unit


D.

13 sq unit

Required area

= 212- x2 + 3x - 2dx= 2- x33+ 3x22 - 2x12= 2- 83 + 6 - 4 + 13 - 32 + 2= 2- 83 + 4 - 76= 2 . 16 = 13 sq unit


2. integral fraction numerator dx over denominator cos space straight x space plus space square root of 3 space sin space straight x end fraction equals
  • 1 half space log space tan space open parentheses straight x over 2 plus straight pi over 12 close parentheses space plus straight C
  • 1 half space log space tan space open parentheses straight x over 2 minus straight pi over 12 close parentheses plus straight c
  • log space tan space open parentheses straight x over 2 minus straight pi over 12 close parentheses plus straight c
  • log space tan space open parentheses straight x over 2 plus straight pi over 12 close parentheses plus straight c

A.

1 half space log space tan space open parentheses straight x over 2 plus straight pi over 12 close parentheses space plus straight C
integral fraction numerator dx over denominator cos space straight x space space plus space square root of 3 space sin space straight x end fraction
space equals space 1 half space integral sec space open parentheses straight x minus straight pi over 3 close parentheses dx
equals space 1 half space log space tan space open parentheses straight x over 2 minus straight pi over 6 plus straight pi over 4 close parentheses space plus straight C
space equals space 1 half space log space tan space open parentheses straight x over 2 plus straight pi over 12 close parentheses space plus straight C
137 Views

3. limit as straight n rightwards arrow infinity of space sum from straight r equals 1 to straight n of space 1 over straight n straight e to the power of straight r over straight n end exponent space is space
  • e

  • e+1

  • e-1
  • 1-e


C.

e-1
limit as straight x rightwards arrow infinity of space sum from straight r space equals 1 to straight n of space 1 over straight n straight e to the power of straight r divided by straight n end exponent
space equals space integral subscript 0 superscript 1 space straight e to the power of straight x space dx
space equals space open square brackets straight e to the power of straight x close square brackets subscript 0 superscript 1
straight e minus 1
134 Views

4.

Area bounded by the curve y2 = 16x and line y = mx is 23, then m is equal to

  • 3

  • 4

  • 1

  • 2


B.

4

Required area

=     016m216x - mxdx = 23 4 . 23x32 - mx22016m2 = 23           83 . 64m3 - m . 2562m4 = 23             1m35123 - 2562 = 23                          1m31283 = 23                                     m3 = 1283 × 32                                     m3 = 64                                      m = 4


5.

The solution for x of the equation integral subscript square root of 2 end subscript superscript straight x fraction numerator dt over denominator straight t square root of straight t squared minus 1 end root end fraction space equals straight pi over 2 space is

  • 2

  • π

  • square root of 3 divided by 2
  • 2 square root of 2

B.

π

integral subscript square root of 2 end subscript superscript straight x space fraction numerator dt over denominator straight t square root of straight t squared minus 1 end root end fraction space equals space straight pi over 2
left square bracket sec to the power of negative 1 end exponent right square bracket subscript square root of 2 end subscript superscript straight x space equals space straight pi over 2
sec to the power of negative 1 end exponent space straight x space minus space straight pi over 4 space equals space straight pi over 2
sec to the power of negative 1 end exponent space straight x space space equals space fraction numerator 3 straight pi over denominator 4 end fraction
straight x space equals space minus space square root of 2
121 Views

6.

The parabolas y2 = 4x and x2 = 4y divide the square region bounded by the lines x = 4, y = 4 and the coordinate axes. If S1, S2, S3 are respectively the areas of these parts numbered from top to bottom; then S1 : S2: S3 is

  • 1 : 2 : 1

  • 1 : 2 : 3

  • 2 : 1 : 2

  • 1 : 1 : 1


D.

1 : 1 : 1

y2 = 4x and x2 = 4y are symmetric about line y = x

space straight y space equals straight x space is space integral subscript 0 superscript 4 space left parenthesis 2 space square root of straight x minus space straight x right parenthesis space dx space equals space 8 over 3
rightwards double arrow space straight A subscript straight s subscript 2 end subscript space equals space 16 over 3 space and space straight A subscript straight s subscript 1 end subscript space equals space straight A subscript straight s subscript 3 end subscript space equals space 16 over 3
rightwards double arrow space straight A subscript straight s subscript 1 end subscript colon space straight A subscript straight s subscript 2 end subscript space colon space straight A subscript straight s subscript 3 end subscript space colon colon space 1 colon 1 colon 1

243 Views

7.

The area of the region bounded by the straight lines x = 0 and x = 2x and the curves y = 2 and y = 2x - x2 is equal to

  • 2log2 - 43

  • 3log2 - 43

  • 1log2 - 43

  • 4log2 - 32


B.

3log2 - 43

Required area

= 022x - 2x - x2dx= 022x - 2x + x2dx= 2xlog2 - x2 + x3302= 4log2 - 4 + 83 - 1log2= 3log2 - 43 sq unit


8.

The area enclosed between the curves y2 = x and y = |x| is

  • 2/3

  • 1/3

  • 1/6

  • 3


C.

1/6



straight A space equals space integral subscript 0 superscript 1 left parenthesis square root of straight x minus straight x right parenthesis dx
space equals space open square brackets 2 over 3 straight x to the power of 3 divided by 2 end exponent minus straight x squared over 2 close square brackets subscript 0 superscript 1
space equals space 2 over 3 space minus 1 half space equals space 1 over 6
127 Views

9.

The area of the region bounded by the curves y = |x – 2|, x = 1, x = 3 and the x-axis is

  • 1

  • 2

  • 3

  • 4


A.

1

space integral subscript 1 superscript 3 space straight y space space dx
space equals space integral subscript 1 superscript 3 space vertical line space straight x minus 2 vertical line space dx
space equals space integral subscript 1 superscript 2 space minus space left parenthesis straight x minus 2 right parenthesis space dx space plus space integral subscript 2 superscript 3 space left parenthesis straight x minus 2 right parenthesis space dx
space equals space integral subscript 1 superscript 2 space left parenthesis 2 minus straight x right parenthesis dx space plus integral subscript 2 superscript 3 space left parenthesis straight x minus 2 right parenthesis space dx
open square brackets 2 straight x space minus straight x squared over 2 close square brackets subscript 1 superscript 2 space plus open square brackets straight x squared over 2 minus 2 straight x close square brackets subscript 2 superscript 3
space equals space left parenthesis 4 minus 2 right parenthesis minus left parenthesis 2 minus 1 divided by 2 right parenthesis space plus left parenthesis 9 divided by 2 minus 6 right parenthesis minus left parenthesis 2 minus 4 right parenthesis space
space equals space 2 minus 3 divided by 2 minus 3 divided by 2 minus 3 divided by 2 space plus 2 space equals space 4 minus 3 space equals space 1
124 Views

10.

The area enclosed between the curve y = loge (x + e) and the coordinate axes is

  • 1

  • 2

  • 3

  • 4


A.

1

Required space area space left parenthesis OAB right parenthesis space equals space integral subscript 1 minus straight e end subscript superscript 0 space In space left parenthesis straight x space plus straight e space right parenthesis thin space dx
space equals space open square brackets straight x space ln space left parenthesis straight x plus space straight e right parenthesis minus integral fraction numerator 1 over denominator straight x plus straight e end fraction straight x space dx close square brackets subscript 0 superscript 1 space space equals 1
258 Views